Proof. For every \(C\in\mathscr H\), \[\frac12\mathop{\mathrm{diam}}(C)\le \mathop{\mathrm{dist}}_{\mathscr H}(C,Z)\le\mathop{\mathrm{diam}}(C).\] The lower bound is the triangle inequality, and the upper bound follows by choosing a singleton in \(C\). If \(d_{\rm match}\le\varepsilon/2\), a member of diameter greater than \(\varepsilon\) must therefore match a positive-diameter member of the other family within \(\varepsilon/2\). Conversely, under \(R_\varepsilon\) each member of diameter at most \(\varepsilon\) is within \(\varepsilon\) of \(Z\), and each larger member has the required match. This proves the inclusions.

The hyperspace of nonempty compact subsets of a compact metric space is compact, and the condition \(Z\subset F\) is closed. For each \(t>0\), \(N_t\) is lower semicontinuous: any finitely many distinct members of diameter greater than \(t\) persist in disjoint small hyperspace neighborhoods under Hausdorff convergence. Hence each constraint \(N_{t_j}\le M_j\) is closed. Their intersection is compact, and choosing \(j\) with \(t_j<t\) proves finiteness above every \(t>0\). Finally, choose the \(M_j\) so that the respective tail probabilities are at most \(\eta2^{-j}\), and use the union bound. This proves compact containment. ◻

Proof. If \(s\) disagrees, the crossing \(N\) edge is forced open and its \(d\) spins agree. If \(s\) agrees and \(W\) is closed, the weight is \(1-t\) and the same agreement is required. Finally, if \(W\) is open, summing the possibilities for the other current gives weight \(t\) whether the transverse spins agree or disagree. This proves (8). Its dependence on \(d\) consists exactly of equality constraints on \(N\) edges, proving the first assertion. Fixing a wired component’s sign removes the constant factor \(2\) associated with that component, independent of its geometry.

On an open \(W\) perimeter only the first line of (8) occurs. The transverse variables on its two sides have no interaction across it. The remaining product factors split by chamber. Along the perimeter \(s\) is constant, and the corner height rule gives constant height between successive same-parity vertices. This proves the second assertion, first for a simple perimeter and then for a boundary walk by keeping its incident faces separate.

For two \(s\) vertices \(v,v'\) incident to a transverse vertex \(f\), \[ K(v')-K(v)=\varepsilon_W d(f)\bigl(s(v)-s(v')\bigr), \tag{9}\] where \(\varepsilon_W=1\) when the \(s\) lattice is primal and \(\varepsilon_W=-1\) when it is dual. This fixed orientation sign has no effect on the independent fair relative signs. The rim facing any specified hole of a connected \(N\) component belongs to one incident \(W\) component, as does its exterior rim. Thus crossing that \(N\) component gives either zero or \(2\) times its \(d\) color times the fixed exterior \(s\) sign. All odd holes give the same relative jump. The third assertion now follows from the first.

For the stopping assertion, inspect no variable beyond a completed open \(W\) rim sealing an unsearched face. Whenever the search queries \((s,W)\), it exposes endpoints before edge states. For each possible finite history, its event is the cylinder of the values queried by the deterministic search. The product factors inside the residual chambers are precisely those already described, even if the rule deciding which completed cluster to examine next used its revealed \(d\) color. Thus the conditional residual kernels, and not just their average, are the stated independent flat laws. Skipping an even hole reveals no new jump; stopping at an odd rim retains its observed jump. A connected exterior-attached seed and all clusters it has met form one connected explored region \(S\). Every edge from \(S\) to an unsearched vertex is \(N\)-closed, since every met cluster has been revealed in full. For one geometric component \(C\) of the unsearched vertices, its complement is connected through \(S\); every other unsearched component attaches to \(S\) in the underlying graph. Thus the edge boundary of \(C\) is a planar bond and its dual is one open \(W\) perimeter, with the stated medial resolution at pinches. It is monochromatic, and all its remote connections are retained. This argument concerns connectedness in the underlying graph; the added seed edges need not themselves be \(N\)-open. The seed may have visited several height levels, but this causes no varying boundary value in a residual chamber: every edge on its component-side \(W\) perimeter joins equal \(s\) spins, and (9) gives zero height increment along that edge. The connected perimeter therefore has one constant height. The history determines that constant using only already revealed clusters and their colors; no interior or future information is used. For a wired current, the incident opposite-parity height differs from the boundary height by one corner increment. After its boundary cluster is removed, no interior \(N\) cluster is wired to the original boundary. This proves the last assertion. ◻

Proof. Summing \(d\) in (8) contributes \(2^{k(N)}\), up to the constant just noted for a fixed wired color. Planar Euler counting gives \[ 2^{k(N)}=C\,2^{k(W)+|W|}, \tag{11}\] where \(C\) depends only on the underlying finite graph and its boundary convention. Indeed the components of the complementary dual graph are the faces of \(W\), whose number is \(|W|-|V(W)|+k(W)+1\). For a local window, contract the actual same-color boundary contacts joined by exterior \(W\) paths and write \(k_\xi(W)\) for the resulting component count. The same identity holds with \(k_\xi(W)\), up to a constant depending on this boundary convention. These identifications are retained in the law even when their paths lie far outside the window.

Put \(r=(1-t)/t>0\). After division by the constant \(t^{|E|}\), an open monochromatic \(W\) edge has weight \(2\), a closed monochromatic edge has weight \(r\), and a disagreeing edge is closed with weight \(1\). Represent \(2^{k(W)}\) by auxiliary colors \(\rho\in\{-1,1\}\). Summing the \(W\) state on a monochromatic edge gives \[ r+2\mathbf 1_{\{\rho_i=\rho_j\}} =r+1+\rho_i\rho_j. \tag{12}\] Consequently the \(s\) marginal is the product of a positive factor \((1+r)^{|E_{\rm mono}|}\) and the even-subgraph partition functions on the two monochromatic induced graphs, with edge parameter \(q=(1+r)^{-1}\in(0,1)\).

For completeness, let \(Z_F(A)\) denote such an even-subgraph partition function on an available edge set \(A\), with ferromagnetic identifications \(F\) retained. It is increasing and log-supermodular in \(A\). One way to see both facts is to write it as the zero-field ferromagnetic Ising partition function divided by its single-edge \(\cosh\) factors. Adding an edge multiplies the normalized partition function by \(1+q\mathbb E[\rho_i\rho_j]\). The expectation is nonnegative and increases when other nonnegative couplings are added, by the two Griffiths inequalities (Griffiths 1967); a complete proof for nonnegative couplings is given in (Friedli and Velenik 2017, sec. 3.8.1, Theorem 3.49). Indeed differentiating an expectation with respect to a coupling gives the corresponding nonnegative covariance. Identifications are limits of positive bonds: after dividing by the common maximal exponential, the finite-state factor is \(e^{-2K D_F}\), where \(D_F\) counts violated identifications. As \(K\to\infty\) it tends to their joint indicator. Thus the same inequalities survive the finite-state limit. The single-edge factors cancel from the mixed inequality, proving \[Z_F(A\cup B)Z_F(A\cap B)\ge Z_F(A)Z_F(B).\] The same incremental-edge formula gives the boundary comparison that will be needed below: if \(A\subseteq B\) and \(F\subseteq F'\), then \[ Z_{F'}(B)Z_F(A)\ge Z_{F'}(A)Z_F(B). \tag{13}\] Indeed each multiplier \(1+q\mathbb E[\rho_i\rho_j]\) increases when the bonds implementing \(F'\) are added. Integrating the Griffiths covariance inequality and then taking their infinite-bond limits proves (13). For plus vertex sets \(S,T\), \(E(S\cap T)=E(S)\cap E(T)\) and \(E(S\cup T)\supseteq E(S)\cup E(T)\). Apply the preceding inequality and monotonicity to both the plus and minus induced graphs. The factor \((1+r)^{|E_{\rm mono}|}\) is itself ferromagnetic. This proves the spin lattice condition, to which the finite association and comparison theorems (Fortuin et al. 1971; Holley 1974) apply.

Conditional on \(s\), the open edges on each monochromatic graph have the ordinary random-cluster law with cluster weight \(2\) and edge weight \(v=2/r\). Given the other edges, its opening probability is \(v/(v+2)\) if the endpoints are disconnected and \(v/(v+1)\) if they are already connected. The probabilities increase with the available connections, so common-uniform edge updates give the edge comparison. Together with the spin lattice condition, this gives positive association and the order (10). Fixed closed edges with prescribed endpoints give constants; fixed open edges give ferromagnetic identifications. Thus the same argument applies after all the allowed prescriptions. The ordered-wiring comparison follows from (13) for plus identifications and its reflected version for minus identifications. In particular, arbitrarily shaped exterior connections cannot be silently discarded.

Use these conditional comparisons at each step of a simultaneous exploration, always preserving each configuration’s residual conditional law. A plus circuit in the lower configuration is also open plus in the upper. By 4, the laws on the sealed side then have the same constant boundary spin and no transverse interaction across the circuit. Couple those laws identically. No \(N\) component crosses the cut, so its free colors may also be coupled identically. The minus case is the reflected argument. ◻

Proof. The imported input is (Duminil-Copin et al. 2026, Theorem 16.4): for \(\mathbf c=1/t\in[1,2]\), an opposite-color spin-percolation circuit in a lattice annulus of radius at least \(4\) has a positive lower bound, uniform in its radius, under the worst constant wired boundary condition. The decoration in that theorem opens an ambiguous edge with probability \(1/\mathbf c=t\), so it is precisely \(W\). Its exterior-conditioning version, (Duminil-Copin et al. 2026, Corollary 16.6), allows arbitrary data of the same spin parity and its edges outside the buffered ball. We only use this statement for one parity. Its conditional FKG hypotheses agree with 5.

After the fixed change from source lattice units to physical units, the lower bound holds for \(r\ge c_*\delta\). Comparison with the worst constant boundary condition gives item 1 for squares; a fixed finite cover gives the other stated aspect ratios and the tube version. Intersect its matching circuit events by positive association. Matching pins are favorable in (10). In the wired \(N\) law before removal, fixing the single wired \(d\) color removed the geometry-independent factor \(2\) in the proof of 5. Its \((s,W)\) comparison is therefore the same. For the unpinned bulk test, choose a deterministic circuit in the unused part of the buffer between the annulus and the buffer boundary. Compare to the law with that circuit pinned to the worst \(W\) color. Its sealed side is a \(W\)-flat kernel by 4, so the same lower bound applies there despite the arbitrary data beyond the buffer. Requested-color pins are then handled by monotonicity. This proves the claimed wired bulk case.

For separated scales, reveal earlier trial neighborhoods before testing the next one. The conditional failure probability remains at most \(1-p\), so multiplying conditional bounds proves item 2 without an independence assumption. Take a geometric sequence of scales between \(r+\delta\) and \(R\), separated enough to have disjoint buffers. A circuit at one such scale meets the wall because the wall crosses the annulus. It separates the inner from the outer scale. This gives (15); the \(\delta\) term absorbs the finitely many scales below the lattice cutoff.

For item 3, apply item 1 in the completed graph. All additional pins in a test buffer have the requested matching color. The exact accessible marginal or (14) transfers the event to the original law. In this transfer an edge of the circuit inside \(F\) is explicitly artificial; every other edge of it is an actual matching edge. Thus a full circuit is available even when an accessible side is slit and has no full circuit by itself. The geometric blocking assertion follows from the definition of the augmented graph. For a connected source the boundary incident to one fixed complementary component is one boundary walk. It can enter the window arbitrarily often, but these entrances form one retained label. This verifies exactly the premise on labels when that bounded comparison is used. ◻

Proof. These tests detect every connected passage of a fixed positive diameter \(b\) in a fixed box. Choose an inner square of side a small fixed multiple of \(b\) from a fixed grid around one endpoint, and stop the connecting path on its first exit from a larger square of diameter less than \(b/2\). The exit side fixes an orientation, and a fixed enlargement of the vertical extent gives a traversal with the required margin. There are finitely many grid squares and orientations. Their working rectangles, intermediate enlargements, and buffers are all chosen before applying the lemma; anchoring is required on those intermediate enlargements. The same construction inside an open band detects a full traversal of the band.

It suffices to prove one left-to-right test. Use the crossing segment from the last left-side hit before its first right-side hit, and then erase loops. The result is a simple proper crosscut: its interior misses both vertical sides. This trimming is needed because a simple path that returns to its starting side can leave a third pocket. A proper crosscut has just a lower and an upper side. Every local obstacle piece disjoint from it is assigned to one of those sides. Include the top and bottom as virtual barriers, which the traversal margin already forbids. Here the pieces are components after restriction to this working rectangle. Connections outside it retain their wire labels in the law but do not identify pieces on opposite sides of an endpoint.

We first reduce these assignments, except for a uniformly small event, to a fixed finite number of possibilities. An anchored piece meeting a narrower central rectangle contains a path to the boundary of the working rectangle: take the part of an anchored path up to its first exit. A piece reaching the top or bottom is already assigned to that barrier. For a piece reaching the left, in the fixed-width left end strip, retain a segment from its last hit of the outer vertical to its first hit of the inner vertical and erase loops. It is a proper crosscut of that strip. Make a deterministic choice if a piece has several such paths. Representatives of distinct pieces are disjoint and can be ordered from bottom to top. The analogous construction is used at the right side and in the rotated windows. Connections of a piece between different sides later impose consistency; they add no choices.

Here is the quantitative reduction in one end strip. Choose a middle horizontal interval \(I\) of length \(\ell>0\) and take \(\rho>0\) below both its horizontal clearance from the vertical ends and the fixed clearance of \(R\) inside the larger window buffer. For consecutive representative arms, let \(g_j(x)\) be the difference of their topmost intersection heights with the vertical at \(x\), and put \(d_j=\ell^{-1}\int_I g_j(x)\,\,\mathrm dx\). These heights have the same order as the arms: the upper crosscut separates the lower one from the top of the strip, so a vertical ray from the top to the topmost point of the lower one first meets the upper one. The arms need not be graphs. In particular, \[ \sum_j d_j\le H, \tag{17}\] where \(H\) is the fixed strip height. The set \(E_j=\{x\in I:g_j(x)\le2d_j\}\) has length at least \(\ell/2\).

At lattice scale the separation convention gives \(d_j\ge a_0\delta\) for a fixed \(a_0>0\). Put \(h_j=d_j+\delta\le\beta d_j\), and choose a fixed large \(K\) so that \(2Kh_j\) exceeds the cutoff in 6. If \(4K\beta d_j<\rho/2\), a maximal selection of points in \(E_j\) at horizontal separation greater than \(16Kh_j\) has at least \(\ell/(64K\beta d_j)\) points, up to a fixed endpoint loss. Center a disk at the midpoint of the two arm intersections at each selected point. Both arms meet its radius-\(2Kh_j\) disk and leave its radius-\(4Kh_j\) disk, since they continue to the strip ends. The radius-\(8Kh_j\) buffers are disjoint by construction. Since \(8Kh_j<\rho\), they stay inside the allowed window buffer and lie between the end strip’s vertical sides. Every augmented matching circuit in the intervening annulus meets both proper arms. Orient such a circuit, stop at its first upper-arm hit after a chosen lower-arm hit, and retain the arc from its last preceding lower-arm hit. Its interior misses both arms. The two proper arms divide the infinite vertical strip into lower, middle, and upper components. Only the closure of the middle component meets both arms, so the arc’s connected interior lies in that component. Its closure is contained in the finite end strip. The retained arc therefore joins the arms geometrically inside the end strip, even if the rest of the circuit extends beyond its horizontal sides. In particular, a remote wire alone is not counted as the joining connection.

Successive conditional circuit tests in the completed graph, followed when needed by the single density comparison (14), now give \[ \mathbb P[\text{the two arms have no such local joining circuit}] \le C_k\exp(-c/d_j). \tag{18}\] Here and below \(C_k\) is the fixed density factor for the allowed completion; it is an absolute constant for either exact completion, including the product class with any number of chambers. Gaps too large for the preceding packing are simply retained. The locations of all tests depend only on the prescribed obstacles, not on the sought traversal. If a matching circuit joins two arms, an avoiding crosscut cannot assign them to opposite sides.

For any \(u>0\), the total probability lost by discarding the gaps \(d_j\le u\) is at most \[ \sum_{d_j\le u} C_k e^{-c/d_j} \le C_k\left(\sum_jd_j\right) \sup_{0<d\le u}\frac{e^{-c/d}}{d} \longrightarrow0\qquad(u\downarrow0). \tag{19}\] At most \(H/u\) larger gaps remain, together with a bounded number adjacent to virtual barriers. A proper traversal can split the ordered representatives at only one of these ranks. Repeating this on the finitely many sides leaves a finite product of deterministic rank lists, bounded in terms of \(u\) and the windows. A piece joined around another side must receive the same assignment there, which only removes rank choices. Trials for different gaps may overlap; the union bound in (19) does not require their independence.

We finish by a compactness argument, keeping its choices deterministic. If (16) failed, there would be deterministic allowed histories and walls at meshes \(\delta_n\downarrow0\), widths \(\eta_n\downarrow0\), and a fixed positive lower bound for the traversal probability. First fix \(u\) so that (19), summed over the sides, is less than one quarter of that bound. One of the finitely many surviving rank choices then has a fixed positive event mass along a subsequence. We do not condition the law on that event. Take Hausdorff limits, in the central buffered rectangle, of the unions assigned below and above, including the virtual barriers.

Suppose first that the two limits touch there. Unless a virtual barrier is involved, there are deterministic points on two oppositely assigned pieces at distance \(r_n\downarrow0\). Anchoring makes both pieces cross the annuli from scale \(r_n+O(\delta_n)\) to a fixed outer scale. Any augmented matching circuit in such an annulus joins them. By (15), the probability of a traversal with this assignment tends to zero. If one limiting piece is a virtual barrier, the fixed traversal margin rules out the opposite assignment: a connected piece reaching that margin lies on the barrier’s side of the proper crosscut.

It remains that the two closed limits are disjoint in the central rectangle. On a central vertical line choose a consecutive lower encounter \(a\) and upper encounter \(b\). Such a transition exists because the virtual bottom and top have those assignments. The open segment \((a,b)\) misses both limiting unions. Fix an integer \(m\). At each actual endpoint choose \(m\) separated circuit radii, all less than \(|a-b|/8\) and the available buffer clearance; call their smallest radius \(r_m>0\). If both endpoints are virtual, take any fixed sufficiently small \(r_m\) in their traversal margins instead. Only after this choice, trim the gap by less than \(r_m/2\) at its ends and call the remaining compact segment \(K_m\). It has positive distance from the whole limiting obstacle union, not just from its intersections with the vertical. This clearance holds for the discrete obstacles for all large \(n\), and then \(K_m\cap F_{\delta_n}^{\eta_n}=\varnothing\).

Center the endpoint tests at discrete points converging to \(a\) and \(b\). An attached circuit at any of the chosen radii meets \(K_m\): it surrounds its center, and the ray into the gap crosses the circuit between that radius and twice that radius. Its other intersection is with the endpoint’s anchored piece. For a virtual endpoint the short segment in the fixed traversal margin supplies the connection instead. Consequently a successful endpoint circuit at each actual endpoint, together with \(K_m\), is a connected set from a lower-assigned piece or barrier to an upper-assigned one, wholly in the rectangle. Every proper left-to-right crosscut with that assignment meets this set. It cannot do so: it avoids the pieces and the attached circuits, and the collar condition excludes \(K_m\).

For fixed \(m\), the probability that either required endpoint connection fails has limit superior at most \(2C_k(1-p)^m\), by the separated-scale estimate and, if needed, one density comparison. Letting \(m\) increase contradicts the retained positive event mass. The order of choices here matters: the finite endpoint radii precede the trimmed segment, so every requested circuit really reaches that segment. This proves uniformity in (16).

Finally let a closed disk be bounded by an actual \(W^{-\sigma}\) loop and disjoint from \(F_\delta\). A connected augmented matching circuit attached to \(F_\delta\) cannot enter this disk. Its artificial edges are in \(F_\delta\), outside the disk, and its actual \(W^\sigma\) edges cannot cross or share a vertex with the opposite-color boundary. A dual path inside the disk therefore automatically avoids all those circuits. The existence event already ranges over every underlying dual path, so the bound holds simultaneously over the possible disks, without conditioning on a future choice of one. Choosing all the windows with fixed margin inside an open band gives the last localization assertion. Monotonicity in \(\eta\) gives the conclusion for \(\eta_\delta\downarrow0\). ◻

Proof. For item 1, start with the connected cell region of the given exact cut history; for a single flat chamber use its exterior side. Explore with a deterministic connected seed attached to that region. Add one seed vertex at a time and immediately reveal the entire previously undiscovered \(N\) cluster it meets. The seed eventually visits every vertex of the box. Record only discoveries with the specified passage, choosing one such passage by a deterministic rule. At every recorded stop, the initial region, seed, and all clusters already discovered form one connected cell region. By 4, the residual laws are exactly the flat kernels, jointly in the class with a common fill when there are several chambers.

A finite family of smaller collar windows detects every passage of diameter at least \(b\). Freeze the cell region \(F_i\) at the \(i\)th recorded stop, with \(F_0\) the initial region, before continuing the search. An actual unsearched \(N\) path is disjoint from \(F_i\) and cannot cross the actual part of an augmented matching circuit. Thus 7 gives an \(\eta>0\) such that, conditionally on any such frozen history, the probability that some still unsearched qualifying passage lies wholly in \(F_i^\eta\) is at most a fixed \(p<1\). This event is existential over future passages; no future location or visibility is conditioned upon. It also bounds the event that the next recorded cluster exists and its selected passage lies wholly in \(F_i^\eta\).

On the complementary event, the next passage contains a point at distance greater than \(\eta\) from \(F_i\). Such points from successive records are mutually \(\eta\)-separated, because each earlier cluster is included in every later frozen region. A bounded box contains at most \(M=M(\eta)\) such points. There is no cost for the number of intervening microscopic discoveries: they only enlarge the same connected region. In the product case reflect the residual chamber spins to one matching color and use the exact realization with a common fill for this single existential collar event. In particular, no union bound over the number of holes is taken. A test buffer meeting the initial wall is covered by the same augmented graph.

If at least \(n\) records occur, at most \(M\) of the first \(n\) can be of the separated-point kind. Conditional iteration for the other indices, including the initial stop, gives \[ \mathbb P[\text{at least }n\text{ recorded discoveries}] \le \sum_{j=0}^{M}\binom{n}{j}p^{\,n-j} \le C e^{-c n}. \tag{20}\] The same proof starting from a later allowed stop bounds its unsearched count. It does not bound the number already fixed by that history. Meshes above the small threshold used by the collar lemma are absorbed by the deterministic number of vertices in the fixed box.

For item 2 we use a crossing exploration followed by a finite-domain boundary comparison. First fix the relative planar dual of the given restricted \(N\) lattice quadrilateral. Its medial boundary follows the four sides of the rectangle. In the auxiliary \(W\) graph, truncate a dual edge at that boundary when it exits, and give each boundary incidence its own degree-one port. There are no edges in an exterior side column and no identifications between distinct ports. Every retained edge or half-edge inherits the state of its actual \(W\) edge. This is the matched dual restriction used for every \(W\) connectivity test below; its placement differs from the geometric rectangle by \(O(\delta)\), which is absorbed by the joining buffer.

If \(J_N\) distinct restricted \(N\) components cross the quadrilateral, choose one simple crossing in each, ordered from bottom to top. Between each two successive crossings, relative planar duality gives a parallel open crossing of this matched \(W\) graph. The separators in two successive gaps belong to different components of that graph: a connection inside it would cross the intervening \(N\) path, and the separate ports give no route around an endpoint. Consequently \[J_N\le J_W+1,\] where \(J_W\) counts crossing components in precisely the matched \(W\) graph. The probabilistic search uses only these \(W\) crossings; it never certifies a separator against an unrevealed upper \(N\) crossing.

Distinct \(W\) crossing components have a consistent vertical order: one component cannot have crossing paths on both sides of another, because its joining path would cross the latter. In each component choose a lowest simple proper crossing, with a deterministic tie rule; the lower boundary of the union of its crossing paths supplies such an open crossing. Enumerate these representatives from below, with all connectivity restricted to the matched graph. For a candidate \(\gamma\), the event that it is the next distinct crossing is determined by the \((s,W)\) configuration in its closed below envelope. All lower candidate crossings lie there, and any connection from a lower component to \(\gamma\) first meets \(\gamma\) from below. A proposed connection between lower pieces through the region above \(\gamma\) would first join \(\gamma\) as well. This is the stopping-line property for the enumeration.

After selecting \(\gamma\), reveal its whole closed below envelope, with endpoint spins before edge states. Reflect the colors if necessary so that \(\gamma\) is plus. We use test rectangles with sides parallel to the coordinate axes of the \(W\) lattice; the primal and dual square lattices have the same axes. Complete only the two endpoint half-edges of \(\gamma\) to their actual \(W\) edges. Their states are already known: each half-edge inherited the state of that actual edge, and its being open forces the additional endpoint to have the same spin. Thus this operation assumes no unexamined connection. The actual endpoints of all left-exiting half-edges lie on one \(W\)-lattice column, and the analogous statement holds on the right. Let \(B\) have these two columns as its lateral boundary and extend its top and bottom a fixed distance into the unprescribed joining buffer. The completed \(\gamma\) is a proper crosscut of \(B\): its only vertices on the lateral boundary are its two endpoints, because the matched graph contains no exterior side-column edges. Write \(U_\gamma\) for the component above this crosscut and \(F_\gamma\) for the closed region below it. All the earlier crossing-search transcripts in \(B\) are below \(\gamma\).

Analytically disintegrate the conditional law over the entire configuration outside \(U_\gamma\), including the unused parts of the buffer, and over its boundary contact spins before interface edge states. This extra disintegration is used only for the estimate; it is not added to the next search transcript. On every positive exterior fiber the finite weight on \(U_\gamma\) is the weight of 5, with the exterior same-color connections retained. All contacts on \(\gamma\) are plus and belong to a single class, since \(\gamma\) itself is open. The other contacts lie on the upper boundary arc of \(B\).

Replace this fiber’s boundary condition by the following adverse one. Keep \(\gamma\), including its endpoints, plus and open. Make all other contacts on the upper boundary arc minus and wire them together; retain only the one plus class supplied by \(\gamma\). The two boundary edges at a change of spin are closed. This is an ordered decrease for the plus order: it retains every required lower plus connection, removes only additional plus connections, and merges all remaining minus connections. Changes of contact spins and their compatible incident edge states use the same order. Therefore the original fiber dominates this fixed adverse law on \(U_\gamma\), by 5. In particular, no event belonging to the original history has been removed from a conditional probability.

Realize the adverse law on the full box \(B\) by filling \(F_\gamma\) with plus spins and open \(W\) edges, including the lower outer boundary arc, while keeping the upper outer boundary arc minus and wired as above. Every edge of the separating crosscut remains an actual open plus edge. The finite edge factors in (8), or equivalently the component-count formula in (11), show that this completion has exactly the adverse law as its \(U_\gamma\) marginal: factors on the filled side cancel, the plus contacts form one class, and the other boundary contacts form the one minus class. There is no factor depending on the length or roughness of \(\gamma\).

The completed law is obtained from the law in \(B\) with the worst constant minus wired boundary by increasing the lower boundary data and imposing compatible plus pins in \(F_\gamma\). 5 therefore makes it dominate that constant-boundary law for every increasing plus-circuit event. The ordinary buffered circuit estimate used in 6, namely (Duminil-Copin et al. 2026, Theorem 16.4), now gives a uniform positive lower bound for each circuit test in a fixed central strip of \(B\). In transferring a circuit to \(U_\gamma\), edges in \(F_\gamma\) are artificial matching edges; all its other edges are actual plus edges. The estimate holds on every original exterior fiber, so averaging over those fibers preserves the same lower bound. This argument uses only the ordinary full-circuit estimate, not the arbitrary-history partial-circuit assertion of (Duminil-Copin et al. 2026, Lemma 19.4).

Choose a fixed chain of buffered circuit tests in this central strip, starting below the counted rectangle and ending above it. All test buffers stay away from the lateral sides of \(B\), and their number depends only on the fixed aspect and buffer ratios. The usual tube gluing from 6, followed by positive association in the completed law, gives a uniform positive probability \(p\) that the augmented matching graph joins the filled lower part to above the rectangle. This tube event is increasing, so the same boundary comparison and fiberwise averaging give probability at least \(p\) under the original search history. Its last exit from \(F_\gamma\) is on \(\gamma\) in the central strip. From there until its first exit through the top of the counted rectangle the connection is actual and lies in the matched graph. A later parallel crossing of the same color would meet this connection and hence join \(\gamma\) inside that graph, whereas a crossing of the opposite color cannot cross it. The tube therefore rules out another distinct \(W\) crossing.

These tube tests are used to bound the conditional probability; their upper data are not added to the next search transcript. For every transcript at the \(i\)th representative, the conditional chance of a further one is at most \(1-p\). Iteration gives a geometric tail for \(J_W\) and hence for \(J_N\). Connections outside the rectangle play no role because both counts use restricted components. The original wired law has the same interior \((s,W)\) comparison by 6.

A fixed finite rectangle cover of the shell between \(Q_0\) and \(Q_1\) assigns each transmitting restricted component to a crossing of one of these rectangles. The small lattice layer in the incidence definition is absorbed by the fixed clearance. This proves the transmission assertion.

For the shell assertion, the clusters counted are unsearched global free clusters of the residual law, rather than distinct restricted components. At shell radius \(r\), a fixed finite cover assigns every enclosing or passing cluster a connected passage of diameter at least \(c_2r\) in one of its boxes. Apply item 1 after rescaling by \(r\). Its collar proof is uniform when \(\delta/r\) is small, including boxes met by matching walls in the realization with a common fill. When \(\delta/r\) is bounded below there are only a bounded number of vertices in the rescaled boxes. The resulting exponential tail gives every fixed moment, uniformly down to lattice order. For disjoint independent chambers apply this assertion separately in each one. A pair of vertices in different chambers shares no cluster, which is the only cross-chamber fact needed for the later moment estimates. ◻

Proof. Cover \(L\) by finitely many inner squares of side a small fixed multiple of \(b\). Around each choose a passage square and a larger buffer, with fixed ratios, in the regular ambient box. If a connected cluster has diameter at least \(b\), choose two of its points at that separation and an inner square containing the first. The path to the second has a first exit from the passage square; its initial portion is a connected passage of diameter at least \(c_1b\) there, for a fixed \(c_1>0\). Let \(\mathcal N_{\delta,i}^{\rm passage}(c_1b)\) be the item 1 count of unsearched global free clusters of the initial residual law having such a passage in the \(i\)th square. Every global cluster is counted in at least one of these counts. Consequently \[\mathcal N_\delta(b) \le \sum_{\text{cover squares }i} \mathcal N_{\delta,i}^{\rm passage}(c_1b).\] The cover is fixed before \(\delta\). Its passage buffers may cross the original boundary or the revealed wired layer: extend the initial cell region to the ambient box and use the augmented matching completion of 6. Item 1 of 8 and a finite union bound give (21). For the original free dual law the matching primal boundary is the initial fill. Behind the wired \(N\) boundary cluster, 4 gives precisely the product law whose common initial fill is that connected cluster joined to the exterior.

For (22), every discovered cluster meets the seed. If its support meets \(Q\), it has diameter at least \(r-O(\delta)\). If instead one of its odd holes meets \(Q\), that hole lies in the convex hull of the cluster’s cell region: a bounded complementary component is contained in that convex hull. A point of the hole is therefore at distance at most \(\mathop{\mathrm{diam}}C+O(\delta)\) from the seed contact on \(C\). This again gives \(\mathop{\mathrm{diam}}C\ge r-O(\delta)\). The listed clusters are distinct members of the initial free collection, proving the pathwise inequality. This argument covers a cluster which surrounds \(Q\) without meeting it. ◻

Proof. For \(x\in U_\delta\), take a segment to a nearest point of \(S_\delta\). Its first exit from the resolved chamber meets that chamber’s own medial perimeter, within \(O(\delta)\) of the corresponding \(W\) edge. The distance to this first exit is at most the length of the segment. Taking suprema proves the inequality. This uses the own perimeter even if the nearest support point belongs to another nearby part of a pinched wired layer. For the original free chamber the same argument uses its polygonal boundary as \(S_\delta\). The Hausdorff conclusion follows from the one-sided inclusion \(C_*\subset S_*\). ◻

Proof. An actual unsearched \(N\) passage avoids \(F_\tau\) and the actual edges of every attached augmented matching circuit. A finite collar-window cover of \(Q\) therefore gives (24) from 7, including windows met by the initial wall. The exact cut factorization identifies the residual marginal for every permitted history. The bound is for the existential future event under that marginal; it does not reveal a particular future passage.

On \(D_\delta(b)\le K\), there are at most \(K+1\) frozen indexed stops. Apply (24) at each one and use a union bound. The complementary event is bounded by (21), which yields (25). Finally, (26) follows from monotonicity of neighborhoods. It gives the stated order of limits even when the width was chosen from a future Hausdorff containment. ◻

Proof. Consider first a single \(Q\) inside its buffer \(B\). Prescribe a complete admissible pattern in four batches: its plus vertices, its minus vertices, its increasing compatible edge prescriptions, and its decreasing compatible edge prescriptions. After the first two batches every edge endpoint is fixed. A disagreeing edge is already determined closed. On monochromatic edges the two remaining batches mean open plus or closed minus, and closed plus or open minus, respectively. Each batch is monotone in one of the two orders of 5. The number of batches is four, independently of the pattern size.

Fix the preceding batches and consider an increasing current batch \(E\). Choose two deterministic annular buffers between \(Q\) and \(\partial B\), and fix a plus circuit on the farther one. Let \(\lambda\) be the law inside that deterministic circuit with the preceding prescriptions retained. Comparison gives \[ \mu(E)\le\lambda(E). \tag{28}\] In the nearer annulus, conditional on the preceding prescriptions, a random plus circuit has probability at least \(p>0\). To see this even if those prescriptions contain both signs, cover that annulus by buffered test boxes missing \(Q\); the exterior bound for one parity and FKG supply its circuit with the same positive constant. Search outside-in, inspecting nothing on the sealed side, and stop at the first such circuit in a fixed exploration. Its interior law is plus wired by 4, and hence dominates the restriction of \(\lambda\), with the same preceding prescriptions. Consequently \[ \mu(E)\ge p\lambda(E). \tag{29}\] The decreasing batch uses minus circuits and the reversed order.

The law \(\lambda\) is common to any two exterior conditions under comparison. Applying (28)– (29) to each of the four conditional batch probabilities yields bounded ratios for the complete pattern probabilities. Zero-probability patterns are exactly the common local incompatibilities and are harmless. These constants remain valid when the comparison law is a flat exhaustion of the plane. The common plane spin/percolation law obtained from flat exhaustions in (Duminil-Copin et al. 2026, Theorem 17.1) therefore gives the comparison with \(\pi_i\) by taking that exhaustion limit. For fixed \(W\) parity this law is invariant under parity-preserving lattice translations and reflections, with the compatible global residue action. An automorphism exchanging the lattices acts also by exchanging which parity is called \(W\); it is not invariance of the same fixed edge decoration under that automorphism. The height-gradient marginal has the full corner-grid symmetries used later.

For several boxes, condition successively on the patterns in the other boxes. These constraints lie outside the current filled buffer, so the single-box estimate applies with unchanged constants. The chain rule gives (27). Conditioning on separated patterns and repeating the same argument proves the last assertion. Notice that the proof is multiplicative at every stage; it is not an inference about rare events from a total variation error. ◻

Proof. In a finite \(W\)-flat domain, planar cut-cycle duality says that two marks disconnected in \(N\) are separated by a simple monochromatic \(W\) loop, allowing portions of the matching outer perimeter. We will replace possibly long loops by loops in successively enlarged cores. Write \(\mathcal O\) for the set of required opposite pairs and \(m=|\mathcal O|\le k(k-1)/2\). If \(m=0\) there is nothing to prove. Choose regular closed cores \[Q'=K_0\Subset K_1\Subset\cdots\Subset K_m\Subset B.\] In each gap fix a middle annular band and a wider band compactly inside the gap, with further positive margins. At lattice scale the core includes every \(W\) edge meeting the closed core and its endpoints. This convention counts a crossing on its boundary; all such edges of \(K_m\) lie in \(B\) for small mesh.

For a complete pattern \(g\) on \(K_j\), let \(\mathcal S(g)\subseteq \mathcal O\) be the pairs separated by some actual monochromatic loop in that lattice core. This set is determined by \(g\) and can only increase when the core pattern is enlarged in the same configuration. A pair outside \(\mathcal S(g)\) is called missing. We prove the following one-step statement. If, under the original law conditional only on \(g\), compatibility has probability at least \(a>0\) and some pair is missing, then \[ \mathbb P[\text{compatibility and }|\mathcal S(P_{j+1})| >|\mathcal S(g)|\mid P_j=g]\ge r(a,k)>0. \tag{30}\] The constants use the fixed bands of this step and are uniform in \(g\) and the mesh. Here \(P_j\) denotes the complete core pattern.

First choose a finite type of witness configuration. On compatibility, among simple loops separating a missing pair choose one whose enclosed set \(I\) of marks is inclusion-minimal, with a deterministic rule for ties. It contains a nonempty proper subset of the marks. No missing pair is wholly inside its disk. To prove this, let \(C\) be the selected loop and suppose a missing pair \(a,b\) lies inside. Choose a witness loop \(D\) for that pair. If it meets \(C\), their colors agree, since opposite-color open \(W\) paths cannot meet. Consider the finite planar graph \[H=(C\cup D)\cap\overline{\operatorname{int}C}.\] Every underlying dual path from \(a\) to \(b\) leaving \(\operatorname{int}C\) crosses \(C\), and every such path staying inside crosses \(D\). Thus \(H\) separates \(a\) from \(b\) in the full plane. Planar cut-cycle duality extracts a simple cycle of \(H\) separating them. Its disk lies in \(\overline{\operatorname{int}C}\) and contains one of \(a,b\) but not the other, so its marked set is a nonempty proper subset of \(I\). All its edges have the common color. If \(C\) and \(D\) are disjoint, the disk of \(D\) must lie inside that of \(C\) and gives the same contradiction. This proves the claim, including shared-edge and tangency cases.

Call \(C\) the target. For every missing pair wholly outside \(I\), choose a witness loop and record its color. Record also the target color and \(I\). Reflect the colors for the description so that the target is plus. A chosen minus witness is disjoint from the target; its disk is either disjoint from the target disk or contains that disk. Of its two marked endpoints record the one on the side away from the target disk. The whole type consists of these data, including the ordered away-endpoint vector and the fixed mark \(i_0=\min I\). There are at most \(L(k)<\infty\) types, independently of the shapes of all loops. If the compatibility probability is at least \(a\), one type event \(\mathcal T\) has probability at least \(a_0=a/L(k)\). Fix that type now. The sampling law remains the original law conditional on \(g\); it is not conditioned on \(\mathcal T\).

From its fixed away marks perform a deterministic exploration of the full components of the underlying dual graph whose edges do not cross an open \(W^-\) edge. These are components of the complement of \(W^-\), not components of \(N\). Explore through already known core edges as well as new ones, revealing endpoint spins before each new edge state, until each component is exhausted. There are at most \(k\) distinct source components. The transcript \(h\) is the cylinder of the queried values, so its residual law \(\mu_h\) is an allowed pin law in 5. In particular it remains positively associated. The interior frontier of a full source component consists of actual \(W^-\) edges; it is not a frontier artificially stopped at the old core. Original outer-boundary conditions are retained, and the middle-band test windows are away from the old core and the outer boundary.

Let \(F_h^0\) be the union of the source cell regions of these full components, without filling any complementary pocket. On \(\mathcal T\), each chosen minus witness prevents the source on its away side from reaching the target disk. Thus the entire \(F_h^0\), including its frontier, misses that disk. The transcript also certifies the selected minus separations: the other endpoint of each such pair is outside its explored complement-of-\(W^-\) component. Every \(N\) edge belongs to that complement graph, so these certificates imply the corresponding \(N\) disconnections. Denote them by \(\mathcal C(h)\).

In the residual law, let \(\mathcal U_h\) be the event that there exists an actual plus target loop enclosing exactly \(I\), whose closed disk misses \(F_h^0\), together with actual plus witnesses for the specified missing outside pairs of plus type. This event is increasing in (10). The minimality rule and the identities of the previously selected loops have been dropped; they would not be increasing. If \(\nu_g\) is the law of the transcript under the original law conditioned on the core, the implication from \(\mathcal T\) gives \[a_0\le\int \mathbf 1_{\mathcal C(h)}\, \mu_h(\mathcal U_h)\,\nu_g(\,\mathrm dh).\] It follows that the histories satisfying \(\mathcal C(h)\) and \(\mu_h(\mathcal U_h)\ge a_0/2\) have \(\nu_g\) mass at least \(a_0/2\). We work separately under each such \(\mu_h\). There is no FKG or circuit estimate under the mixture conditioned on this choice of histories.

For this fixed history choose the accessible component \(U_h\) of the complement of the raw region \(F_h^0\) containing the already fixed mark \(i_0\). Histories with \(i_0\in F_h^0\) have \(\mu_h(\mathcal U_h)=0\) and do not occur in the retained family. Every target disk in \(\mathcal U_h\) is connected, contains \(i_0\), and misses \(F_h^0\), so it lies in this one component \(U_h\). Only after choosing \(U_h\), add the other complementary pockets to the forbidden cell region and call the resulting region \(F_h\). This addition is for the marginal collar test alone. It does not discard any actual residual variable from \(\mu_h\); plus witnesses for other pairs may still lie in those pockets.

For each individual connected source, the connected set \(U_h\) lies in one component of its complement and hence faces one incident boundary walk of that source. It may face that walk at many entrances, but there are at most \(k\) source labels in total. Against \(U_h\) the frontier is actual minus in the middle band. The exact completion of one component of 6 therefore applies there; any use of a partition comparison has the same \(k\) indivisible labels. The added pockets attach to these sources, so every local piece in the band is anchored back to a source mark in the old core or to the outer boundary. A target disk remains disjoint from \(F_h\).

Put \(b_0=a_0/2\). Choose a collar width \(\eta>0\), uniformly in the retained histories, so that 7 bounds by \(b_0/2\) the existence of a full underlying dual traversal of the middle band which lies in \(F_h^\eta\) and avoids \(F_h\) and its attached minus circuits. Use the selected \(U_h\) marginal for this estimate. The full residual law \(\mu_h\) is retained for all other events. The virtual-disk clause makes this a simultaneous bound for traversals inside any target disk in \(\mathcal U_h\). In particular, we subtract this error from \(\mu_h(\mathcal U_h)\); we do not condition the law on absence of the collar event.

We spell out the increasing event supplied by this subtraction. Write \(K\) for the enlarged lattice core, \(A\) for its fixed middle band, and \(A^+\) for the wider band. Choose a fixed regular outer dual graph \(R\) inside \(B\), containing \(K\) with a complete connected annular shell outside \(K\). In the path tests below every underlying dual edge of \(R\) is available, regardless of its \(N\) state or whether it crosses a minus edge or the forbidden region. For each required pair across \(I\), orient paths from its \(I\) mark, and require every simple underlying path \(\gamma\) between the two marks to satisfy \[ \begin{array}{c} \gamma\text{ crosses an actual }W^+\text{ edge in }K,\\ \text{or}\\ \gamma\text{ has a subpath }\beta\subset A \text{ from the inner to the outer boundary of }A\\ \text{which visits a point }z\text{ with }\mathop{\mathrm{dist}}(z,F_h)>\eta . \end{array} \tag{31}\] Intersect this requirement with the actual plus separations for the specified outside pairs, and call the event \(\mathcal V_h\). The path family is finite and fixed by \(R\); \(F_h\) is fixed by \(h\). Only the first alternative changes with \(W^+\). Thus \(\mathcal V_h\) is increasing. Neither \(F_h\) nor any minus edge is a real barrier in this definition.

On \(\mathcal U_h\) outside the collar event, (31) holds. Indeed a path avoiding the actual \(W^+\) edges in \(K\) must exit \(K\) before its first crossing of the target loop. An edge meeting the closed core is counted in \(K\), so a first crossing on its boundary cannot be lost. The path prefix before that target crossing lies in the target disk. Before its first exit from \(K\), take the segment from its last hit of the inner band boundary to its first subsequent hit of the outer one. This is a full subpath \(\beta\subset A\), even if the original path made earlier excursions. It lies in the target disk and therefore avoids \(F_h\) and all attached minus circuits. Absence of the collar event forces it to visit a point at distance greater than \(\eta\). Hence \[ \mu_h(\mathcal V_h)\ge b_0/2. \tag{32}\]

We replace the second alternative by actual plus circuits. Choose a deterministic grid in \(A\) of mesh much smaller than \(\eta\) and than the separation of its two boundary curves. For each cell whose filled buffer is disjoint from \(F_h\) and contained in \(A^+\), request an actual plus circuit surrounding the cell in that buffer. Choose the grid and buffers so that every point of \(A\) at distance greater than \(\eta\) from \(F_h\) lies in an inner cell surrounded by one requested circuit, and the disk of every such circuit has diameter smaller than the distance between the two boundary curves of \(A\). This is possible by taking the cell size a sufficiently small fixed multiple of \(\eta\) and of the band width. There are at most \(M=M(\eta)\) cells. Their buffers miss all source queries and the old core; for small mesh they exceed the lattice cutoff.

Under the full residual pin law \(\mu_h\), each requested actual plus circuit has probability at least \(p_1>0\) by 6. Positive association intersects all these increasing events with \(\mathcal V_h\), giving probability at least \((b_0/2)p_1^M\). On their intersection, no across-\(I\) path can avoid the actual plus edges in \(K\). Otherwise (31) supplies a full band subpath visiting the interior of one requested circuit. A connected path avoiding that circuit and visiting its interior must stay inside its disk. Its two band-boundary endpoints cannot both lie there, by the diameter choice. This contradiction realizes the temporary alternative using actual plus edges only.

For completeness, separation in the graph \(R\) gives actual closed witness loops in \(K\). Any path using a more distant exterior can have each exterior excursion replaced in the complete connected shell of \(R\) outside \(K\). Thus separation in \(R\) is separation in the full underlying dual graph by \(W^+\cap K\). Planar cut-cycle duality yields a simple plus loop in \(K\) for each across pair; a boundary-to-boundary cut is impossible because these blocking edges lie strictly inside \(R\). Artificial fill and minus edges play no role in this final separation.

The history certificates protect the outside minus pairs. The outside plus pairs are protected by \(\mathcal V_h\), the across pairs by the new local plus loops, and there was no missing pair wholly inside \(I\). Previously short loops are fixed in \(g\). Therefore compatibility holds and at least one missing pair has joined \(\mathcal S(P_{j+1})\). The retained histories have \(\nu_g\) mass at least \(b_0\), so the construction proves (30) with, for example, \[r(a,k)=\frac{b_0^2}{2}\,p_1^{M(\eta)}>0, \qquad b_0=\frac{a}{2L(k)}.\] The collar width, grid, and mesh threshold were chosen from \(a,k\) and the fixed bands, not from the particular core pattern. Replace \(r(a,k)\) by its minimum over the finitely many band groups.

We finally iterate without retaining the long source histories. Let \(\mu\) be the original law conditional on the initial pattern, and set \[\mathcal G_j=\{\text{compatibility and }|\mathcal S(P_j)|\ge j\}.\] Initially \(\mu(\mathcal G_0)\ge p\). If \(\mu(\mathcal G_j)\ge q_j>0\), disintegrate this original law only by the complete pattern \(P_j\). Patterns with \(|\mathcal S(P_j)|\ge j\) and posterior compatibility at least \(q_j/2\) have total mass at least \(q_j/2\): the other patterns contribute at most \(q_j/2\) to \(\mu(\mathcal G_j)\). If such a pattern already has \(j+1\) short pairs, its conditional contribution is at least \(q_j/2\); otherwise apply (30). Hence \[\mu(\mathcal G_{j+1})\ge \frac{q_j}{2}\min\left\{\frac{q_j}{2}, r(q_j/2,k)\right\} =:q_{j+1}>0.\] At the next step all source transcripts and selected type events have been integrated out. The new conditional law is again the original law given only its complete core pattern, so its unused buffers have the pin/FKG law required above. After \(m\) steps all pairs have short witnesses in \(K_m\Subset B\), with probability at least \(q_m\). All widths and the finitely many mesh thresholds are chosen before taking the mesh limit. This proves the lemma. ◻

Proof. Choose \(Q\Subset Q'\Subset B'\) with \(B'\Subset B\), leaving the fixed buffers of 13. Let \(P\) be the complete \((s,W)\) pattern in \(Q'\), and let \(T(P)\) be its number of terminals. By 8, uniformly in both domain and plane laws, \[ \mathbb P[T(P)>M]\le C e^{-cM}. \tag{34}\] Choose \(M\) so that the right side is less than \(\varepsilon/4\) for the domain law.

On \(T(P)\le M\), write \(b\) for an assignment of signs to its terminals, and let \(v_\mu(b\mid P)\) be its conditional exterior compatibility probability. By 4, the actual terminal-color probability \(q_\mu(b\mid P)\) satisfies \[ 2^{-M}v_\mu(b\mid P) \le q_\mu(b\mid P)\le v_\mu(b\mid P). \tag{35}\] Indeed, given a compatible exterior partition, specifying these terminal signs costs \(2^{-j}\) for some \(j\le M\) distinct free global components. The same calculation applies in the plane because (Duminil-Copin et al. 2026, Theorem 17.1) gives no unbounded complementary component and independent fair transverse colors. Components of the restricted pattern not incident to an unprescribed edge carry independent fair colors. Their distribution is the same under the two laws once \(P\) and the terminal signs are given. The lower bound in (35) uses freeness; a terminal wired to a prescribed incompatible color would not satisfy it.

Choose \(p>0\) with \(2^M p<\varepsilon/4\). The total domain probability of assignments with \(v_\mu(b\mid P)<p\) is at most \(2^M p\), by the upper bound in (35). For every remaining assignment, 13 gives short witnesses in \(B'\) with conditional probability at least \(c(p,M)>0\). Fix a realized pattern \(g\) for \(P\), and let \(\mathcal W_b\) be this short-witness event, determined by a complete pattern in \(B'\). If \(C_{B'}\) and \(C_{Q'}\) are the unconditioned pattern density constants in 12, summing over the larger patterns and then dividing gives \[\pi_\delta(\mathcal W_b\mid P=g) =\frac{\pi_\delta(\mathcal W_b\cap\{P=g\})}{\pi_\delta(P=g)} \ge (C_{B'}C_{Q'})^{-1} \mu_\delta(\mathcal W_b\mid P=g).\] This uses only the original laws on one parity on the two pattern sets, not a comparison after conditioning on \(d\) or on the witness event. Hence there is \(c_0=c_0(p,M)>0\) such that \[ v_\pi(b\mid P)\ge c_0, \qquad q_\pi(b\mid P)\ge 2^{-M}c_0 \tag{36}\] whenever \(v_\mu(b\mid P)\ge p\) and \(T(P)\le M\). There is no conditioning on \(d\) in this comparison: the short witness event concerns only \(W\) and \(s\).

Let \(C_0\) be the uniform density-ratio constant for \(P\) in 12. Conditional on \(P\) and a terminal assignment, integrate the identical independent internal colors. On the retained assignments, (36) bounds the ratio of terminal-color probabilities by \(2^M/c_0\). Therefore every tested event satisfies \[ \mu_\delta(E) \le \frac{\varepsilon}{2} + C_0\frac{2^M}{c_0}\,\pi_\delta(E). \tag{37}\] Choosing \(\zeta<\varepsilon c_0/(2C_0 2^M)\) proves (33). All constants were chosen from \(\varepsilon\), the fixed geometry, and \(t\) before the mesh limit, so the modulus is uniform in the stated domain laws. Enlarging the tested data by \(W\) does not change the conditioning argument. ◻

Proof. First use a finite reflection-symmetric flat exhaustion. Across an axial row, plaquette interactions on the two sides factor after conditioning on the row heights. Across a diagonal row, a straddling plaquette has the following conditional weight matrix. If its two row heights are equal, the other heights are independently one unit above or below that value. Equal transverse heights have weight \(c\), and unequal ones have weight one. Consequently the matrix, after multiplication by \(t\), is \[\begin{pmatrix}1&t\\t&1\end{pmatrix}.\] It is positive semidefinite because \(0<t\le1\). If the row heights differ by two, the transverse heights are forced and there is only a scalar factor. Factoring each positive matrix as a Gram matrix proves positivity after summing the row data. Independent local decorations are included in the factors on their own sides. Spin residues are determined by heights and one compatible global sign choice, which can be fixed at an anchor on the row. For a phase \(U\exp(i\alpha(K(x)-K(o)))\), also condition on its value at that anchor; \(U\) has independent uniform law. Thus these augmentations preserve the same Gram representation. Circuit comparison passes each reflected exhaustion to the same plane law.

Let \(\Theta\) be reflection in the row and set \(\langle F,G\rangle=\mathbb E[\overline{\Theta F}G]\) for functions on its inward half-plane. Denote the pure normal translation through two row spacings by \(T\). Stationarity gives symmetry of \(T\) for this form. Reflection positivity at the intermediate row gives \(\langle F,TF\rangle\ge0\). To prove boundedness before quotienting, put \(b_j=\|T^jF\|\). Reflection Cauchy–Schwarz gives \(b_j^2\le b_{j-1}b_{j+1}\), whereas ordinary Cauchy–Schwarz and stationarity give \(b_j\le\|F\|_{L^2}\) for all \(j\). A bounded nonnegative log-convex sequence cannot increase at its first step. Hence \(\|TF\|\le\|F\|\), also when \(\|F\|=0\). Thus \(T\) descends to a positive self-adjoint contraction. Tangential translation preserves the form, has a form-preserving inverse, and commutes with \(T\), whenever it preserves the chosen state space. For the height-increment law this includes one row period. With a fixed one-parity decoration it includes two axial periods or one diagonal period; the normal transfers used above already preserve that parity. ◻

Proof. Let \(V_x\) be ordinary translation on the plane probability space, \(E(\,\mathrm dp)\) its spectral resolution (Williams 2018, Theorem 5.6), and \(\xi_x=K(x)-K(0)\). The cocycle identity gives \[(V_y-I)\xi_x=(V_x-I)\xi_y.\] On any spectral neighborhood where \(e^{ip\cdot y}-1\ne0\), define \[C(\,\mathrm dp)= \frac{\langle\xi_y,E(\,\mathrm dp)\xi_y\rangle_{L^2}} {|e^{ip\cdot y}-1|^2}.\] The identity makes the definitions agree on overlaps; two grid directions cover the punctured torus. The invariant part is additive and has the form \(A\cdot x\) for a random vector \(A\in L^2\). Smearing at scale \(L\) against a neutral test of first moment \(m\) gives invariant part \(LA\cdot m\). Its norm is uniformly bounded by [eq:plane-neutral-moments], since orthogonal projection cannot increase norm. Two independent choices of \(m\) give \(A=0\).

Choose two real neutral smooth tests whose Fourier transforms have no common zero on \(1\le|p|\le2\); derivatives in the two coordinate directions of a sufficiently narrow smooth bump suffice. Their scale-\(r^{-1}\) interpolation transforms converge uniformly on this annulus to those transforms. The sum of their squared absolute values is therefore bounded below on \(r\le|p|\le2r\). Apply [eq:increment-spectral] and the uniform second moment bound. Summing over dyadic annuli proves the last assertion. ◻

Proof. The row increment \(F=K(\ell u)-K(0)\) belongs to the reflection Hilbert space. The joint spectral theorem for its commuting tangential unitary and positive normal contraction (Williams 2018, Theorem 5.6) gives, for \(j\ge0\), \[\mathbb E[F V_{n\ell u+jdv}F] =\int e^{in\ell k}\lambda^j\,\gamma(\,\mathrm dk,\,\mathrm d\lambda), \qquad 0\le\lambda\le1.\] Reflection gives the same expression with \(|j|\). Set \(\lambda=e^{-ds}\), including \(s=\infty\) when \(\lambda=0\). These are precisely the vertical Fourier coefficients of \(Q_s\). Uniqueness of Fourier coefficients for finite measures identifies the ordinary increment measure as this mixture. That measure is \(|e^{i\ell k}-1|^2C'\). Division for \(k\ne0\) proves the formula. Since every \(Q_s\) has mass one, the integral in [eq:weighted-mixture] equals \(\int|k|C'(\,\mathrm dk,\,\mathrm dl)\) over the same band. Near the physical origin it is finite by [eq:spectral-annulus]; all other contributions have finite mass. ◻

Proof. Integrate [eq:angular-poisson] over \(\varepsilon<|k|<\kappa\). Its error is bounded independently of \(\varepsilon\) by [eq:weighted-mixture]. Replacing \(C'\) outside the small physical disk by the original measure costs bounded outer and alias mass. In the radial annulus \(\varepsilon<|p|<\kappa\), the omitted endcap \(|k|\le\varepsilon\) has negative fourth harmonic only where \(|p|\le C\varepsilon\): negativity implies \(|l/k|<1+\sqrt2\). Its negative part is bounded by a fixed number of the annuli in [eq:spectral-annulus]. Thus, in a frame of angle \(\theta\), \[ \int_{\varepsilon<|k|<\kappa}D\,\,\mathrm d\nu \le \int_{\varepsilon<|p|<\kappa} \cos(4(\arg p-\theta))\,C(\,\mathrm dp)+C_2. \tag{45}\] Choose the same physical inner and outer radii for frames \(\theta\) and \(\theta+\pi/4\), making the fixed outer radius smaller if necessary. Their radial integrands are negatives pointwise. Adding the inequalities therefore bounds the sum of the two nonnegative defect integrals. Monotone convergence proves finiteness for each. Apply the same argument to the other frame pair. No rotation of the measure \(C\) through \(\pi/4\) has been used.

On \(r\le|k|\le2r\), the part with \(s\notin[r/2,4r]\) has \(D\ge1/9\), so its mass is bounded. For \(s\in[r/2,4r]\), a fixed fraction of \(Q_s\) lies in \(|l|\le2r\). The remaining mixing mass is bounded by a constant times \(C'\{r\le|k|\le2r,\ |l|\le2r\}\), which is bounded by the annular estimate and finite alias mass. This proves the horizontal-band bound for sufficiently small \(r\). For the remaining \(r\), the bands lie in a fixed horizontal region bounded away from zero, whose \(\nu\) mass is finite by 17; bands beyond the reciprocal interval are empty. Increasing \(C_1\) gives the bound for every \(r>0\). ◻

Proof. Describe the same region in a frame at angle \(\pi/4\). Its new coordinates \(k',l'\) both have size comparable to \(r\). At fixed \(k'\), the old restriction \(|k|\le\varepsilon\) confines \(l'\) to an interval of length \(O(\varepsilon)\). On this interval each central Cauchy density is at most \(C/r\), since \(s/(s^2+l'^2)\le1/(2|l'|)\), and the folded tails are bounded. The \(s=0\) atom is outside the interval; the uniform law also satisfies the bound. The horizontal-band mass in the new frame is bounded by [eq:dispersion-defect]. Projection only adds nonnegative alias mass, so its bound bounds the original physical measure. Letting \(\varepsilon\) decrease to zero gives the last statement. ◻

Proof. Rescale the physical measure by \(p\mapsto p/\delta\). The annular bound gives vague subsequential compactness off zero; write \(M\) for one such limit. Fix \(0<b<B\) in one reflection frame. Let \(\nu_\delta^{b,B}\) be the pushforward of \(\mathbf 1_{\{b\delta\le |k|\le B\delta\}}\nu\) under \((k,s)\mapsto(\xi,\sigma)=(k/\delta,s/\delta)\), with \(\infty/\delta=\infty\). The horizontal-band bound in [eq:dispersion-defect] bounds its total mass uniformly. Homogeneity of \(D\) and its integrability there give \[\int D(\xi,\sigma)\,\nu_\delta^{b,B}(\,\mathrm d\xi,\,\mathrm d\sigma) =\int_{b\delta\le |k|\le B\delta}D(k,s)\,\nu(\,\mathrm dk,\,\mathrm ds) \longrightarrow0.\] Take a weakly convergent further subsequence on the compact space \(\{b\le|\xi|\le B\}\times[0,\infty]\), where the last interval is compactified at infinity. On this space \(D\) is continuous, equals one at \(\sigma=0\) and \(\sigma=\infty\), and vanishes exactly at \(\sigma=|\xi|\). Every limiting mixing measure is therefore supported on that graph. In particular, the defect estimate excludes mass at both ends of the rescaled normal parameter.

Let \(\widehat Q_{\delta,\sigma}\) be the pushforward of \(Q_{\delta\sigma}\) under \(l\mapsto l/\delta\). On a compact vertical interval, uniformly for \(\sigma\) in a compact subset of \((0,\infty)\), \[\widehat Q_{\delta,\sigma}(\,\mathrm dl) =\left(\frac{\sigma}{\pi(\sigma^2+l^2)}+O(\delta)\right)\,\mathrm dl.\] The first term is the rescaled central Cauchy density; the bound on the folded tails used in [eq:angular-poisson] gives the error. The vanishing defect makes the mixing mass outside any fixed neighborhood of \(\sigma=|\xi|\) tend to zero. We may therefore pass this kernel formula against compactly supported vertical tests and tangential tests in the interior of the closed band. This restriction avoids any issue from atoms on its boundary.

For these tests the projected measure \(C'\) and the physical measure have the same limit. Indeed, the preimage near an alias \(p_a\) lies in \(B(p_a,c_*\delta)\setminus\{p_a\}\) for a constant \(c_*\) depending on the test support: the nonzero scaled tangential coordinate excludes \(p_a\) itself. The measure \(C\) is finite near \(p_a\), so its mass on this shrinking punctured neighborhood tends to zero. There are only finitely many aliases. Exhausting \(k\ne0\) by interiors of the bands above now gives \[M(\,\mathrm dk,\,\mathrm dl)=\frac{|k|}{\pi(k^2+l^2)}\,\eta(\,\mathrm dk)\,\,\mathrm dl\] for a nonnegative measure \(\eta\). The tangential measures agree on overlapping bands, since integrating the displayed positive density over a fixed vertical interval recovers them uniquely.

Thus \(|p|^2M\) is locally invariant under normal translations where \(k\ne0\), without assuming that \(\eta\) has a density. At every nonzero point at least two of the four frames have nonzero tangential coordinate and independent normals. The two corresponding distributional derivatives of \(|p|^2M\) vanish in a neighborhood. Convolving locally with a smooth kernel shows it is a constant multiple of Lebesgue measure there. Constants agree on overlaps in the connected punctured plane. We obtain \(M(\,\mathrm dp)=b_0|p|^{-2}\,\mathrm dp\), with \(0\le b_0<\infty\).

To pass from vague convergence to tests, let \(A_{\delta,f}\) be the transform of the interpolation coefficients. They have exactly zero sum. Near zero, for every fixed integer \(q\ge3\), \[|A_{\delta,f}(p)|\le C_q \min\{|p|/\delta,(1+|p|/\delta)^{-q}\}.\] The first bound uses the first absolute moment of the coefficients; the second follows by \(q\) lattice summations by parts against the smooth test convolved with its nodal hat. Away from zero the same summations give \(O(\delta^q)\). Thus the low-frequency tail is \(O(\varepsilon^2)\), the high-frequency tail within the small disk is \(O(R^{-2q})\), and the remaining tail vanishes by local finiteness. On the intervening compact annulus use vague convergence. The Fourier convention \(\widehat f(p)=\int e^{-ip\cdot x}f(x)\,\mathrm dx\) gives \[b_0\int\frac{\widehat f(p)\overline{\widehat g(p)}}{|p|^2}\,\mathrm dp =2\pi b_0\mathcal L(f,g).\] This proves the assertion with \(\beta=2\pi b_0\), including varying tests.

Finally choose two disjoint small disks and a difference of nonnegative mass-one tests in them. Choose two fixed nested annuli enclosing the first disk and avoiding the second, with positive margins. For all sufficiently small meshes, the nodal supports remain in their respective disks. Apply 6 successively in the two annuli. With probability bounded below, they contain \(W\) circuits of opposite prescribed colors. This uses the conditional one-parity bound, so it does not require positive association between the two opposite-color events.

Reveal the outer circuit by an outside-in stopping search and condition on it and its exterior. No \(N\) cluster crosses this cut. Conditional also on the complete \((s,W)\) data in its finite interior, 4 gives independent fair colors to the enclosed \(N\) clusters. The opposite \(s\) colors of the two circuits imply that an odd number of their intervening cluster jumps are nonzero. At least one such free cluster has an odd hole containing the first support, while the second support is outside the outer cut. Flipping its fair color shifts the first average by two in one direction or the other relative to the second, and hence changes their difference by four. Its conditional variance is four. The resulting strictly positive lower variance bound is uniform in mesh. It bounds every possible \(\beta\) away from zero. The upper bound follows from [eq:plane-neutral-moments] applied to these tests. ◻

Proof. Every subsequence has a further covariance limit in 20. The squared reflection norm then tends to the Green pairing of \(\Delta\phi\) with its reflected copy, times \(\beta\). This pairing is zero: after integration by parts it is a constant times the integral of the two gradients, whose supports are disjoint. The coefficient need not yet be unique. Positivity and the subsequence argument prove convergence to zero along the full mesh limit. Decorations not used by \(X_\delta\) do not change this covariance, and converging placements are covered by the varying-test part of 20. ◻

Proof. We give the count argument, so that this estimate does not depend on the Gaussian limit proved later. Write the coefficients of the insertion as \(b_v\), where \(\sum_v b_v=0\), \(|b_v|\le C\delta^2\), and \(\sum_v|b_v|\le C\). If both corner parities occur, transfer the coefficients on the other parity to nearest \(s\) sites. Each corner increment has absolute value one, so this changes the insertion by a uniformly bounded remainder \(R_\delta\) and preserves exact neutrality. Conditional on the complete \((N,s)\) data, the finite cut identity gives \[ Y_\delta=R_\delta+2\sum_C \epsilon_C q_C, \qquad q_C=\sum_{v\in O_C}b_v. \tag{50}\] Here \(O_C\) is the union of the odd holes of \(C\), and the signs \(\epsilon_C\) are independent and fair. An additive constant disappears because the coefficients sum to zero. If a fixed wired color is present, its single contribution is bounded by \(2\sum_v|b_v|\) and is treated separately. The same argument can first be made in finite exhaustions; the estimates below are uniform in the exhaustion.

Fix a connected neighborhood of the support with positive clearance inside a pin-free buffer. For a cluster of diameter \(r\) contained in that buffer, its filled hull lies in a disk of radius \(r+O(\delta)\), so \(|q_C|\le C(r+\delta)^2\). A cluster with \(q_C\ne0\) must meet the connected neighborhood: otherwise its odd-hole indicator is constant there and neutrality gives \(q_C=0\).

We localize the count of such clusters before applying pattern comparison. Fix a sufficiently large constant \(L\). For \(L\delta\le r\) below a fixed buffer scale, cover the one-mesh enlargement of the insertion neighborhood by \(O(r^{-2})\) inner squares of side \(r/20\). Assign each relevant cluster of diameter in \([r,2r)\) to one such square containing a deterministically chosen vertex of the cluster. Some other vertex is at distance at least \(r/2\) from that vertex, so a path in the cluster exits the concentric square of side \(r/2\). Its first exiting segment gives a component of the relative restricted \(N\) graph meeting the inner square and a vertex incident to an edge outside the restriction. Distinct global clusters give distinct restricted components. The fixed rectangle cover in item 2 of 8 therefore gives \[A_{r,j}\le\sum_{\ell=1}^{L_0}T_{r,j,\ell},\] where \(A_{r,j}\) is the assigned global count in square \(j\), and each \(T_{r,j,\ell}\) counts restricted crossings of one rectangle, using the separate side ports of that proposition. The number \(L_0\) is fixed. Each \(T_{r,j,\ell}\) is determined by a complete local pattern in an enlarged box of size \(O(r)\). Choose the buffer scale so that these boxes and all their joining buffers lie inside the pin-free neighborhood.

The transmission bound gives moments of every fixed order for these local counts. Since they are local-pattern measurable, the density comparison in 12 transfers their moments under all the remote patterns, with constants uniform in \(r\). It follows that the conditional \(L^q\) norm of each \(A_{r,j}\) is bounded by \(C_q\). For \(\delta\le r<L\delta\), use boxes of side \(L\delta\) instead: each contains only a bounded number of lattice vertices, so its assigned cluster count is bounded deterministically. For every \(q\ge1\), Minkowski’s inequality now gives, under the law conditioned on the remote patterns, \[\left\|\sum_{C:\,r\le\mathop{\mathrm{diam}}C<2r}|q_C|^2\right\|_q \le C_q r^2,\qquad r\ge\delta,\] as long as \(r\) is smaller than a fixed buffer scale. Summing dyadic scales is harmless.

The remaining clusters with \(q_C\ne0\) meet the insertion neighborhood and either have diameter at least the fixed buffer scale or leave the buffer. In either case they transmit across a fixed intermediate scale. The same first-exit argument bounds their number by a fixed finite sum of local restricted crossing counts, with joining buffers inside the pin-free neighborhood, regardless of how their pieces join farther out. Transfer these local counts by the same density comparison. Each coefficient is at most \(\sum_v|b_v|\), so these contributions also have bounded moments. Conditioning on \((N,s)\) and applying the elementary moment bound for a sum of independent signs now proves (49). All comparisons used only \((s,W)\) patterns, not \(d\)-color prescriptions. The same buffer and counting proof works in a flat domain, and after reflection of the entire separated arrangement. ◻

Proof. Set \(P=\bigcap_iP_i\) and \(J=\prod_i p_i\). The square of the left side is \[\frac{\mathbb E[\mathbf 1_{P\cap\Theta P}\overline{\Theta A}A]}{J^2}.\] The reflected and original pattern neighborhoods are disjoint. 12 bounds their joint probability by \(CJ^2\). If \(A\) contains a neutral insertion, apply conditional Cauchy–Schwarz and 23 to the insertion and its reflection. Otherwise the numerator is bounded directly by the pattern probability. Thus the squared denominator is canceled. No lower bound on any individual \(p_i\) is required. ◻

Proof. Choose \(\eta>0\) smaller than the projection gap. Back the high group up by \(\eta_\delta=N_\delta\tau_\delta\to\eta\), where \(\tau_\delta\) is the positive normal step in 15. Choose a reflection row inside the remaining gap. Reflect and conjugate the normalized low-side factor, obtaining a positive-side vector \(A_\delta\); call the backed-up high-side factor \(B_\delta\). By 24, both reflection norms are at most a constant independent of the patterns. If \(T_\delta\) is the positive normal contraction, define \[F_\delta(z)= \langle A_\delta, T_\delta^{(z+\eta_\delta)/\tau_\delta}B_\delta\rangle_\Theta, \qquad \mathop{\rm Re}z>-\eta_\delta.\] For \(\mathop{\rm Re}w>0\), the spectral power is \(T_\delta^w=\int_{(0,1]}e^{w\log\lambda}\,\,\mathrm dE_\delta(\lambda)\), with value zero on the zero spectral subspace. This is a holomorphic contraction by bounded spectral calculus (Williams 2018, Proposition 5.3 and Corollary 5.7): on \(\mathop{\rm Re}w\ge b>0\), each derivative is bounded by \(\sup_{0<\lambda\le1}\lambda^b|\log\lambda|^m<\infty\). Hence \(F_\delta\) is bounded on the common half-plane \(\mathop{\rm Re}z>-\eta/2\) for all sufficiently small \(\delta\). At nonnegative lattice-step arguments it equals \(M_\delta\) exactly. Cauchy’s estimate makes lattice rounding negligible on compact sets. Every subsequence has a locally uniformly convergent further subsequence; its holomorphic limit vanishes on the given interval and hence everywhere by the identity theorem. In particular it vanishes at zero. The norm bounds and the common backoff remain valid for all the sequences in the statement. Moving the low group in direction \(-n\) is equivalent. ◻

Proof. Use the normals \(e_y=(0,1)\), \(u=(1,1)/\sqrt2\), \(e_x=(1,0)\), and \(v=(-1,1)/\sqrt2\). First translate \(T\) upward and \(D\) downward until \(T\cup R\) and \(D\cup L\) are strictly above and below the insertion in \(u\)-projection. Next translate \(T\cup R\) in direction \(u\) and \(D\cup L\) in direction \(-u\). Choose these distances large enough that the first group is above the insertion and wholly to its right, and the second group wholly to its left. Move the latter farther left until it is strictly above every other support in \(v\)-projection. This additional horizontal move is needed because \(u\cdot v=0\). Finally move that group in direction \(v\) until all factors are above the insertion.

Every move is rigid on a complete side of a strict projection gap. Within-group clearances are preserved, and between-group gaps increase. The distances can be chosen in open intervals with positive margins. At the final arrangement, reflection Cauchy–Schwarz in a horizontal row gives the bound \(C\|X_\delta\|_\Theta\), which tends to zero by 21. Undo the moves in reverse order using 25, each time first varying the last distance over an open interval. There is no motion through overlapping supports. This proves convergence along every sequence of patterns and factors. If the displayed supremum did not tend to zero, a sequence within half its positive limsup would contradict precisely this conclusion. ◻

Proof. For each block pattern, form the restricted \(N\) graph from precisely the dual edges whose \(W\) states that pattern specifies, as in 3. An observed restricted component with no vertex incident to an unprescribed \(N\) edge is sealed. Its \(d\) color is independent and fair by the color law there; its filled hull lies in the block’s filled isolating neighborhood, so that color cannot affect \(X_\delta\). Integrate these colors out. Every remaining observed component meets the core and contains a vertex incident to an unprescribed edge; call these components terminals.

Under \(\mathbb P_\delta\), before fixing a realization of \(S_\delta\), the fixed core and block clearances meet the unprescribed-interior hypotheses of item 2 of 8. A finite union of its transmission bounds gives a geometric tail for the total number of terminals.

Truncate this number at \(M\). For each pattern, choose one representative in each terminal, in a deterministic order. An assignment of their at most \(M\) signs is the product of bounded local indicators, one factor in each block. It need not have a positive probability bounded below: global joining outside the blocks can forbid assignments. The quantity in (53) is the sum of absolute unnormalized expectations over the data. Thus 26 bounds its retained part by \[2^M e_\delta \sum_{P_1,\ldots,P_m}\prod_i\mathbb P_\delta(P_i) =2^M e_\delta.\] Integrating the independent internal colors contributes a factor one. On the discarded event, Cauchy–Schwarz and 23 bound the contribution by a constant times the square root of the count-tail probability. First let \(\delta\downarrow0\) at fixed \(M\), then let \(M\to\infty\). ◻

Proof. Reveal previous trials completely before testing the next. Their buffered neighborhoods are disjoint, so this is exterior \((s,W)\) data for the current trial. The tube version of 6 gives \(p(q)\); the aspect ratios are fixed once \(q\) is fixed. A stack of separated annuli between \(Cw\) and \(R/10\) gives circuit failure probability at most \(\eta(q)\le Cq^\alpha\) for some \(\alpha>0\).

Write \(A_j\) for the increasing two-arm event, \(G_j\) for the surrounding plus-circuit event, and \(\mathscr H_j\) for the full previous trial history. Positive association in the remaining law gives \[ \mathbb P(A_j\cap G_j^c\mid\mathscr H_j) \le \eta(q)\mathbb P(A_j\mid\mathscr H_j). \tag{55}\] If \(J\) is the first successful trial, sum this inequality over the disjoint events \(\{J=j\}\). Their total probability is at most one; there is no factor \(M\) in the resulting shielding error. Iterating the conditional arm bound gives the failure probability. Revealing all remaining block patterns afterward does not change either averaged estimate. We do not assert (55) after conditioning on each individual complete pattern. ◻

Proof. Fix a block pattern \(S\), and write \(H(S)\) for success at all four gaps. On this event let \(G_i(S)\) be the plus-circuit event around the selected hole at gap \(i\), and let \(\mu_i^S\) be the conditional law after the first \(i\) selected bridges have been pinned, with \(\mu_0^S\) the original conditional law. On unsuccessful patterns put \(\mu_i^S=\mu_0^S\) and \(G_i(S)=\Omega\). For two successive laws use the monotone stopping coupling of 5, searching outward from the newly pinned hole inside its shielding disk. Stop at the disk’s outer boundary and declare this adjacent coupling unsuccessful if no surrounding plus circuit has been found. All previously inserted pins are identical in these two laws. In the lower law the first plus circuit surrounding that hole is also open in the upper law. All changed data are inside this common cut. By 4, the configurations outside it can be sampled identically, including complementary-component colors. In particular their exterior color connections are unchanged. The exact neutrality of the insertion removes any change in the chosen additive height anchor. If \(G_i(S)\) occurs in the lower law, this cut is found inside the shielding disk, so the required insertion, exterior data, and core colors outside that disk agree.

To realize these adjacent couplings in the plane, first run each disk-stopped search in finite flat exhaustions at fixed \(\delta\) and \(S\), retaining a binary failure marker. Each conditioning event involves finitely many prescriptions and has positive plane probability. To verify positivity, place \(S\) and the compatible bridge pins in a larger buffered flat box with an open plus \(W\) perimeter. Keep every prescribed state, put plus on all remaining \(s\) sites, close all remaining interior \(W\) edges, and take all \(d\) spins equal. Every factor in [eq:spatial-edge-weight] is positive because \(t\in[1/2,1)\). Let \(\mu_{\mathrm{box}}\) denote the flat law in this box. The restriction \(p\) of this filling to a regular inner core containing the prescriptions is a complete pattern with \(\mu_{\mathrm{box}}(p)>0\). The upper bound \(\mu_{\mathrm{box}}(p)\le C\pi(p)\) in 12 gives \(\pi(p)>0\), and hence positive plane probability for the conditioning event containing it.

The conditional marginals therefore converge on cylinder events. Extract a joint limit on the countable product of spin and edge coordinates, together with the failure marker. Its probability bound passes to the limit, as does equality of each fixed exterior coordinate on success. Taking their countable intersection gives equality of the full exterior configuration; its computable height increments agree as well. Thus this limit has the plane marginals and the coupling properties just proved in finite volume.

For fixed \(S\), the event \(G_i(S)\) is now fixed and increasing. The previous plus pins can only increase its probability. With \(\mathbb E_S\) denoting expectation over the original pattern marginal, this gives \[\mathbb E_S\!\left[\mathbf 1_{H(S)} \mu_{i-1}^S\bigl(G_i(S)^c\bigr)\right] \le \mathbb E_S\!\left[\mathbf 1_{H(S)} \mu_0^S\bigl(G_i(S)^c\bigr)\right] \le \eta(q).\] The first inequality is the comparison at each fixed pattern; the second is 28 after averaging over the selected trial. At a fixed mesh the pattern space is finite, so choose these adjacent conditional couplings measurably in \(S\). Apply the displayed estimate to each of their marginal failure probabilities, then glue the four couplings and use a union bound. Their searches stay in disjoint buffers, and every intermediate law retains the original pattern marginal. This proves the asserted joint coupling.

On a successful pattern, join the outer arm ends through compact cores of the blocks. The core portions together with the open arms and the new bridges give a closed walk winding once around \(B\) and lying inside \(Q\). Extract a simple winding cycle, with its lattice-width incident sites, by deterministic loop erasure. On every non-open core edge, condition on both incident \(d\) sites; include the incident sites at the transitions. Along the arms and bridges no \(d\) observation is needed. Indeed the local factor there is \(t\mathbf 1_{s_i=s_j}\), independent of transverse colors. Along a remaining core edge all its \(s,W\) data and the two incident \(d\) values are fixed. Every local factor crossing the cycle is consequently either fixed or independent of the transverse variables. The Gibbs weight factors into an inside weight and an outside weight after these observations. This proves conditional independence. This factorization is made with colors as actual spin variables, before summing them into cluster-count factors; no hidden connectivity weight remains across the separator. All observed \(d\) sites lie in the compact cores and outside the shielding disks. ◻

Proof of 22 in the plane. Truncate the insertion to \([-T,T]\), writing the result as \(X_\delta^{(T)}\), and set \[r(T)=\sup_\delta\mathbb E_{\mathbb P_\delta} |X_\delta-X_\delta^{(T)}|.\] The moment bound gives \(r(T)\to0\). Suppose the coupling mismatch probability is \(\varepsilon\), and the unsuccessful-pattern probability is \(\zeta\). Duality against bounded functions gives \[\begin{align*} \|\mathbb E_{\mathbb P_\delta}[X_\delta^{(T)}\mid\mathscr E_\delta]\|_1 &\le \|\mathbb E_{\mathbb Q_\delta} [X_\delta^{(T)}\mid\mathscr E_\delta]\|_1 +2T\varepsilon\\ &\le \|\mathbb E_{\mathbb Q_\delta}[X_\delta^{(T)}\mid Y_\delta]\|_1 +2T\varepsilon+2T\zeta\\ &\le \|\mathbb E_{\mathbb P_\delta}[X_\delta^{(T)}\mid Y_\delta]\|_1 +4T\varepsilon+2T\zeta. \tag{56}\end{align*}\] For the middle inequality use conditional independence on success and bound the unsuccessful histories by their probability times \(2T\). The outer inequalities use agreement of the joint insertion and conditioning data. They do not require equal laws of \(Y_\delta\).

By 27, at each fixed layout the final conditional norm has limsup at most \(r(T)\). The coupling and trial lemmas give \(\varepsilon\le4\eta(q)\) and \(\zeta\le4(1-p(q))^M\). Returning from \(X_\delta^{(T)}\) to \(X_\delta\) therefore gives \[\limsup_{\delta\downarrow0} \|\mathbb E[X_\delta\mid\mathscr E_\delta]\|_1 \le 2r(T)+16T\eta(q)+8T(1-p(q))^M.\] Given an error tolerance, first fix \(T\) large, then \(q\) small, then \(M\) large, and finally fix a layout as in (54). Taking its mesh limit proves (48). Only original-law moments have been used; there is no moment assumption for the artificially pinned law. ◻

Completion of the proof of 22. Choose a regular, fixed-width spin ring \(A\) between the insertion and \(\partial Q\), with positive clearance from both. Let \(R_\delta\) denote both spins on a lattice-width enlargement of this ring. For a ring assignment \(r\) admitting an interior filling, let \(\mathsf K_\delta^r\) be the finite interior spin Gibbs kernel obtained from [eq:spin-constraint,eq:spin-weight], with \(r\) fixed on the ring and all factors meeting the interior retained. Define \[g_\delta(r)=\int X_\delta\,\,\mathrm d\mathsf K_\delta^r,\] and put \(g_\delta(r)=0\) when no interior filling is admissible. The neutral insertion is determined by the interior height increments, so its value in this formula is independent of an additive height anchor. The finite product weights give this same conditional interior kernel in every flat domain; the identity also passes from flat exhaustions to the plane on cylinder events. Thus \(g_\delta(R_\delta)\) is a version of the conditional mean of \(X_\delta\) under each of these laws, including on admissible ring assignments that may have zero plane probability. All these expectations are finite at each mesh.

Apply the separated-block proof to a slightly smaller surrounding square, whose exterior contains the ring. All its strict projection clearances persist under this small change of the surrounding square. Since the ring data belong to that exterior sigma-field, conditional Jensen gives \(\mathbb E_{\mathbb P_\delta}|g_\delta(R_\delta)|\to0\), hence convergence in plane probability. The uniform full-spin contiguity of 14, on a fixed larger box, transfers this convergence in probability to the flat laws, even for mesh-dependent tests of the ring data. The neutral moment estimate in 23, with no patterns, gives uniform integrability of these conditional means by conditional Jensen. Consequently their absolute means tend to zero uniformly over the flat laws. Finally the Markov property across the ring gives \[\mathbb E[X_\delta\mid\mathscr E_\delta(Q)] =\mathbb E[g_\delta(R_\delta)\mid\mathscr E_\delta(Q)],\] and conditional Jensen proves the asserted exterior estimate. Uniformity can equivalently be checked by taking an arbitrary contradicting sequence of flat domains; all buffers remain fixed and the same argument applies. At a random exact cut, the statement is used for the realized conditional flat law. Subsequent conditioning is not included in this claim. ◻

Proof. Adjoin the exterior of \(B_0\) in the sphere. The complements remain connected, and so does their Hausdorff limit. The planar criterion just stated gives simple connectivity of every open component. Hausdorff convergence of compact sets gives uniform convergence of their distance functions, which proves compact containment.

For the estimate, choose a conformal map \(f:\mathbb D\to W\) with \(f(0)=x\), and write \(f(z)=y\). The growth bound and the quarter theorem recalled in 10.3 give \[|x-y|\le |f'(0)|\frac{|z|}{(1-|z|)^2} \le \frac{4d_W(x)}{(1-|z|)^2}.\] Under the stated hypothesis this implies \(|z|\ge1/2\). The conformal Green identity and \(-\log t\le2(1-t)\) for \(1/2\le t\le1\) now give [eq:flat-green-boundary].

For convergence, domain monotonicity gives the common bound \(0\le G_{V_n}\le G_{B_0}\), with zero understood between components. The upper kernel is bounded on compact sets away from the diagonal and has only an integrable logarithmic singularity on the diagonal. Fix \(y\in V\) and a component \(C\) of \(V\). On compact subsets of \(C\setminus\{y\}\), harmonic compactness gives subsequential limits in the first variable. If \(y\in C\), a fixed disk about \(y\) is eventually contained in one component of \(V_n\). On that disk subtract \(\log(1/|x-y|)\). The difference is harmonic and uniformly bounded on a fixed interior circle by the common upper bound; the Poisson formula therefore preserves the unit logarithmic pole in the limit. If \(y\notin C\), there is no pole.

For \(x\in V_n\), its distance to the boundary of its component equals \(\mathop{\mathrm{dist}}(x,F_n)\). Passing [eq:flat-green-boundary] to a subsequential limit shows that this limit has zero trace on \(\partial C\), since \(\mathop{\mathrm{dist}}(x,F_n)\to\mathop{\mathrm{dist}}(x,F)\). The maximum principle now identifies it as \(G_C(\,\cdot\,,y)\) when \(y\in C\), and as zero when \(y\notin C\). The removable difference is bounded near its pole and near the boundary, so the maximum principle applies on the whole component. Every subsequential limit is the same. Interior harmonic estimates in both variables give the asserted local uniform convergence. The common logarithmic upper bound also gives convergence of \(G_{V_n}(f,g)\) for fixed smooth tests with supports compactly contained in \(V\). On the space of connected complements, the local uniform assertion gives continuity of \(G_{B_0\setminus F}(x,y)\) where \(x,y\notin F\) and \(x\ne y\). These positive-clearance sets form a countable cover of the interior off-diagonal pairs. Extension by zero off the open set and the usual diagonal value therefore give a Borel kernel, and hence Borel smeared covariances. ◻

Proof. Condition on the current \(N\) and the \(s\) spins. By 4, the centered height at a wall-parity site is a sum of jumps of size two with independent fair signs, one sign per contributing cluster. A cluster’s coefficient at a site is zero or has absolute value two. Its sign is shared by all its odd holes.

The same signed expansion controls opposite-parity sites. For such a site \(x\), choose an incident wall-parity site \(v\), and let \(C_x\) be the free \(N\) component containing \(x\). The corner rule gives \[K(x)=K(v)+\varepsilon_W s(v)d(C_x),\] where the fixed sign \(\varepsilon_W\) records which parity carries \(W\). Absorb this term into the coefficient of \(d(C_x)\). Conditional on \((N,s)\), every height is still a sum of the independent fair component colors, now with coefficients of absolute value at most three. If a component has a nonzero coefficient at a site, that site lies in the component or its filled hull up to distance \(O(\delta)\): wall-parity contributions come from odd holes, and the extra contribution is incident to the component. Consequently a component shared by two sites has diameter at least their separation minus \(O(\delta)\).

Expanding an even product and summing the independent signs leaves only terms in which every sign appears an even number of times. Pair the occurrences of each such sign. Overcounting by all pairings gives an upper bound by \(C_m\) times sums of products of shared-cluster counts \(M(z_i,z_j)\). This argument does not assert independence of the counts.

A contributing cluster surrounds or is incident to the relevant site. The preceding support observation gives the required diameter lower bound for a shared cluster. At a fixed site only boundedly many components of diameter at most \(C\delta\) can contribute, including isolated vertices; put all of these contributions in the lattice cutoff. Above that cutoff, a fixed number of boxes at a cluster’s diameter scale detect its passage or enclosure. By 8, the count in each such dyadic shell has uniformly bounded moments of every fixed order. Minkowski’s inequality therefore gives \[\|M(z_i,z_j)\|_{L^m} \le C_m\left(1+\log^+\frac{R_0}{|z_i-z_j|\vee\delta}\right).\] Holder’s inequality for the product of the \(m\) counts proves [eq:flat-log-moments]. Height reflection about the flat value proves vanishing of odd moments. The argument is unchanged in several independent chambers; shared-cluster counts between different chambers are zero.

Integrate the estimate against the interpolation weights, or the atomic Riemann weights, of the tests. Each logarithmic factor is uniformly integrable in its two variables, including its lattice diagonal: the latter contributes \(O(\delta^2\log(1/\delta))\). Since the pairs have disjoint sets of variables, this proves [eq:flat-smearing-moments]. Rescaling a fixed profile by \(x=z+ru\) gives [eq:flat-scaled-moments]. Here \(\delta/r\le1\) is essential for an atomic smearing: after the rescaling the diagonal part has the additional uniformly bounded factor \((\delta/r)^2\log(r/\delta)\), while the common scale contributes \(1+\log^+(R_0/r)\). ◻

Proof. The relevant wall component reaches from the small scale to the outside of the fixed testing window. A shared cluster either is incident to the distinguished site or encloses it. In the incident case its connected passage reaches the outer scale forced by the distant spectator. In both cases it must avoid matching circuits attached to this wall at every scale between \(r\) and \(R\): such a circuit separates the distinguished contribution from the distant one. The conditional circuit estimate in 6 gives a uniformly positive success probability at a fixed fraction of these separated scales. An attached circuit blocks the cluster, so iteration gives \(C(r/R)^\eta\). The argument is within the accessible chamber with its wire retained; no regularity of the geometric wall is used.

In every pairing contributing to a mixed even moment, the distinguished site is paired with one spectator. The corresponding shared cluster has a fixed positive minimum diameter. Its count, restricted to surviving the attached-circuit trials, tends to zero in every smaller \(L^p\) norm: apply Holder to the survival indicator and the higher count-moment bound from 8. The remaining counts are bounded by [eq:flat-log-moments]. This proves vanishing of the mixed moment as the distinguished site approaches the wall. For fixed distinct spectators the pairing bound itself is uniformly bounded near the wall. Smearing first at a radius smaller than the site’s wall distance gives the same result for distributional moment limits. Once harmonicity is established below, the mean-value property identifies these smearing limits with the pointwise boundary trace. The proof also applies when the wall develops a pinch: only the actual connected wall crossing the testing scales is used. ◻

Proof. Pair each site of one parity with an incident site of the other parity by a fixed lattice translation. The paired height difference is \(Z_e\in\{-1,1\}\). Conditional on \((N,s)\) it is, up to a deterministic orientation, a fixed \(s\) spin times the fair color of the adjacent \(N\) component. Thus for two well-separated paired edges, \[|\mathbb E[Z_eZ_{e'}]|\le \mathbb P(\text{their adjacent sites belong to the same }N\text{ component}).\] A matching \(W\) circuit around the first edge separates these components. Bulk circuit trials between the lattice scale and their separation imply that, for separation at least \(r\) and fixed wall clearance, \(|\mathbb E[Z_eZ_{e'}]|\le C(\delta/r)^\eta\). This is equivalently the conditional fair-color coupling in disjoint pin-free buffers. In particular it does not infer smallness merely from the deterministic bound \(|Z_e|=1\).

For a smooth test, the variance of its averaged paired differences is bounded by \(C r^2+C(\delta/r)^\eta\): separate pairs at distance below \(r\) from all other pairs. First let \(\delta\) decrease to zero and then let \(r\) decrease to zero. Changing the test’s evaluation point within a paired cell gives an \(O(\delta)\) smooth-test error in \(L^2\) by [eq:flat-smearing-moments]. This proves convergence of the difference to zero in \(L^2\). Uniform higher moments and interpolation give convergence in every finite \(L^p\). Nodal interpolation is an average of the same local pairings with smooth weight errors, so the argument includes it. ◻

Proof. Put a cutoff \(\chi\) in a fixed square regarded as a torus, with normalized Fourier basis \(e_n\). The uniform second-moment bound for \(K_\delta(\chi e_n)\) is independent of \(n\). Hence, for \(q>1\), \[\sup_\delta\mathbb E\|\chi K_\delta\|_{H^{-q}}^2 \le C\sum_{n\in\mathbb Z^2}(1+|n|^2)^{-q}<\infty.\] For \(1<q<-s\), compactness of \(H^{-q}\hookrightarrow H^s\) gives tightness. Use a countable exhaustion of cutoffs and summable exceptional probabilities. Higher versions of [eq:flat-smearing-moments] give uniform integrability of all finite test products, proving convergence of moments. These statements also hold for a finite independent collection of chambers or for fields set to zero on their already exposed complement.

Here is a direct verification of the last assertion, independent of any pointwise choice on diagonals. Mollify each variable of a \(j\)th moment distribution at the same radius \(r\). For each fixed \(r\) the mesh limit is taken first, so eventually \(\delta\le r\). Holder and [eq:flat-scaled-moments] bound the mollified kernel by \(C_j(1+|\log r|)^{j/2}\) on compact sets, even at collisions. Away from pairwise distances at most \(2r\), separate harmonicity and radial mollifiers make it exactly the smooth unmollified kernel. Taking the mollification radius to be a fixed fraction of the smallest pair distance also bounds that kernel by a power of its logarithm. The collision neighborhood has volume \(O(r^2)\), so both integrals over it tend to zero. The mollified kernels therefore converge locally in \(L^1\) to the smooth kernel extended across the diagonals. Their distributional limit is the original moment distribution, proving the assertion. ◻

Proof. Condition on the stopped data and apply the second-moment case of 32. If the ambient diameter is at most \(R_0\), a rearrangement bound for the logarithmic kernel gives \[\mathbb E\bigl[|R_\delta(f\mathbf 1_{E_\delta})|^2\mid\mathcal E_\delta\bigr] \le C\|f\|_\infty^2 m_\delta^2 \left(1+\log^+\frac{R_0^2}{m_\delta}\right),\] with value zero at \(m_\delta=0\). To verify the discrete bound, for each fixed site sum the decreasing logarithmic kernel first over its nearest \(O(m_\delta/\delta^2)\) cells, allowing the fixed cell-area factor of either parity. A disk of radius comparable to \(\sqrt{m_\delta}\) bounds this sum by \(C m_\delta(1+\log^+(R_0^2/m_\delta))\); its central cell uses the cutoff \(|x-y|\vee\delta\). Sum once more over \(E_\delta\). The right side is a bounded function of \(m_\delta\) tending to zero at zero, so averaging proves the claim. A mean bounded by \(M\) contributes at most \(M\|f\|_\infty m_\delta\). In particular the lemma applies to shrinking neighborhoods of an explored set once their areas have been shown to vanish; it does not presume that area estimate or thinness. ◻

Proof. A harmonic function on a punctured disk has constant angular mode \(a+b\log r\) and nonconstant modes \(a_n r^n+b_n r^{-n}\), \(n\geq1\). Polylogarithmic growth forces every \(b_n\) to vanish. This proves the expansion and removability of its remainder. To obtain parameter uniformity, choose a fixed radius \(q\) avoiding all other insertions. The coefficient is the outward flux \[ L_j(y,\mathbf z) =-\frac1{2\pi}\int_{|w-y|=q} \partial_{n_w}T_j(w,y,\mathbf z)\,\,\mathrm ds_w. \tag{63}\] After writing \(w=y+q\exp(i\vartheta)\), the integrand varies smoothly on a collision-free compact set. This proves smoothness of \(L_j\). Subtract its logarithm from the boundary values on the circle and apply the Poisson formula. Interior derivative estimates give the claimed uniform bounds on \(R_j\). This obtains uniformity from a fixed circle, not from a potentially nonuniform singular limit. ◻

Proof. Here are the stopping and coloring details of the coupling. Rescale \(R\) to one and take \(\rho=r^{1/2}\); all sufficiently small \(r\) have both local supports in \(B(y_0,Cr)\subset B(y_0,\rho)\). Search inward using only \((s,W)\) for a monochromatic open \(W\) circuit in the stack of annuli between radii \(\rho\) and \(R\). The search stops on the first complete circuit and does not inspect its interior. By [prop:circuits,prop:comparison], the probability of exhausting the stack without finding a circuit is at most \(C\rho^\eta\), for some \(\eta>0\). The exponent and constants depend on the fixed model parameter and the buffered shapes, not on the flat domain.

On success reveal the entire exterior of the stopped circuit, including its transverse colors. No \(N\)-cluster crosses an open \(W\) circuit, so this revelation determines the outside color identifications without imposing an interior color constraint. By 4, the unexplored interior is a flat law, with its constant \(s\) boundary value and independent fair colors on its free \(N\)-clusters. The sign of the wall causes no change to the law of neutral height increments: reflect a spin sign if necessary.

The spectator heights are measurable in the revealed exterior. For a bounded flat piece, read each of them by summing corner increments from its prescribed wall along paths avoiding \(B(y_0,R)\). Such paths exist: any path entering this interior disk can be detoured through an annulus contained in the domain. The same detour works in all sufficiently fine approximations, and averaging uses only nearby exterior sites. For plane increments, choose the integration paths between their two spectator buffers outside the disk. Thus we have not conditioned on absolute heights before obtaining a genuine stopping cut.

Inside the circuit apply the common-cut monotone coupling from [prop:comparison,prop:circuits] to the flat law and the plane law. Between scales \(Cr\) and \(\rho\) there are order \(\log(\rho/r)\) buffered trials. Each successful common signed \(W\) circuit makes the two unexplored interiors identical in law. After this cut, couple their \((s,W)\) configurations identically and assign the same fair colors to the corresponding interior \(N\)-clusters. Failure has probability at most \(C(r/\rho)^\eta\). This construction is valid for every exposed exterior history; in each conditional coupling its second marginal is the same unconditioned plane law. Consequently that marginal is independent of the spectator observations. On a failed initial search choose the plane copy independently. If necessary, first perform the construction in finite flat exhaustions and then pass to the plane law. More explicitly, a completed \(W\) cut removes every transverse interaction, so fixing exterior colors leaves a kernel determined only by its signed wall, with no interior color pins. For each history, bracket this kernel and a plane exhaustion by the same ordered extreme boundary laws on an interior buffered region. The common-cut coupling preserves both marginals; after its cut use the identical free interior colors. Its plane marginal does not depend on the history. Thus the circuit trials use only \((s,W)\) conditioning, never a residual law with arbitrary transverse-color pins.

The coupled local increments therefore agree outside an event \(\mathcal B_{\delta,r}\) satisfying \[ \limsup_{\delta\downarrow0}\mathbb P(\mathcal B_{\delta,r}) \leq C\{\rho^\eta+(r/\rho)^\eta\} \leq C r^{\eta/2}. \tag{65}\] Agreement determines local heights up to a constant, which the exactly neutral weights remove. In particular both local smeared variables agree on the successful event.

For completeness, the unbounded factors cost only logarithms. For each fixed \(r\), the exactly neutral lattice coefficients are discretizations of fixed scaled smooth profiles, with the same bounds once \(\delta/r\) is sufficiently small. Thus [eq:flat-scaled-moments] gives, for every fixed finite \(p\), \[ \limsup_{\delta\downarrow0} \|K_\delta(\alpha_r)\|_{L^p} +\limsup_{\delta\downarrow0} \|K_\delta(\beta_r)\|_{L^p} \leq C_p(1+|\log r|)^{1/2}. \tag{66}\] The logarithmic diagonals are integrable, including their lattice cells. The corresponding plane bounds are uniform in \(r\), by the scale-invariant regularity estimate (Duminil-Copin et al. 2026, Theorem 4.5). The fixed, smoothed spectator product has bounded moments of every finite order. Hölder’s inequality with four factors, namely the two insertions, the spectator product, and the failure indicator, now bounds the discrepancy by \[C r^{\eta/8}(1+|\log r|)=o(1).\] Only original-law and plane-law moments were used. No uniform pointwise height moments or moment bounds under an artificially conditioned law are required. The mesh limit is always first, with the two continuum radii fixed. ◻

Proof. Choose two smooth zero-mass tests \(\alpha,\beta\) with disjoint supports and \[J=\iint\alpha(u)\beta(v)\log\frac1{|u-v|}\,\,\mathrm du\,\,\mathrm dv\ne0.\] For an explicit choice let \(\varrho_b\) be a smooth radial unit bump supported within distance \(1/10\) of \((b,0)\) and obtained by translating one fixed bump. Take \(\alpha=\varrho_0-\varrho_1\) and \(\beta=\varrho_3-\varrho_4\). Radial harmonic averaging gives \(J=\log(8/9)\ne0\) exactly.

Take fixed radial unit mollifiers in disjoint disks around the spectator points \(\mathbf z\). By separate harmonicity, averaging \(T_{j-2}\) in those disks gives exactly \(T_{j-2}(\mathbf z)\). The same is true of \(T_j\) when its first two points lie in a small disk around \(y_0\) disjoint from the spectator disks. Apply 38 and (61). For each fixed \(r\), the latter plane pairing is \(\kappa J\): the added constant \(\log(1/r)\) vanishes against neutral tests. Thus \[ \lim_{r\downarrow0} \iint\alpha(u)\beta(v) T_j(y_0+ru,y_0+rv,\mathbf z)\,\,\mathrm du\,\,\mathrm dv =\kappa J T_{j-2}(\mathbf z). \tag{68}\] There is no additional large-\(L\) limit in this choice of tests.

On the other hand insert 37 in the left side. The term \(L_j(y_0+rv,\mathbf z)\log(1/r)\) integrates to zero in \(u\) exactly, before any continuity estimate. The remaining logarithmic term tends to \(L_j(y_0,\mathbf z)J\), uniformly for spectator configurations in compact sets. For the remainder, subtract \(R_j(y_0;y_0+rv,\mathbf z)\) inside the \(u\) integral. The subtracted term again integrates to zero, and the uniform first-variable derivative bound gives an \(O(r)\) error. Comparison with (68) proves the formula. ◻

Proof. Fix \(x_2,\ldots,x_j\) and subtract the right side of (69). As a function of \(x_1\) the difference is harmonic away from these points. 39 cancels every logarithmic pole in its component; 37 removes the remaining punctures. Points in other components do not produce a pole. The resulting harmonic function has zero boundary trace by 33. The Dirichlet Green functions have the same zero trace by 31. The difference is bounded, since it is bounded near the boundary and its finitely many interior punctures are removable. The maximum principle therefore makes it zero on each component. This proves the recursion, including mixed moments when discrete components have separated only in the limit.

Starting with \(T_0=1\) and \(T_1=0\), the recursion gives zero odd moments and the sum over perfect pairings of products \(\kappa G_U(x_i,x_j)\) for even moments. The products are locally integrable. Since 35 excludes diagonal mass, the identity holds for the full smeared moment distributions. These moments determine the law: if a real linear combination \(Z\) of smeared fields has variance \(s^2\), then \[\mathbb EZ^{2n}=(2n-1)!!s^{2n},\qquad \mathbb EZ^{2n+1}=0,\qquad \mathbb E|Z|^n\leq s^n(2n)^{n/2}.\] The exponential series is integrable for every real argument, so the characteristic function is \(\exp(-s^2u^2/2)\). Apply this to all finite linear combinations and then to a countable determining family of tests in the local Sobolev space. The zero cross-component covariance proves independence. ◻

Proof. First consider two positive endpoints. Use the neutral tests \(\alpha,\beta\) from the proof of 39, whose logarithmic pairing is \(J=\log(8/9)\). Smooth every remote endpoint in a small fixed radial disk. Integrating the two colliding endpoints against \(\alpha_r,\beta_r\) cancels their two remote reference terms exactly. By the smearing identification, the resulting expression is the limit of the two neutral lattice insertions times a product of smoothed plane increments. These spectators have the bounded moments required by 38. That lemma and [eq:subsequential-kappa] give the limit \[\kappa J\,Q_{m-2}(d_2,\ldots,\widehat{d_i},\ldots,d_m) \qquad(r\downarrow0).\] Separate harmonicity removes the spectator mollifiers. In the uniform logarithmic expansion, the term containing \(\log(1/r)\) integrates to zero against \(\alpha\) before any limit; smoothness of the coefficient and the uniform first-variable derivative bound for the remainder leave the coefficient times \(J\). Comparing the two expressions proves the positive sign. Antisymmetry under reversal of either increment gives the negative signs. This calculation uses only smoothed spectator moments. ◻

Proof. The cases \(m=0,1,2\) have been established. For even \(m\ge4\), induct on \(m\) and subtract \[ \sum_{i=2}^m C_\kappa(d_1,d_i) Q_{m-2}(d_2,\ldots,\widehat{d_i},\ldots,d_m) \tag{71}\] from \(Q_m\). In the variable \(x_1\), 41 cancels all logarithmic poles at \(x_i\) and \(y_i\), \(i\geq2\), and its harmonic remainders remove the punctures. The regularity estimate gives continuous extension through \(x_1=y_1\), with value zero there; this is the collapse of the first increment, not an additional collision between different factors. The remainder is consequently entire harmonic. With all other endpoints fixed, the same estimate bounds its growth by a fixed power of \(1+\log(2+|x_1|)\), in particular by \(o(|x_1|)\). An entire harmonic function of sublinear growth is constant: the interior gradient bound on disks of radius \(M\), followed by \(M\to\infty\), makes its gradient zero. At \(x_1=y_1\) both \(Q_m\) and (71) vanish. The constant is therefore zero. This proves the plane Wick recursion.

The argument initially identifies the kernels where all the endpoints are distinct. The quoted equicontinuity and the increment-collapse estimate extend the identity locally uniformly to the full configuration spaces allowing the two endpoints of one increment to coincide. Thus all orders, not only covariance or smooth-test laws, have been identified on one common subsequence. ◻

Proof. The coefficient is strictly positive by 20.

Set \(\sigma^2=2\pi\kappa\). The plane Wick kernels just proved are precisely those of \(\sigma\) times a plane GFF in the normalization with singularity \(-(2\pi)^{-1}\log|x-y|\). The fixed rotation and dilation used above leave the increment cross-ratio in [eq:increment-covariance] unchanged. Their compact-uniform convergence at every order is the exact hypothesis of (Duminil-Copin et al. 2026, Theorem 8.1). That convergence criterion, whose input is the regularity estimate, supplies the full-plane subsequential scaling limit in the sense required by its Definition 2.7. We may therefore apply (Duminil-Copin et al. 2026, Theorems 4.3 and 4.4): their free-energy correspondence and explicit computation give \[\sigma^2=\frac1{\arcsin(c/2)}.\] The source uses the column slope \(I_{\rm col}/L\) and per-site free energy, as reconciled in 10.2; that appendix also records the finite-error factors before their final limits. All three cited inputs apply for \(1\leq c\leq2\); the present fixed parameter satisfies \(1<c\leq2\). This use does not invoke the narrower-range height convergence theorem of that paper. It follows that \(2\pi\kappa=1/\arcsin(c/2)\). Finally \(a^2=g(\pi/2)^2=2\pi\arcsin(c/2)\), proving (72).

The logical order is important. The spectral argument supplied only a subsequential coefficient; the local collision argument identified every Gaussian correlation with that coefficient; the convergence criterion established the requisite plane GFF limit; only now has the free energy fixed the coefficient. ◻

Proof of 30. By 31, every compact subset of \(U\) eventually lies inside the discrete open set. On a finite list of interior buffers, 35 gives joint subsequential field and moment convergence. The screened identity applies there with their fixed positive clearances. The boundary estimate uses the connected accessible wall in each chamber, and remains valid at every boundary point of the limiting components, as proved in 33. The Green boundary estimate in 31 supplies the matching zero trace. Thus the proofs of [prop:collision-coefficient,prop:gaussian-moments] apply to these components without a smooth-boundary assumption. They identify all mixed moment kernels, including those in different components, with the block Dirichlet covariance \(\kappa G_U\).

The plane coefficient on any simultaneous subsequence is \(\kappa=1/a^2\) by 43. Hence all subsequential laws agree. Tightness proves full convergence, and the moment bounds give convergence of moments. 34 supplies the asserted parity and interpolation equivalences. For a mesh-dependent number of independent chambers, the moment proof is unchanged: each retained site belongs to at most one chamber, and shared-cluster counts between different chambers are zero. In particular the constants do not accumulate with the number of chambers. The same uniform bounds, with zero extension, prove the final tightness assertion. That assertion does not identify an extra field supported on a possibly positive-area limiting exposed set; subsequent uses of thinness will address the full decomposition across that set. ◻

Proof. For a deterministic realized chamber configuration, let its flat product law supply the characteristic function in the first line of [eq:conditional-flat-characteristic]. The error from the displayed Gaussian characteristic function tends to zero uniformly over all such configurations with the stated clearance. Otherwise choose a sequence with error bounded away from zero. Compactness of the closed complements in \(\overline B_0\) gives a Hausdorff-convergent further subsequence, and the clearance persists in its limit. The deterministic theorem gives convergence of the flat characteristic function to the Gaussian one for that limit. 31 gives \(G_{U_\delta}(f_i,f_j)\to G_U(f_i,f_j)\) on the same subsequence. The two characteristic functions therefore have the same limit, a contradiction.

Condition now on the stopped data. By 4, the conditional kernel is exactly the deterministic product kernel just considered, with its constants subtracted. The uniform error proves [eq:conditional-flat-characteristic]. Multiplication by any bounded stopped-measurable variable preserves convergence to zero.

Assume the displayed joint convergence and write \(d(F)=\mathop{\mathrm{dist}}(\bigcup_i\mathop{\mathrm{supp}}f_i,F)\). Choose a bounded continuous function \(\chi\) of the clearance whose support is contained in \([d_0,\infty)\) for some \(d_0>0\), and a bounded continuous function \(v\) on \(\mathsf E\times\mathcal K(\overline B_0)\). Multiply the conditional difference by \(\chi(d(F_\delta))v(e_\delta,F_\delta)\), and use the tower property for the term involving \(R_\delta\). The estimate just proved makes the resulting expectation tend to zero. Evaluation of \(R\) on a smooth test and the clearance \(d(F)\) are continuous in the displayed state space. The connected complements containing \(\partial B_0\) form a closed subspace supporting every \(F_\delta\) and \(F\). On that subspace, 31 makes the Gaussian expression continuous where \(\chi\) is supported. Joint convergence therefore gives \[\begin{align*} &\mathbb E\left[\chi(d(F))v(e,F) \exp\left(i\sum_i u_iR(f_i)\right)\right]\\ &\qquad=\mathbb E\left[\chi(d(F))v(e,F) \exp\left(-\frac1{2a^2} \sum_{i,j}u_i u_jG_U(f_i,f_j)\right)\right]. \end{align*}\] The monotone class theorem on the Polish space \(\mathsf E\times\mathcal K(\overline B_0)\) extends this identity to bounded measurable \(v\). A countable family of cutoffs \(\chi\) whose positive sets cover \(d(F)>0\) now gives [eq:conditional-flat-limit]. A countable determining family of smooth tests then identifies the regular conditional law on the component interiors. The zero cross-component Green kernel gives their independence, including when one precursor separates into several limiting components. Increasing the list of buffers and decreasing their required clearance gives the full interior statement. Jointly retained constants can then be added. ◻

Proof. Fix the initial exact-cut data. Suppose a seed stop satisfies \(d_\delta(Q)\ge\rho\) and its revealed set comes within \(\eta\) of \(z\in Q\). Up to one lattice cell, a newly revealed cluster then meets \(B(z,O(\eta+\delta))\) and contains a current path from there to its seed vertex at distance at least \(\rho\). Thus this event is contained in the event of an \(N\)-arm from the small disk to radius \(\rho\). This inclusion holds for every terminal rule, including rules using the transverse colors.

Test this arm in the \((s,W)\) marginal of the initial flat kernel. In a stack of separated buffered annuli between these radii, every prescribed-color \(W\) circuit blocks the arm. Reveal only \((s,W)\) in successive annuli, with endpoints before edges. The conditional bound of 6 gives probability at least \(p>0\) for each blocking circuit. The arm probability is therefore at most \((1-p)^{c\log(\rho/(\eta+\delta))}\). This calculation never conditions that circuit estimate on the full terminal history. Integrating the event inclusion over \(z\in Q\) proves (75) for seed stops, including the product of initial flat chambers in the composite construction.

For a carpet of odd depth \(r\), use successive pairs of annuli with opposite prescribed \(W\) signs. Between the two circuits in each successful pair, the jump rule of 4 forces an odd rim. Nested successful pairs force distinct odd layers. Reaching the inner disk before depth \(r\) therefore implies fewer than \(r\) successes among logarithmically many one-parity trials. Their conditional lower bounds give a binomial tail; its polynomial factor is absorbed by decreasing the exponent. For the first wired layer, a \(W\) circuit blocks the boundary current’s arm, using the bulk pre-removal clause of 6. These arguments take place in the original or fresh exact-cut kernel before any future color revelations.

Condition now on the terminal data, only to estimate the remainder. The residual fields are independent flat fields by 4. For every terminal-measurable union \(E\) of cells in \(Q\), the logarithmic covariance bound gives \[ \mathbb E[|R_\delta(\phi\mathbf 1_E)|^2\mid\mathcal E_\delta] \le C\|\phi\|_\infty^2 \iint_{E\times E}\left(1+\log^+ \frac{R_0}{|z-w|\vee\delta}\right)\,\mathrm dz\,\,\mathrm dw \le C'\|\phi\|_\infty^2|E|. \tag{78}\] The final bound follows from the uniform integral of the logarithmic kernel on a bounded set; its atomic version includes the diagonal cell cutoff. Multiplying by \(\zeta_\rho(d_\delta(Q))\), averaging, and using (75) proves (76).

The revealed values and wall labels at fixed odd depth are bounded by a deterministic depth-dependent constant. At a seed stop, on the stated truncation the exact jump rule instead gives \[|M_\delta|\le B+2aM+a\quad\hbox{on }Q,\] where \(B\) bounds the initial labels. Each contributor gives either zero or one jump of size \(2a\); on its own sites the extra parity discrepancy is at most \(a\). The area estimate proves (77). The pathwise contributor bound above and the initial whole-box tail remove its truncation in probability. ◻

Proof. Use a Skorokhod representation of the joint subsequence for this proof. We first derive zero area before using any continuum neighborhood bound. Let \(d_\delta=d_\delta(Q)\) and \(d=\mathop{\mathrm{dist}}(Q,F_0\cup S)\). Their convergence follows from Hausdorff convergence. For fixed \(\eta\), every point of \(A\cap Q\) eventually lies in \(A_\delta^\eta\). Fatou’s lemma and (75) therefore give \[\mathbb E[\zeta_\rho(d)|A\cap Q|] \le \liminf_{\delta\downarrow0} \mathbb E[\zeta_\rho(d_\delta)|Q\cap A_\delta^\eta|] \le C\eta^\alpha.\] Letting \(\eta\downarrow0\) proves zero area where \(d\ge2\rho\). Exhausting positive rational \(\rho\) proves it on \(\Lambda_Q\). The same holds for \(F\cap Q\), because \(A\) and \(F\) agree in \(D\).

We next prove the conditional characteristic function across the cut. Fix a small \(\eta>0\). Choose a finite nonnegative smooth partition \((\psi_i)_{i\le n_\eta}\) equal to one in sum on \(Q\), with each \(\psi_i\) supported in a rational ball and with a fixed larger rational closed ball \(B_i\Subset D\) of diameter at most \(4\eta\). Choose a continuous \(\chi:[0,\infty)\to[0,1]\) equal to zero on \([0,1/4]\) and to one on \([1,\infty)\), and put \[ w_{i,\delta}=\chi\!\left(\frac{\mathop{\mathrm{dist}}(B_i,F_\delta)}{\eta}\right), \qquad \phi_\delta^\eta=\sum_i w_{i,\delta}\phi\psi_i. \tag{81}\] Use the corresponding \(w_i,\phi^\eta\) with \(F\) in the limit. These weights are continuous functions of the closed-set coordinate. If \(w_{i,\delta}>0\), the whole buffer \(B_i\) has clearance at least \(\eta/4\). If \(1-w_{i,\delta}>0\), every point in \(\mathop{\mathrm{supp}}(\phi\psi_i)\) is within \(5\eta\) of \(F_\delta\). Consequently \(\phi-\phi_\delta^\eta\) is bounded by \(\|\phi\|_\infty\) and supported in \(Q\cap F_\delta^{5\eta}\).

Stratify the finitely many possible active index sets \(\{i:w_{i,\delta}>0\}\). On each stratum, [eq:conditional-flat-characteristic], multiplied by \(a\) in the field normalization, applies to the fixed tests \(\phi\psi_i\). Its fixed real coefficients may be replaced by the stopped-measurable vector \((w_{i,\delta})\): approximate the compact coefficient cube by a finite rational net. The conditional second-moment bound makes the characteristic function uniformly Lipschitz in those coefficients, and \(G_{U_\delta}\le G_{B_0}\) gives the same uniform continuity for the Gaussian expression. Summing over the active strata yields \[ \mathbb E\left| \mathbb E[e^{\,iR_\delta(\phi_\delta^\eta)}\mid\mathcal E_\delta] -e^{-\frac12G_{U_\delta}(\phi_\delta^\eta,\phi_\delta^\eta)} \right|\longrightarrow0. \tag{82}\] This used only terminal-measurable coefficients and ancillaries, never information from a later interior revelation.

For indices with positive limiting weight, their buffers retain positive clearance. Green convergence in 31 applies to every pair of their fixed tests. Indices whose weights vanish are harmless by the uniform bound from \(G_{B_0}\). Hence the Gaussian exponent in (82) converges to \(G_U(\phi^\eta,\phi^\eta)\). The left test converges jointly by (79) and the finite continuous weights. Multiplying by a bounded continuous function of \(\mathbf e_\delta\) and by \(\zeta_\rho(d_\delta)\) is therefore legitimate.

There are two neighborhood errors to remove. The one-cell relation between \(F_\delta\) and \(A_\delta\), (78), and (75) imply \[\lim_{\eta\downarrow0}\limsup_{\delta\downarrow0} \mathbb E[\zeta_\rho(d_\delta) |R_\delta(\phi-\phi_\delta^\eta)|^2]=0.\] For the limiting kernel, positivity and domain monotonicity give \[ \|\phi-\phi^\eta\|_{G_U}^{\,2} :=G_U(\phi-\phi^\eta,\phi-\phi^\eta) \le C\|\phi\|_\infty^2|Q\cap F^{5\eta}|. \tag{83}\] Indeed \(0\le G_U\le G_{B_0}\), whose logarithmic singularity has a uniformly bounded integral. The right side tends to zero on \(\Lambda_Q\) by the zero-area result. Energy Cauchy–Schwarz now shows that \(G_U(\phi^\eta,\phi^\eta)\to G_U(\phi,\phi)\) there; the other energy factors are bounded by the same outer kernel.

The inequality \(|e^{ix}-e^{iy}|\le|x-y|\) removes the lattice error in the tested conditional identity. Then (83) and bounded convergence remove the continuum error. We obtain, for every bounded continuous \(Z\) of the retained state, \[\mathbb E[\zeta_\rho(d)Z(\mathbf e)e^{\,iR(\phi)}] =\mathbb E[\zeta_\rho(d)Z(\mathbf e)e^{-\frac12G_U(\phi,\phi)}].\] Bounded continuous functions determine the law on the retained metric state. A monotone-class argument gives the same equality for all bounded \(\mathscr E\)-measurable \(Z\). Exhausting positive clearances proves (80), because \(R=h-M\) in the joint limit. A countable determining family fixes one conditional version; continuity then gives every real linear combination in the finite-vector assertion, including independence across components that shared a discrete precursor.

It remains to identify \(M\). On an active buffer in (81), \(M_\delta\) is exactly its flat chamber label. The protected rational-label coordinates and their agreement along compact connecting paths show that \(M_\delta(\phi_\delta^\eta)\) converges to \(\int_U m_O\phi^\eta\). Terms of zero limiting weight vanish by tightness of the fixed \(M_\delta(\phi\psi_i)\) pairings. The joint extraction also identifies this limit with \(M(\phi^\eta)\). We spell out the bound on the limiting labels. A finite deterministic cover of \(Q\) by rational balls of radius at most \(\rho/8\) can be chosen so that every point lies in one of them. On \(d_\delta(Q)\ge\rho\), any initial chamber meeting \(Q\) contains such a ball with clearance at least \(\rho/2\) from \(F_{0,\delta}\). Thus the maximum of its initial labels is bounded by the maximum of finitely many protected initial-label coordinates and is tight. On this clearance event, a contributor bound \(N_\delta(Q)\le M\) and an initial-label bound \(B\) bound every terminal chamber label meeting \(Q\) by \(B+2aM+a\). Transfer this bound first to any finite family of protected limiting buffers, using their joint convergence and a slightly smaller clearance. The probability of its failure is at most the initial count or label tail. Exhaust the countable rational buffers; the same tail bound holds for the supremum of the limiting labels on \(Q\setminus A\). Sending \(M,B\) to infinity proves that this supremum is finite almost surely on each positive-clearance localization.

On these truncations the omitted finite contribution vanishes by (77); the omitted limiting integral vanishes by the local label bound and \(|Q\cap F^{5\eta}|\to0\). Joint convergence of the two \(M_\delta\) pairings transfers the finite omission bound to \(M(\phi)-M(\phi^\eta)\) in probability. Letting \(\eta\downarrow0\) therefore gives \[M(\phi)=\int_U m_O(z)\phi(z)\,\,\mathrm dz \quad\hbox{on }\Lambda_Q.\] The initial whole-box tail of 9 and label tightness remove the truncations in probability. This also proves local \(\mathscr E\)-measurability of \(M\). No mean or centered distribution remains on the cut in the tested region. Countable rational localizations and a separate retained state for each stop give the simultaneous assertion. ◻

Proof. Take a joint subsequence as in (79), also retaining the raw closed carpets. Compactness of closed subsets, field tightness, and the finite label sets permit this extraction. The exact cuts give residual labels \(\pm a\) for the wired layer and \(\pm2a\), relative to the parent, for the first free odd carpet. Their closed limits are boundary-connected. Every open component is represented by rational interior buffers: if a disk lay outside the closed limit but in no pointed component, its compact containment in the discrete complement would give a contradiction.

46 gives the full local-set decomposition and its bounded label mean on tests throughout the parent. It also gives zero area inside that parent. Hence the limit is a boundary-connected bounded thin local set, including across its cut. The uniqueness theorem of (Aru et al. 2019, Proposition 2) identifies the two limits as \(A_{-a,a}\) and \(A_{-2a,2a}\), since \(2a>2\lambda\) and \(4a>2\lambda\).

The original wired domain is Jordan. Its limiting holes are Jordan by the two-valued-set geometry recalled above. The first odd holes have label magnitudes \(2a>2\lambda\), so their loops avoid even a rough parent boundary and are Jordan as well. Restart in such a hole after subtracting its observed label. The conditional characteristic-function estimate is uniform on deterministic interior buffers; a countable exhaustion handles the random hole. Induction gives every fixed finite odd depth, with a single global closed carpet at that depth and pointed compact restrictions inside its holes. In the wired case, the local applications are on compact subsets of pointed wired holes, and the already identified \(W_0\) is supplied by the preceding layer. Write \(K_r\) for the connected global limit at depth \(r\). If \(V\) is a pointed Jordan hole of \(K_{r-1}\), then \(\partial V\subset K_{r-1}\subset K_r\), and \(K_r\cap\overline V\) is connected. Indeed, a separation of this compact intersection would put the connected \(\partial V\) on one side and leave the other side compactly inside \(V\). Adjoining \(K_r\setminus V\) to the first side would then separate \(K_r\). Thus the child restriction is boundary-connected. Inside each pointed parent, uniqueness identifies that restriction; outside the union of those parents the global closed set is the preceding layer. This identifies the whole global closed limit even when one precursor splits. The sets are canonical functions of the same limiting field, so every subsequential law is the stated one. ◻

Proof. The initial wall, the attached seed, and all clusters met by it form one boundary-connected revealed union. Each residual face has one sealed monochromatic \(W\) perimeter with its actual connections, by 4. Its flat label differs from that of its initial parent by an element of \(2a\mathbb Z\). Hausdorff limits preserve the connected revealed union. The planar connected-complement criterion then makes every open component simply connected. In the wired composite case \(A\) contains \(W_0\), so each open component lies in one component of \(D\setminus W_0\).

Before the limit, reflection of a centered flat law makes its mean zero. Thus on an unsearched site the exact jump rule gives \[ M_\delta(z)=b_{0,\delta}(z)+ 2a\sum_{C\ \mathrm{discovered}} \epsilon_C\mathbf 1_{\mathcal O_\delta(C)}(z), \qquad z\notin A_\delta, \tag{85}\] where \(b_{0,\delta}(z)\) is its initial chamber label and \(\epsilon_C\in\{-1,1\}\) is the cluster’s relative jump sign. The contributor bound and 45 give the finite local truncation of this formula. On a connected compact ball away from \(W_0\), the initial label is one constant. On exposed sites the parity discrepancy from an incident wall is bounded and is removed with the cut neighborhood.

46 now proves both the full mean identity and the full residual law. Its interior restriction is (84); the block Green kernel gives independence even when one discrete chamber pinches into several limiting components. The label tightness was proved in the joint extraction. For the composite wired stop, all inputs are those of the single global finite exploration, so the same argument covers such a pinch without a future choice of lattice chamber. ◻

Proof. Point each component at its first rational point and use the normalized Riemann map from the disk. The kernel theorem makes this map Borel in the pointed domain. For each \(c\in\mathcal L\cap(-p,q)\), choose one Borel version of the canonical \(A_{-p+c,q-c}\) map in the disk, apply it to the pulled-back remainder, transport its interior set, and adjoin the component boundary. This gives jointly measurable insertions in the random pointed components for the countably many fixed pairs. Enumerate the selected components by rational interior points and let \(E_N\) insert only the first \(N\) complete two-valued sets. Conditional local-set gluing in these distinct components makes \(E_N\) local. Its mean distribution is the old mean plus the zero-extended relative means of those insertions. On a new hole the value is \(c_H\) plus its relative value, hence is \(-p\) or \(q\). The old density on the inserted set contributes nothing, because a two-valued set has zero area inside its open parent (Aru and Sepúlveda 2018, Proposition 3.1). This says nothing about the area of the parent boundary. Thus \(E_N\) is thin, field-measurable, boundary-connected, and a \(K\)-BTLS for \(K=\max(p,q)\).

Here is the needed justification for passing through countably many holes. Choose an integer \(M\) with \(K\le2M\lambda\). By (Aru et al. 2019, Propositions 1 and 3), all \(E_N\) lie in the same larger canonical set \(Z=A_{-2(M+1)\lambda,2(M+1)\lambda}(f)\). The compact-interior dimension bound of (Aru and Sepúlveda 2018, Proposition 3.1) gives \(|Z\cap P|=0\) by exhaustion. Let \(E_\infty=\overline{\bigcup_N E_N}\), with closure in \(\overline P\). The increasing-local-set result (Aru et al. 2020a, Lemma 2.3(3)) applies: the sets are field-measurable, and after adjoining \(\partial P\) each has one nontrivial connected component. It makes \(E_\infty\) local and gives convergence of the conditional mean distributions.

For every smooth compactly supported \(\varphi\), thinness gives \(|f_{E_N}(\varphi)|\le K\|\varphi\|_{L^1(P)}\). Take an almost surely convergent further subsequence on a countable dense family of tests, pass this bound to the limiting distribution, and then use continuity. By \(L^1\) duality that distribution has an \(L^\infty(P)\) density bounded by \(K\). Locality identifies it with the harmonic mean off \(E_\infty\). Since \(E_\infty\cap P\subset Z\cap P\) has zero area, this is the full mean distribution, so \(E_\infty\) is thin. Every complementary component is an untouched terminal hole or a hole of one inserted set: a compact subset of one original \(H\) cannot be affected by insertions in the other components. Its value is therefore \(-p\) or \(q\). Uniqueness in (Aru et al. 2019, Proposition 2) identifies the union with \(A_{-p,q}(f)\).

This also proves the nested-interval rule for each fixed target interval and a partial local set whose labels range over a fixed deterministic countable set: after centering the incoming field, in \([L,U]\) with \(L<0<U\) and \(U-L\ge2\lambda\), a hole of value \(c\in(L,U)\) is continued by the relative set \(A_{-c-L,U-c}\), while a hole already at an endpoint is left alone. The uses below have finite label sets. This is a same-field statement, consistent with the monotonicity and nested construction in (Aru and Sepúlveda 2018, sec. 3.2). ◻

Proof. Write \(P\) for the current parent. Each retained set is a two-valued set inside a Jordan \(\mathrm{CLE}_4\) hole, with that hole’s boundary included. Its closed set is connected and compact. Distinct outer \(\mathrm{CLE}_4\) loops are disjoint, and every subsequent restart is strictly inside a complementary hole of the previous retained set. Induction gives disjointness at all finite generations; their union is countable because open components each contain a rational point. Canonical measurability follows at each restart from local-set measurability. None of these observations asserts all-generation local finiteness.

Subtract the incoming \(b\) and condition on the outer CLE data. Consider a positive \(\mathrm{CLE}_4\) hole \(V\). The field there is \(2\lambda+f_V\), and the retained split is \(A=A_{-2\lambda,2a-2\lambda}(f_V)\), with total values \(0\) and \(2a\). The first-passage set \(F=A_{-2\lambda}(f_V)\) is the increasing limit obtained by sending the upper two-valued barrier to infinity. Thus it contains \(A\), and \(F\) together with the outer CLE rim is the positive excursion inside that loop (Aru et al. 2020a, 2023).

The FPS decomposition on \(V\) reads \(2\lambda+f_V=\nu+f^F\), where \(\nu=(f_V)_F+2\lambda\,\,\mathrm dz\) is a positive measure whose relative support is all of \(F\cap V\) (Aru et al. 2023, Definition 8 and Theorem 9). The tower property for the nested sets \(A\subset F\) gives, for a nonnegative smooth test \(q\) compactly supported in \(V\), \[ \mathbb E[\nu(q)\mid V,A\text{ and their labels}] =2a\int_{\mathcal O(A)}q(z)\,\,\mathrm dz, \tag{86}\] where \(\mathcal O(A)\) is the union of its odd holes. Indeed the centered remainder after \(F\) has conditional mean zero, and thinness of \(A\) makes the conditional mean of \(2\lambda+f_V\) exactly the density \(2a\mathbf 1_{\mathcal O(A)}\). A countable family of nonnegative tests, multiplied by the event that their supports miss the odd holes, shows from positivity that \(\operatorname{supp}_V\nu\) is contained in their closure. Since that support contains \(A\cap V\), every interior neighborhood of a point of \(A\cap V\) meets an odd hole. The point itself lies outside those interiors, so their boundaries accumulate there.

The relative support statement does not include the outer rim. Condition on \(V\) and fix a conformal map from the disk onto this Jordan domain. On a closed angular subarc take a smooth nonnegative probability density \(w\) relative to \(\,\mathrm d\vartheta/(2\pi)\), and push its averages on circles of radii \(r\uparrow1\) into \(V\). Smooth radial approximations give compactly supported smooth tests \(q_r\) with the same limiting estimates. The disk Green kernel is nonnegative, and its average against uniform angular measure on both circles of radius \(r\) is \(-\log r\). Hence the Green energy of these weighted averages is at most \(\|w\|_\infty^2(-\log r)\), which tends to zero. In particular \(f_V(q_r)\to0\) in \(L^2\).

The centered remainder \(f^F\) after the FPS continuation has conditional covariance bounded above by \(G_V\). Therefore \(f^F(q_r)\to0\) in \(L^2\) as well. If an ambient neighborhood of the chosen boundary arc missed the interior FPS, its positive measure would vanish on \(q_r\) for all sufficiently large \(r\). On this event the full FPS decomposition would give \(2\lambda+f_V(q_r)=f^F(q_r)\). Both centered terms tend to zero in probability, whereas their difference on that event is the nonzero constant \(-2\lambda\). The event consequently has probability zero. A countable basis of angular arcs and ambient neighborhoods, using the Jordan boundary extension (Garnett and Marshall 2005, I, Theorem 3.1), proves \(\partial V\subset\overline{F\cap V}\). This is the boundary-average argument behind FPS nontriviality (Aru et al. 2020a, Proposition 4.6), localized at the outer rim; the closed excursion and its ambient support are also described in (Aru et al. 2023, Theorem 1 and Section 3).

By [eq:fps-odd-support], the interior FPS lies in the closure of the odd holes. The preceding inclusion thus puts the outer rim in that closure too. Since a point of \(A\) is outside every odd-hole interior, approaching it from an odd hole also approaches its boundary. This proves the claimed closure identity on the whole retained cluster.

For the upper FPS containment, expose the outer \(\mathrm{CLE}_4\) set and this positive split. Finite conditional gluing makes it a field-measurable boundary-connected BTLS with relative labels among \(\{-2\lambda,0,2\lambda,2a\}\). Fix \(R>2\lambda\). All but \(2a\) lie strictly between \(-R\) and \(2a\), since \(a>\lambda\). By 49, completing these holes to \(-R,2a\) yields \(A_{-R,2a}(f)\), containing the split. As \(R\uparrow\infty\) through a deterministic sequence, reflection of the FPS construction (Aru et al. 2020a, Remark 4.2 and Proposition 4.4) identifies the closed increasing union as the upper FPS of \(f\) at \(2a\). This last set is generally nonthin; the preceding bounded-density argument is not used as the barrier grows. Its metric approximants connect points to the parent boundary by sublevel paths, with coupled FPS convergence given by (Aru et al. 2020b, Proposition 4.7). For excursions, 55 below proves directly the one-sided same-field approximation needed for connected paths between strict interior hits. In the total field the two path levels are \(b\) and \(b+2a\). Reflection gives levels \(b\) and \(b-2a\) for a negative cluster.

For the sign assertion, let \(\xi\) be the fair outer \(\mathrm{CLE}_4\) sign. The zero-boundary remainder \(f_V\) is independent of it. The field \(\xi f_V\) has the same law and is still independent of \(\xi\). Reflection carries the negative split into the positive split, preserving its unlabelled geometry and the distinction between odd and even holes. Conditional remainders in the holes can be reflected in the same way before restarting. This proves the independent-fair-sign assertion against every finite recursive geometry cylinder. A monotone-class argument gives the assertion conditional on the entire countable unlabelled geometry. It does not assert independent labels on arbitrary two-valued-set loops.

Finally, fix the incoming \(b\), subtract it, and put \(T=A_{-2a,2a}(f)\). At a completed first-odd stage, holes labelled \(\pm2a\) are terminal and holes labelled zero are active. In every active hole insert its complete outer CLE set, then the appropriate complete split in each CLE hole. Define each layer by enumerating its holes and taking the closed limit of finite insertions. Every finite partial layer is a field-measurable boundary-connected BTLS by finite gluing. Its labels lie among \(0,\pm2\lambda,\pm2a\), hence inside \([-2a,2a]\). Its completion under 49 is the same \(T\), so the partial layer lies in \(T\). For a full layer, the increasing-local-set and bounded-density proof of that lemma applies with \(Z=T\) and \(K=2a\). It remains a BTLS contained in \(T\).

Let \(B_0=\partial P\), let \(B_n\) be the completed stage after \(n\) rounds of even continuation, and let \(B_\infty=\overline{\bigcup_n B_n}\). The same increasing-local-set result makes \(B_\infty\) local and gives convergence of the mean distributions. Their absolute values on a test \(\varphi\) are bounded by \(2a\|\varphi\|_1\), so the limiting mean has an \(L^\infty\) density bounded by \(2a\). Since \(B_\infty\subset T\) and \(|T\cap P|=0\), that density agrees with the harmonic mean as a full distribution. This proves thinness of the increasing limit.

The active region has vanishing area. Conditional on a positive CLE hole, thinness and mean zero for \(f_V\) give, for Lebesgue-a.e. \(z\), \[0=-2\lambda(1-p(z))+(2a-2\lambda)p(z), \qquad p(z)=\lambda/a,\] where \(p(z)\) is the conditional probability that the split assigns the odd value. Reflection gives the same probability in a negative hole. Fubini therefore bounds the expected area still active after \(n\) rounds by \((1-\lambda/a)^n|P|\). Monotone convergence makes the area surviving all rounds zero almost surely. Terminal labels persist, so the limiting harmonic mean takes values \(\pm2a\) almost everywhere off \(B_\infty\). Harmonic continuity and connectedness make it one of these constants throughout each open component. BTLS uniqueness gives \(B_\infty=T\). At the next odd depth the observed \(b\pm2a\) becomes the new incoming label before the argument is restarted.

At \(a=2\lambda\) the split has width \(4\lambda\) and both relative barriers remain positive. The FPS support proof (86), the sign argument, and the containments still apply, and the odd probability is \(1/2\). Thus the endpoint is included. ◻

Proof. The covariance limit is for the precise clamped form just defined. The deterministic set \(C_{B,n}\) is optional for the cable field, connected and nontrivial, and converges in Hausdorff distance to \(\overline B\). The metric-local-set convergence theorem (Aru et al. 2020b, Lemma 4.9) therefore applies under the stated domain hypotheses. It uses the ambient harmonic extensions of the revealed and residual fields. Every limit is the deterministic local-set decomposition at \(\overline B\), so the residual limit is the zero-extended GFF on \(D\setminus\overline B\). Its finite pairing vectors and their prelimit vectors are centered Gaussian. Convergence of their characteristic functions forces convergence of their covariance matrices, proving the displayed limit. This argument retains the whole zero cable trace; it makes no assertion that deleting perimeter edges commutes with harmonic extension.

Let \(\mathbf f=(f_i)_{i\le\ell}\) be a tuple of smooth tests, put \(F_n=(\widehat\Phi_n(f_i))_{i\le\ell}\) and \(F=(\Phi(f_i))_{i\le\ell}\), and let \(Q_n=\mathbb E[F_n\mid\mathcal G_n(B)]\). Linearity of the ambient extension and the cable Markov decomposition give the exact identities \[ \mathop{\mathrm{Cov}}(F_n-Q_n)=\mathsf K_n^B(\mathbf f,\mathbf f),\qquad \mathop{\mathrm{Cov}}(Q_n)=\widehat G_n^D(\mathbf f,\mathbf f) -\mathsf K_n^B(\mathbf f,\mathbf f), \tag{89}\] where \(\widehat G_n^D\) is the covariance form of \(\widehat\Phi_n\). The residual is independent of \(\mathcal G_n(B)\), hence of \(T_n\). Both matrices in (89) converge. In the continuum, conditioning on \(\mathcal G(B)\) has the corresponding killed covariance: the functions in \(H^1_0(D)\) that vanish on \(B\) are the zero extensions of \(H^1_0(D\setminus\overline B)\). Polygonal regularity makes this the usual domain Markov decomposition.

Choose a countable dense family \((f_i^B)\) of interior smooth pairings for the Gaussian Hilbert space \(\mathcal G(B)\), discarding dependent coordinates in each finite list. Every cell meeting the support of a fixed \(f_i^B\) belongs to \(B_n^\square\) for all sufficiently large \(n\). Thus its ALS face pairing is exactly \(\mathcal G_n(B)\)-measurable. Let \(Q_n^{(r)}\) be the Gaussian projection of \(F_n\) onto the first \(r\) such pairings. For fixed \(r\), convergence of the covariance matrices gives joint convergence with the field to the corresponding finite projection \(Q^{(r)}\). If \(Q=\mathbb E[F\mid\mathcal G(B)]\), density gives \(Q^{(r)}\to Q\) in \(L^2\). Orthogonality and (89) yield \[ \lim_{r\to\infty}\limsup_{n\to\infty} \mathbb E|Q_n-Q_n^{(r)}|^2 =\lim_{r\to\infty}\mathbb E|Q-Q^{(r)}|^2=0. \tag{90}\] This convergence of projection norms is the information needed for an arbitrary local test.

For a bounded Lipschitz function \(v\) of \(F_n\), Gaussian conditioning writes \(\mathbb E[v(F_n)\mid\mathcal G_n(B)]=\Psi_n(Q_n)\), where \(\Psi_n\) convolves \(v\) with the residual Gaussian law. Its covariance limit and (90) approximate this expression in \(L^1\), uniformly for fine mesh, by bounded continuous functions of finitely many interior pairings. The limit is \(\mathbb E[v(F)\mid\mathcal G(B)]\). Multiply this identity by \(T_n\) and by a bounded continuous function of finitely many interior pairings. The \(L^1\) error remains valid because \(T_n\) is bounded. Passing to the two limits proves the conditional factorization with \(T\) for every full-field Gaussian cylinder given \(\mathcal G(B)\). A monotone-class argument proves (88). The vector and harmonic mean versions follow by the same argument. ◻

Proof. For the reference field, take a joint subsequential limit of \((\widehat g_n^P,T_n)\). The reverse buffer lemma makes its conditional mean a function \(k(g^P|_B)\). We show that this selected function also governs the mixed expectations in other domains.

First translate the reference cable field by the linear cable interpolation \(u_n\) of a fixed \(u\in C_c^\infty(P)\). Write \(X_n=(\widetilde g_n^P,u_n)_{\nabla,n}\) for the Gaussian Cameron–Martin coordinate; this is the isonormal pairing, not a Dirichlet integral of a rough cable sample. The likelihood is \[ L_{n,u_n} =\exp\left\{X_n-\tfrac12\|u_n\|_{\nabla,n}^2\right\}. \tag{92}\] We need strong identification of \(X_n\), since \(T_n\) is arbitrary. Let \(f=-\Delta u/(2\pi)\), and let \(\widehat u_n\) be the same harmonic face extension of \(u_n\). The Gaussian reproducing identity and linearity of the extension give \[\mathop{\mathrm{Var}}X_n=\|u_n\|_{\nabla,n}^2,\qquad \mathop{\mathrm{Cov}}(X_n,\widehat g_n^P(f))=\int \widehat u_n f,\qquad \mathop{\mathrm{Var}}(\widehat g_n^P(f))=\widehat G_n^P(f,f).\] The three expressions tend respectively to \(\|u\|_\nabla^2,\int uf=\|u\|_\nabla^2\), and \(G_P(f,f)=\|u\|_\nabla^2\). Here the energy is a Riemann sum, \(\widehat u_n\to u\) uniformly on compact subsets, and the last limit is Green convergence. Hence \[ \mathbb E|X_n-\widehat g_n^P(-\Delta u/(2\pi))|^2\longrightarrow0. \tag{93}\] The likelihood coordinate therefore converges jointly with every bounded \(T_n\), not only marginally. For \(r>1\), \[ \mathbb E[L_{n,u_n}^{\,r}] =\exp\left\{\tfrac12r(r-1)\|u_n\|_{\nabla,n}^2\right\}. \tag{94}\] These bounds give uniform integrability on bounded energy sets. Tilting the reference joint law by (92) is exactly the law of the translated total field and the test evaluated on that translated cable field. The limiting likelihood is a function of the continuum field. Such a tilt preserves its conditional kernel \(k\), evaluated at the total restriction.

For two shifts in one reference law, the Gaussian relative entropy bound and Pinsker’s inequality give \[ \|L_u-L_v\|_{L^1} \le \min\{2,\|u-v\|_\nabla\}. \tag{95}\] The same inequality holds on the cables. Finite smooth nets in the Dirichlet norm, together with (94), therefore make the translated mixed-expectation limit uniform on compact families of smooth shifts. This is equality of limiting measures tested first on bounded continuous field functions. It does not require evaluating a possibly discontinuous \(k\) on the discrete fields.

Now take a compared ambient graph containing the identical subdivided \(\mathsf P_n\). Split at its boundary cut points and condition on the outside cables and those boundary vertex values. The Markov decomposition on \(\mathsf P_n\) is exactly \[\widetilde\Phi_n=\widetilde g_n^P+H_n,\] where \(\widetilde g_n^P\) has the reference cable law and is independent of \(H_n\). The mean \(H_n\) is discrete harmonic at the interior vertices and linear on inside cables. Choose a smooth cutoff \(\chi\) supported in \(P\) and equal to one near \(\overline B\). At vertices set \(u_n=\chi H_n\) and interpolate linearly on cables. It equals \(H_n\) on all cables needed for the pairings and tests in \(B\). The reference and ambient harmonic extensions also agree on those complete cells.

Interior discrete harmonic estimates and Green convergence make the values and scaled finite differences of \(H_n\) converge in law on the support of \(\chi\) to those of a smooth harmonic limit. They also give convergence of the cutoff energies. This is a statement about interior finite differences; the piecewise harmonic face extension need not have a global classical smooth norm. Uniformly over the domains, the covariance of \(H_n\) on a smaller buffer is bounded in quadratic-form order by the harmonic covariance obtained from the fixed outer box. Interior estimates consequently bound fixed smooth norms of the limiting harmonic parts and their discrete analogues in moments. Truncate such a bound, approximate the resulting family by a finite smooth net in a weaker norm controlling the cutoff energy, and apply (95). Removing the truncation proves the mixed-expectation limit on the restriction to \(B\). A prescribed harmonic mean is treated by the same cutoff.

For completeness, the mean premise is automatic for the constant labels used below in the regular inner class. For a constant \(t\), the full continuum ALS extension \(u_t\) equals \(t\) on \(\overline D\cap\mathcal B\), is harmonic on \(\mathcal B\setminus\overline D\), and has value zero on \(\partial\mathcal B\). The prelimit mean equals \(t\) on \(\Gamma_n\cap D_n\); its ambient extension is \(t\) times the planar probability of hitting that cable domain before \(\partial\mathcal B\). The maximum principle bounds its absolute value by \(|t|\). For \(z\in\partial D\cap\mathcal B\), fix a radius \(r>0\) below its distance to the box boundary and to a fixed interior point. For every small \(\eta>0\), connectedness of \(D\) gives a compact path in \(D\) from \(B(z,\eta)\) to outside \(B(z,r)\). Inner exhaustion supplies a connected cable path with the same property, up to a vanishing mesh error, for all sufficiently large \(n\). The planar Beurling estimate (Aru et al. 2020b, Lemma 4.1) bounds the probability of avoiding that path until leaving \(B(z,r)\) by \(C(\eta/r)^\beta\). First let \(n\) tend to infinity and then \(\eta\) to zero. The extended mean thus tends to \(t\) at every such boundary point. Away from \(\partial D\), convergence is (Aru et al. 2020b, Lemma 4.2(1)). Bounded convergence now gives convergence to \(u_t\) in \(L^2(\mathcal B)\), hence in \(H^s(\mathcal B)\), even when \(\partial D\) has positive area. On the buffer cells the mean is exactly constant, so the interior finite-difference requirement also holds.

Thus the conditional kernel on that restriction is \(k\). Applying 51 in the actual ambient domain promotes it to conditioning on the full ambient field. The restricted Gaussian laws are mutually absolutely continuous: conditional on \(H\), each is a Cameron–Martin translate of the reference restriction with a strictly positive likelihood, and integrating such translates preserves equivalence. The representative of \(k\) is therefore unambiguous under all these restrictions. The subsequence principle gives the mixed-expectation convergence in item 2 even if a general bounded test has several joint subsequential laws with the same conditional mean.

For item 3, apply that deterministic mixed-expectation conclusion for almost every fixed \(\Theta=\theta\). The full-field convergence of the centered fields and the extended means identifies the limiting conditional field marginal with the selected parent GFF law. Integrating against bounded functions of \(\Theta\) gives \((\Theta,\Phi_D^{\mathrm{aux}})\stackrel{\mathrm d}= (\Theta,\Phi_D)\). The cutoff comparison identifies the conditional mean given \((\Theta,\Phi_D^{\mathrm{aux}}|_B)\); then the reverse buffer argument, applied conditionally after centering the selected mean, identifies it given \((\Theta,\Phi_D^{\mathrm{aux}})\). Integrate the bounded mixed expectations against bounded functions of \(\Theta\), and use equality of measures and a monotone class. This proves the displayed conditional identity on the auxiliary limit; the displayed equality of marginals relates it to the separately given original field. A countable choice of polygons is handled by partitioning according to that choice. The reference subsequence is never reselected for a realized domain.

Finally use a diagonal extraction for the countable tests and thresholds. On \(C_{K,n}\) the exact identity \(T_{n,t}^{\pm}(\widetilde g_n+t)=T_{n,0}^{\pm}(\widetilde g_n)\) holds. The corresponding extensions differ by \(t\) on the complete cells near \(B\), though they need not do so outside the selected domain. Apply the two selected kernel statements to these identical joint restricted laws. Their conditional means give (91). The same mixed-expectation argument conditional on \(\Theta\) gives its random-domain version. ◻

Proof. Choose a fixed polygon \(P\) with \(\overline B\subset P\Subset D_r\) for all large \(r\). In the Markov coupling, the field in \(D\) restricted to \(D_r\) is an independent zero-boundary field in \(D_r\) plus a harmonic Gaussian \(H_r\), with covariance \(G_D-G_{D_r}\). Interior Green convergence and harmonic estimates make \(H_r\to0\) in mean square in each fixed smooth norm near \(\overline B\). Cut it off inside \(P\). The cutoff Dirichlet norm tends to zero in mean, and (95), conditional on \(H_r\), bounds the total variation distance by \(\tfrac12\mathbb E[\min\{2,\|\chi H_r\|_\nabla\}]\). This tends to zero. ◻

Proof. This is a second, explicit planar embedding convention for auxiliary fields. It preserves the entire source cable field, including its exit-edge bridges. On every fixed compact subset of \(D\), its face extension agrees with the earlier whole-cable convention for fine mesh. In particular every test on \(C_{K,n}\) is unchanged.

Let \(F_n\) be the set of vertices in \(C_{B,n}\). The full vertex covariance is the killed random-walk Green kernel on \(V_n\); the residual vertex covariance after the revelation is the killed kernel on \(V_n\setminus F_n\). Conditional on endpoint values, revealing the bridges in \(C_{B,n}\) supplies no further information about vertices outside it. All unrevealed edge bridges remain present. In particular an unrevealed edge with two pinned endpoints still contributes its independent zero-endpoint bridge.

The two killed vertex kernels converge off the diagonal to \(G_D\) and \(G_{D\setminus\overline B}\), respectively. We recall the boundary argument to specify its applicability to these full exit stubs. The active vertex sets exhaust compact subsets of the indicated open sets. Stop the walk in a compact inner domain; the invariance principle identifies its limiting interior walk and harmonic functions. The discrete Beurling estimate then bounds the probability that a walk which has entered a thin neighborhood of a killing boundary returns a fixed distance into the domain before being killed. This bound tends to zero uniformly in fine mesh. The outer complement is connected and nondegenerate; clamping \(B_n^\square\) adds the disjoint nondegenerate polygonal obstacle. Apply the estimate at the component approached by the walk. This identifies killing at the limiting boundary, and the potential-kernel representation gives Green convergence. This is the Green-convergence proof of (Aru et al. 2020b, Lemmas 4.1–4.2), applied to the killed vertex walk. Realizing its killing transition by a full cable stub does not change its vertex transition matrix.

All these kernels are bounded by the killed Green kernel of one fixed containing square. The square-lattice potential-kernel estimate gives \[ 0\le G_n(v,w)\le C\left(1+\log_+\frac{R}{|v-w|\vee\varepsilon_n}\right), \tag{97}\] with fixed \(R,C\); see also the potential-kernel references in (Aïdékon et al. 2023, Appendix B). If \(v_n\) tends to a boundary point and \(w_n\) to a distinct point, Beurling applied before reaching a fixed neighborhood of \(w_n\), followed by (97), gives \(G_n(v_n,w_n)\to0\). Thus the piecewise constant vertex kernels converge off the diagonal to the zero-extended continuum kernel even when the outer boundary has positive area. The logarithmic bound is integrable, and the integrated contribution of a shrinking neighborhood of the diagonal vanishes uniformly. Dominated convergence proves convergence of the paired vertex Green forms for smooth box tests.

We next retain, rather than discard, the cable bridges. Split the completed cable field into the piecewise linear interpolation of its vertex values and its centered edge bridges, and extend both parts harmonically into the square faces. The harmonic basis function of a vertex is nonnegative, supported on its four incident faces, and has integral \(\varepsilon_n^2\). Hence its coefficient against \(f\) differs from \(\varepsilon_n^2 f(v)\) by \(O_f(\varepsilon_n^3)\), and the total absolute coefficient error is \(O_f(\varepsilon_n)\). The containing-square diagonal Green bound therefore bounds the variance of this error by \(O_f(\varepsilon_n^2\log(1/\varepsilon_n))\). A centered edge bridge contributes through at most two faces. Its paired variance is \(O_f(\varepsilon_n^4)\), since its covariance is bounded at the fixed resistance normalization and those faces have area \(O(\varepsilon_n^2)\). There are \(O(\varepsilon_n^{-2})\) independent retained bridges, independent also of the vertex field. Consequently, for the full field and for the clamped residual, \[ \mathbb E\left|\widehat X_n(f) -\varepsilon_n^2\sum_v f(v)X_{n,v}\right|^2 \le C_f\varepsilon_n^2 \bigl(1+\log(1/\varepsilon_n)\bigr)\longrightarrow0. \tag{98}\] For the residual use its pinned vertex values in this display. Unrevealed bridges between two pinned endpoints are included in the bridge estimate. Thus no perimeter cable has been deleted or replaced. The use of the completed grid makes this estimate valid for global box tests, including boundary-adjacent faces. Together with the vertex Green convergence it proves (96).

The nonnegative face basis also carries (97) to the planar covariance, with the logarithm truncated at mesh distance. The bridge covariance is bounded and supported on pairs of adjacent faces. Integrating against the logarithmic kernel for the \(H^{-1}\) norm gives a uniform bound on the expected squared \(H^{-1}\) norm. Compact inclusion into \(H^s\), \(s<-1\), proves the asserted tightness.

Finally \(\overline P\subset D\) implies that a whole neighborhood of \(\overline P\) lies in \(V_n\) for all sufficiently fine mesh: its compact pieces have lattice connections to \(r\) with a fixed positive clearance. Subdivide the prescribed cables at \(\partial P_n^\diamond\). The reference \(\mathsf P_n\) is then literally the same graph, with the same inherited half-edge resistances and unpinned cut points in the ambient field. Conditioning on its exterior gives exactly the original reference residual.

After covariance convergence, the proof of 51 uses Gaussian projection, the cable Markov decomposition, and measurability of compact interior face pairings. All these statements hold for the present extension, so that proof applies unchanged. The comparison proof of 52 uses the same reference \(\mathsf P_n\) and harmonic estimates on fixed interior buffers, where the operators and cables agree exactly. It therefore gives the same \(k\), including the conditional-parent conclusion by integrating the bounded mixed expectations. For a constant label the added deterministic function \(u_t\) is bounded, agrees exactly at every mesh with the prescribed full continuum mean, and is identically \(t\) near the buffers. Thus it has the required full-field and interior mean convergence, without altering any local threshold test. ◻

Proof. Fix one \(\gamma\in(0,2)\). Couple the cable GFF to a critical cable loop soup \(\widetilde{\mathcal L}_n\) by the signed isomorphism (Lupu 2016). Write \(\mathcal L_n\) for the random-walk soup obtained by taking its vertex trace. Vertex occupation times are preserved by this trace construction; see also (Aru et al. 2020b, sec. 2.2). Write \(\mathcal M_{\gamma,n}\) for the normalized unsigned thick-point measure at vertices and \(\mathcal M_{\gamma,n}^{\pm}\) for its restrictions to vertices with the indicated cable-field sign. Thus \(\mathcal M_{\gamma,n} =\mathcal M_{\gamma,n}^{+}+\mathcal M_{\gamma,n}^{-}\). We use exactly the source normalization (Jego et al. 2023, (2.5), (2.10), Theorem 2.1), whose limiting expectations are \(\mathbb E\mathcal M_\gamma(\,\mathrm dz)=2\,\mathrm dz\) and \(\mathbb E\mathcal M_\gamma^+(\,\mathrm dz)=\,\mathrm dz\). All measure limits below may be read in the local weak topology on \(D\). The deterministic conformal-radius factor in this normalization is continuous, positive and bounded with bounded inverse on each compact subset of \(D\), so that changing the GMC convention there causes no boundary reweighting issue.

The cable version of (Jego et al. 2023, Corollary 8.4 and the paragraph following it), in the vertex-trace coupling, gives \[ (\widetilde{\mathcal L}_n,\mathcal M_{\gamma,n}^{+}) \longrightarrow(\widetilde{\mathcal L}, \widetilde{\mathcal M}_\gamma^{+}), \tag{99}\] where \(\widetilde{\mathcal L}\) is a Brownian loop soup and \(\widetilde{\mathcal M}_\gamma^{+}\) is its chaos restricted to independently positively marked clusters. Retain separately the projected random-walk soup coordinate \(\mathcal L_n\), whose joint limits with unsigned and cluster-restricted chaos are given by (Jego et al. 2023, Theorem 8.3). Call its limiting soup \(\mathcal L\). We will not need to identify the two limiting time parametrizations.

Their multisets of individual loop ranges do agree. Indeed, drawing each vertex-trace jump along its edge changes the range of a cable loop visiting vertices by at most \(2\varepsilon_n\) in Hausdorff distance. Erased excursions stay on edges incident to visited vertices, and loops with no vertex have diameter at most \(\varepsilon_n\). At every positive diameter cutoff which is not the diameter of a limiting loop, the limiting soups have finitely many loops. The finite point-process matching in the loop-soup topology, followed by the range map, therefore passes this trace matching to the limit; compare (Aru et al. 2020b, sec. 4.1.2). Exhausting such cutoffs gives equal multisets of individual ranges, and hence identical cluster closures. This uses equality of the limiting range families, not continuity of a cluster-formation map.

The positive and negative thick-point calculations in (Aïdékon et al. 2023, sec. 9, proof of Theorem 1.5) also give convergence of each signed measure jointly with its underlying vertex GFF to the corresponding GMC of that same GFF. Since both limiting GMCs are measurable functions of the field, this also identifies their joint limit. In the source occupation-time normalization the vertex field is \(X_n(v)=\sigma_v\sqrt{4\pi\ell_v}\), so its logarithmic covariance coefficient is one, as required by (87). The whole-cable estimate (98) identifies this vertex-field limit with the completed-grid extension in 54.

In particular, every joint limit has \(\mathcal M_\gamma^{\pm}=\operatorname{GMC}_{\pm\gamma}(\Phi)\), where the GMC notation includes the stated deterministic normalization. The reconstruction theorem (Berestycki et al. 2020, Theorem 1.1) supplies a measurable map \(R_\gamma\) such that \[\Phi=R_\gamma(\mathcal M_\gamma^{+}) \quad\hbox{almost surely}.\] The deterministic conformal-radius factor between the GMC conventions is included in \(R_\gamma\). Thus (99) and the same-field GMC limit identify the entire joint law \((\Phi,\widetilde{\mathcal L},\mathcal M_\gamma^+)\). This step uses measurability after passage to the limit, not continuity of the reconstruction map.

There is also one common unsigned measure in the two soup coordinates. The exact prelimit identity and the two same-field limits give \[\mathcal M_\gamma =\operatorname{GMC}_{\gamma}(\Phi) +\operatorname{GMC}_{-\gamma}(\Phi).\] In the just identified cable-soup coupling this is its unsigned chaos \(\widetilde{\mathcal M}_\gamma\), by (Jego et al. 2023, Theorem 1.4); in the projected-soup coordinate it is the unsigned chaos supplied by (Jego et al. 2023, Theorem 8.3). Thus the two limiting soups have both the same cluster geometry and the same unsigned measure. Equality of the latter follows from their common vertex observable and the two GMC identities, without assuming that chaos is determined by unparametrized ranges.

We next identify the cluster marks in this joint law as the canonical excursion signs. The continuum results (Aru et al. 2023, Theorems 1 and 2) describe the joint law of the closed cluster supports, their Minkowski measures, their independent symmetric signs, and their signed sum. The Minkowski construction in (Jego et al. 2023, Theorems 1.9 and 1.10) normalizes the same Euclidean neighborhood measures. Since both normalizations have finite nonzero limits, their deterministic ratio converges to one positive constant, common to every cluster. For example, compare the total masses for one cluster: the ratio can have neither zero nor infinite subsequential limits, and two distinct finite limits would give two different multiples of a nonzero limiting measure. The signed sums therefore differ by that same constant; their GFF covariance normalizations fix it. The order of summation can be changed to a common sign-independent enumeration: Hilbert-space \(L^2\) convergence and independent-sign orthogonality give \(\mathbb E\sum_i\|\mu_i\|_{H^{-q}}^2<\infty\) for a sufficiently large \(q>1\), hence conditional square summability and unconditional convergence of the signed series. Consequently the joint law just identified has precisely the canonical excursion supports and signs of \(\Phi\). Only the continuum parts of (Aru et al. 2023) are used here.

For the fixed \(\gamma\), every soup cluster \(C\) selected by a rational-point label satisfies \[ \mathcal M_\gamma(C)>0 \quad\hbox{almost surely}. \tag{100}\] Here is a useful argument that does not presume positivity of a restriction to a random thin set. Flip only the independent sign of \(C\), keeping the soup and all other signs fixed. This transformation preserves the signed-soup law. Hence the original and flipped samples both satisfy the same reconstruction identity with \(R_\gamma\), on the intersection of two full-probability events. On \(\{\mathcal M_\gamma(C)=0\}\) the positive-chaos measure is unchanged, so the reconstructed field is unchanged. Its signed-Minkowski representation, however, changes by a nonzero multiple of the nonzero positive measure of \(C\), by (Jego et al. 2023, Theorems 1.4, 1.9 and 1.10). This contradiction proves (100).

Fix a rational \(x\in D\), and let \(C_n(x)\) be the outermost ordinary random-walk loop cluster surrounding \(x\), with projected loops drawn along the grid edges. Its trace is contained in a cable sign cluster; call that sign \(\sigma_n(x)\). If no such cluster exists, use the empty set and assign sign \(+1\); its restricted measure below is then zero. When nonempty, it is a connected one-sign path witness. Retain also \[\mu_n(x)=\mathcal M_{\gamma,n}|_{C_n(x)}.\] The printed random-walk statement (Jego et al. 2023, Theorem 8.3) gives, jointly with the soup, \[C_n(x)\longrightarrow C(x),\qquad \mu_n(x)\longrightarrow\mathcal M_\gamma|_{C(x)},\] in the Hausdorff and weak-measure topologies, respectively. Take an arbitrary further joint limit including \(\sigma_n(x)\). At every mesh, \[\begin{split} \sigma_n(x)=+1&\ \Longrightarrow\ \mathcal M_{\gamma,n}^{+}\ge\mu_n(x),\\ \sigma_n(x)=-1&\ \Longrightarrow\ \mathcal M_{\gamma,n}-\mathcal M_{\gamma,n}^{+}\ge\mu_n(x). \end{split}\] These inequalities pass to the limit because the positive-measure cone is closed. In the identified continuum coupling, the positive chaos restricted to \(C(x)\) is either its entire unsigned chaos or zero, according to the canonical sign \(\tau(x)\). By (100), the limiting inherited sign must therefore equal \(\tau(x)\). No independence of the signs of different random-walk clusters is asserted: they may share a cable cluster.

The rational labels cover every outermost continuum cluster. For each fixed label, strict hits in the two open target sets persist under Hausdorff convergence, and the inherited sign has just been identified. A path in the projected cluster from its inner hit to its outer hit is a cable sign path. Since \(\overline V\subset\operatorname{int}K\), an initial portion reaches a point outside \(\overline V\) before any exit from \(K\). This portion forces the corresponding test. Applying Skorokhod representation to any further joint subsequence proves the assertion, first for each rational label and then simultaneously for all labels by countability. ◻

Proof. First work in an indexed canonical parent \(D_{\mathrm{par}}\) whose total field is \(b+f\). For a positive first-generation cluster \(C\), 50 supplies a containing positive excursion at total level \(b\) and a containing upper FPS with sublevel paths at total level \(b+2a\). For a negative cluster the containing paths are below \(b\) and above \(b-2a\). The parent is the initial domain or a canonical Jordan hole; its auxiliary approximations are the source killed graphs for the excursion test and the regular metric domains for the FPS test.

Suppose \(C\) meets both \(O\) and \(\partial O\), and choose a rational \(Q\Subset O\) whose interior it meets. In a regular inner exhaustion as above, connectedness of \(C\) makes it meet \(\partial O_{j+1}\) for every large \(j\). This gives the two strict neighborhoods for the excursion test: the second lies in \(O_{j+2}\setminus\overline O_j\). The FPS test needs only the strict inner hit, because its boundary-connected path must exit the fixed collar. On the relevant-parent event \(O\subset D_{\mathrm{par}}\), all buffers are contained in this canonical parent.

The stopped label and this parent label belong to the same affine lattice \(b_0+2a\mathbb Z\). For a positive cluster, either \(m\le b\), so the excursion forces the above-\(m\) test, or \(m\ge b+2a\), so the upper FPS forces the below-\(m\) test. For a negative cluster, either \(m\ge b\), using the below-\(m\) excursion, or \(m\le b-2a\), using the above-\(m\) FPS. These alternatives exhaust the lattice. The absolute threshold \(m\) belongs to \(a\mathbb Z\).

Calibrate both tests at every fixed threshold in \(a\mathbb Z\) in the canonical parent, conditional on its ancestor data. For the excursion, use 55 for each rational-point label and take the countable union; for the FPS use the whole-set identity. The path implications and (101) then hold simultaneously for the countable choices of parent index, rational-point label, threshold, and collar. They are identities on the original field space and survive adjoining the stop. To state the bound globally, fix an indexed candidate \(C\) and a rational \(Q\), and select the collar from the stopped state by the rule described below. Let \(\mathsf H_j\in\mathscr E\) record that the selected pointed component and its \(j\)th collar are defined with the required clearances inside \(O\). Let \(\mathsf L_j\) record that the indexed candidate and its parent \(D_{\mathrm{par}}\) exist and that \(\overline P_j\subset D_{\mathrm{par}}\). This latter event may depend on canonical data. Declare it false when \(\mathsf H_j\) fails, use the empty set for an absent candidate or component, and put \[\mathsf C_Q=\{C\cap\operatorname{int}Q\ne\varnothing,\, C\cap\partial O\ne\varnothing\}.\] The right side below is defined to be zero outside \(\mathsf H_j\). Substituting the stopped-measurable \(m\) in the simultaneous forcing identities gives the global pointwise bound \[ \mathbf 1_{\mathsf H_j\cap\mathsf L_j\cap\mathsf C_Q} \le \mathbf 1_{\mathsf H_j} \bigl(k_{j,m}^{+}(h|_{B_j})+k_{j,m}^{-}(h|_{B_j})\bigr). \tag{104}\] The direction selected by the unknown parent label \(b\) has disappeared from this inequality. In particular \(b\) need not be measurable in \(\mathscr E\).

Condition now only on \(\mathscr E\). The event \(\mathsf H_j\), the collar, and \(m\) are stopped-measurable. The component law gives \[\begin{align*} \Pr(\mathsf H_j\cap\mathsf L_j\cap\mathsf C_Q\mid\mathscr E) &\le \mathbf 1_{\mathsf H_j}\mathbb E\!\left[ k_{j,m}^{+}(m+g^O|_{B_j})+ k_{j,m}^{-}(m+g^O|_{B_j})\,\middle|\,\mathscr E\right] \\&= \mathbf 1_{\mathsf H_j}\mathbb E\!\left[ k_{j,0}^{+}(g^O|_{B_j})+ k_{j,0}^{-}(g^O|_{B_j})\,\middle|\,\mathscr E\right], \end{align*}\] where (91) is applied under that conditional Gaussian law. For each realized \(O\), (103) makes the right side tend to zero as \(j\to\infty\). On the branch where this parent is relevant, a crossing with witness \(Q\) belongs to \(\mathsf H_j\cap\mathsf L_j\) for every sufficiently large \(j\). Conditional Fatou applied to the left indicators therefore gives zero conditional probability for that relevant crossing. No conditioning on \(\mathsf L_j\) is used.

All random choices can be made within the countable families already selected. An open hit has a rational polygon \(Q\) as witness. The closed-set coordinate of the stop determines whether a rational polygon has positive clearance inside \(O\). Choose successively the first regular rational polygon containing the preceding one and the next finite list of rational compact buffers. Simple connectedness ensures such a nested exhaustion exists, and the clearance tests make the choice \(\mathscr E\)-measurable. Partition by each finite collar and threshold choice before conditioning. The random-domain clause of 52 and bounded convergence justify the displayed conditional integrals. There are countably many components and canonical clusters.

It remains to justify that the same argument reaches every recursive parent. Assume it for the ancestors of a candidate cluster. If an ancestor lies wholly in \(O\), its filled disk lies in \(O\): its outer Jordan loop is inside the simply connected \(O\), so the connected complement of \(O\) cannot enter that loop without crossing it. All descendants then stay in \(O\). Otherwise, an ancestor that does not cross \(O\) and does not lie inside it is disjoint from \(O\). Connectedness puts \(O\) in one complementary canonical hole, unless the candidate descendant is outside \(O\) already. In the relevant hole the incoming label is some \(b\in b_0+2a\mathbb Z\). The same absolute threshold argument applies there. On this branch \(O\subset D_{\mathrm{par}}\), so \(\mathsf L_j\) holds for every sufficiently large collar and the global bound applies. Calibration conditions on that parent’s canonical data, while the upper bound continues to condition only on the original retained stop. Induction proves the claim.

For a global composite wired stop, \(A\) contains the wired layer, so \(O\) lies in one pointed wired hole. Its label satisfies \(m-b_0\in2a\mathbb Z\) with \(b_0=\pm a\), and all canonical descendants have baselines in the same affine lattice. The preceding proof applies in the original field using those absolute labels. No further exploration is introduced inside the selected limiting hole. ◻

Proof. Let \(\phi_O:\mathbb D\to O\) be normalized by \(\phi_O(0)=q\) and \(\phi'_O(0)>0\). Rational polygonal inner exhaustions, normal-family compactness, and uniqueness of the normalized limit make \(\phi_O\) measurable in \((O,q)\). Push uniform angular probability measure \(\,\mathrm d\vartheta/(2\pi)\) on \(|z|=r\) forward by \(\phi_O\). Conformal invariance of the Green kernel gives covariance \(-\log\max(r,s)\) for the zero-boundary averages at radii \(r,s\). The radius-\(r\) average of \(f\) thus has mean \(m\) and variance \(-\log r\).

For \(r_j=1-2^{-j}\), approximate the angular measure by a smooth radial probability test \(\nu_j\) compactly supported in the disk, with squared Green-energy error at most \(2^{-j}\). Such choices are deterministic: the covariance \(-\log\max(r,s)\) is continuous near each positive \(r_j\), so radial smoothing converges in energy. Push \(\nu_j\) forward by \(\phi_O\) to obtain a smooth compactly supported test \(\psi_j(O,q)\). This test is measurable in the pointed domain, and evaluation \(f(\psi_j(O,q))\) is measurable in \((O,q,f)\). Its conditional mean is \(m\), and its conditional variance is at most \[2(-\log r_j)+2^{1-j}.\] The sum of these variances is finite. Chebyshev’s inequality and Borel–Cantelli therefore give \(f(\psi_j(O,q))\to m\) almost surely. Define the function in the statement by this limit when it exists, and by zero otherwise. The construction uses conformal maps only on compact subsets of the disk, so no boundary extension of \(O\) is needed. The conformal identities are recalled in (Ahlfors 1979, chap. 6, Sections 1.1 and 5.2). ◻

Proof. We first show \(i\sim j\Rightarrow i\sim_*j\). All odd holes of one retained canonical cluster have the same odd depth \(r\) and the same parent at depth \(r-1\): entering any odd hole increases the depth, and later clusters lie inside it. Let \(V_1,V_2\) be two marked odd holes of such a cluster \(C\), and suppose that their lattice suppliers are distinct along the retained subsequence. If \(V_1=V_2\), a compact path inside that hole already puts both marks in the same discrete hole for all sufficiently small mesh. We may therefore assume that \(V_1\) and \(V_2\) are distinct.

Use the following fixed connected search. Starting from the exterior, repeatedly add the lexicographically first unvisited current-lattice vertex adjacent in the underlying graph to the explored region, and reveal its entire current cluster. For the wired family start from the completely exposed wired layer; the same rule interleaves its residual chambers. The explored union remains connected to the original boundary. For the two fixed marks count the surrounding odd rims already revealed, and stop when either counter first reaches \(r\). This rule uses only the revealed history; it never asks for the identity of a future hole. If neither counter reaches \(r\), give the stop a cemetery value; on the event under consideration it is eventually active. Adjoin the retained stopped states for all these countably many fixed rules to the joint subsequence. By 48, each has the stated component laws conditional on its own limiting stopped state. The future supplier relation is not included in that state.

The two marks lie in one pointed parent at depth \(r-1\). A compact path inside that parent joining them is eventually inside the same discrete parent, by pointed convergence. This implication does not assert the converse through a pinch. The exterior seed must cross all of their shared enclosing odd rims before it can reach either \(r\)th rim. Suppose the stop discovers the rim of \(V_1\) first. The other mark has then seen exactly the common \(r-1\) odd jumps; any intervening even holes contribute zero. Its stopped value is the common parent label \(b\).

The newly revealed supplier and its sealing boundary put \(\partial V_1\) in the limiting explored set \(A\). Hence \(C\) meets \(A\). If a connected cluster meeting \(A\) entered any component of the complement, it would also meet that component’s boundary. 56, applied at the appropriate canonical generation and after subtracting the parent label, forbids this. Therefore \(C\subset A\), in particular \(\partial V_2\subset A\).

The entire discrete interior of the second odd hole is still unsearched. A seed vertex could enter it only after meeting its enclosing supplier, which would have raised the second counter to \(r\). A different current cluster inside that hole is trapped there by the supplier and its sealing rim, so it could have been discovered only from a seed vertex in the same interior. Thus the whole explored set, not only the seed, misses the second discrete hole. Every compact subset of \(V_2\), with a slightly larger compact neighborhood, eventually lies in that untouched hole. It follows that \(A\) misses \(V_2\). Since \(\partial V_2\subset A\), the domain \(V_2\) is itself a stopped complementary component.

Its stopped conditional law has flat label \(b\). The labelled odd carpet gives the same \(V_2\) the label \(b+2a\epsilon(i)\), which differs from \(b\). Both are conditional descriptions of the same restricted field on the same pointed domain, so 57 makes this impossible. The conditional laws were used at the fixed observable stop, not after conditioning on the hypothesis of distinct suppliers. This proves the inclusion at every finite odd depth, including clusters reached after any number of even restarts.

To exclude extra identifications, use the relative signs. Conditional on the unsigned lattice data \((N,s)\), the signs of different free current clusters are independent and fair, and one cluster has the same sign at all its odd holes. The validity of a marked address is also unsigned. With our zero sign and false relation at \(\dagger\), 4 therefore gives, for every fixed pair of addresses in the same current family, \[ \mathbb E[\epsilon_\delta(i)\epsilon_\delta(j)] =\mathbb P(i\sim_\delta j). \tag{105}\] The canonical fair-sign statement in 50 gives the same identity with \(\epsilon\) and \(\sim\) for each fixed pair in that same family. Joint labelled-carpet convergence passes the bounded sign product to the limit, while the retained relation bits pass the right side to \(\mathbb P(i\sim_*j)\). Hence \[\mathbb P(i\sim_*j)=\mathbb E[\epsilon(i)\epsilon(j)]=\mathbb P(i\sim j).\] The inclusion already proved makes \(\mathbf 1_{\{i\sim_*j\}}-\mathbf 1_{\{i\sim j\}}\) nonnegative. Its expectation is zero, so it vanishes almost surely. This argument uses global fixed addresses and does not require equality of discrete parent components to be continuous through a pinch. The countable intersection over all pairs proves the asserted partition equality; the relative signs are already the limits of the marked labels.

Finally choose, for each canonical cluster, one address \(i\) supplying one of its odd holes. Every other marked odd hole of that cluster has the same lattice supplier as \(i\) for all sufficiently small mesh. The pointed-boundary observation above therefore puts all its odd-hole boundaries in every Hausdorff limit of \(C_\delta(i)\). Their union is dense in the canonical cluster by 50. This proves containment. For finitely many distinct canonical clusters, the corresponding relation bits are eventually false, so their suppliers are eventually distinct. ◻

Proof. Consider any finite family of \(M\) canonical clusters of diameter greater than \(\varepsilon\). In each cluster choose two points more than \(\varepsilon\) apart. Density of odd rims gives two nearly as separated rim points, witnessed by finitely many nearby odd holes and dense interior test points. The individual holes need not have diameter comparable to \(\varepsilon\). By 58, the two witnesses in each cluster have one precursor, and the \(M\) precursors are distinct. For sufficiently small mesh each has diameter greater than \(\varepsilon/2\). This finite choice involves only finitely many odd depths and marks.

Thus, on the joint representation, the event of at least \(M\) canonical clusters of diameter greater than \(\varepsilon\) implies at least \(M\) lattice clusters of diameter greater than \(\varepsilon/2\) eventually. Finite choices from the countable canonical family and the dense test list suffice; no measurable choice from an uncountable family is needed. Fatou’s lemma and 9 make the probability of this event tend to zero as \(M\to\infty\). Countably many rational \(\varepsilon\) give the result. The same finite-witness argument can be performed in the wired descendants, with the boundary cluster counted once. This proof does not assume nested all-generation local finiteness in advance. ◻

Proof. Fix a nonnegative smooth unit-mass mollifier \(\rho\) supported in the unit disk, and put \(h_u=h*\rho_u\). For a nonnegative \(b\in C_c^\infty(B)\) set \[ X_u=u^{-1}\int b(x)\exp(i\beta h_u(x))\,\,\mathrm dx, \qquad \beta^2>2. \tag{106}\] On the compact support of \(b\), the normalized Green kernel gives \[\mathbb E\exp\bigl(i\beta(h_u(x)-h_u(y))\bigr) \leq C\left(\frac{u}{|x-y|\vee u}\right)^{\beta^2}.\] Consequently \[ \mathbb E|X_u|^2 \leq C u^{-2}\int_0^{O(1)} \left(\frac{u}{r\vee u}\right)^{\beta^2}r\,\,\mathrm dr \leq C. \tag{107}\] For a smooth Gaussian exponential cylinder \(Y=\exp(ih(\phi))\), the Gaussian characteristic function and the bounds \(\mathop{\mathrm{Var}}(h_u(x))=\log(1/u)+O(1)\) and \(|\mathop{\mathrm{Cov}}(h_u(x),h(\phi))|\leq C_\phi\) imply \[|\mathbb E[X_uY]|\leq C_\phi u^{\beta^2/2-1}\longrightarrow0.\] The linear span of these cylinders is dense in \(L^2\) of the field. Together with (107), this proves \(X_u\rightharpoonup0\) weakly in \(L^2\). It also proves weak convergence on the full probability space with extra randomness: replace any \(L^2\) test variable by its conditional expectation given \(h\).

We next bound the conditional mollified variance near \(A\). Let \(U\) be a simply connected component of \(O\setminus A\) and write \(r=|f^{-1}(w)|\), where \(f:\mathbb D\to U\) maps zero to \(z\). The conformal Green formula and the growth and quarter estimates proved or cited in 10.3 give \[G_U(z,w)=-\log r, \qquad |z-w|\leq\operatorname{cr}_U(z)\frac{r}{(1-r)^2}.\] For \(r\geq1/2\) the Green function is at most \(\log2\); for \(r<1/2\) the second inequality gives \[G_U(z,w)\leq\log2+ \log^+\frac{4\operatorname{cr}_U(z)}{|z-w|}.\] Koebe’s quarter theorem gives \(\operatorname{cr}_U(z)\leq4\mathop{\mathrm{dist}}(z,\partial U)\). If \(\mathop{\mathrm{dist}}(x,A)\leq u\) and \(z\) belongs to the mollifier at \(x\), then \(\mathop{\mathrm{dist}}(z,\partial U)\leq2u\). Different components have zero covariance, so the killed Green kernel satisfies on the mollifier supports \[ G_{O\setminus A}(z,w) \leq\log2+\log^+\frac{32u}{|z-w|}. \tag{108}\] Rescaling \(z=x+u\zeta\), \(w=x+u\omega\) and integrating the locally integrable logarithm against the two mollifiers yields \[ \mathop{\mathrm{Var}}(h_u^A(x)\mid\mathcal F_A)\leq C_\rho \quad\text{if }\mathop{\mathrm{dist}}(x,A)\leq u. \tag{109}\] This is uniform in the exploration. It does not assert a finite pointwise variance for the GFF itself.

Choose \(B_0\Subset B_1\Subset B\) and take \(b=1\) on \(B_1\). Let \(E\) be the event that \(h_A=m\) throughout \(B\) and \(A\cap\overline B_0\ne\varnothing\). This is an exploration- measurable event: obtain \(m\) by testing against one fixed smooth unit-mass function, and check equality of distributions on a countable dense test family. In the localized version replace \(E\) by \(E\cap\Lambda\). On \(E\), for small \(u\), \[ e^{-i\beta m}\mathbb E[X_u\mid\mathcal F_A] =u^{-1}\int b(x) \exp\left(-\frac{\beta^2}{2} \mathop{\mathrm{Var}}(h_u^A(x)\mid\mathcal F_A)\right)\,\,\mathrm dx. \tag{110}\] All integrands are nonnegative. Since \(A\) is connected to \(\partial O\) and meets \(B_0\), its \(u\)-neighborhood in \(B_1\) has area at least \(cu\), with deterministic \(c>0\). To verify this without assuming path connectedness, use the distance from a point in \(A\cap\overline B_0\): its image on the connected set \(A\cup\partial O\) is an interval. Choose points at radial spacings \(3u\) up to a fixed positive radius small enough that they lie in \(B_1\). Since \(B_1\Subset O\), these points cannot lie on \(\partial O\) and therefore belong to \(A\). Their disks of radius \(u/2\) are disjoint, lie in \(B_1\), and are within distance \(u\) of \(A\). There are at least \(c/u\) of them.

Using (109) in (110) therefore gives \[\mathbb E[\mathbf 1_E e^{-i\beta m}X_u]\geq c'\mathbb P(E).\] The left side tends to zero by weak \(L^2\) convergence, since \(\mathbf 1_Ee^{-i\beta m}\) is a fixed bounded random variable. Thus \(\mathbb P(E)=0\). A countable exhaustion of \(B\) proves the lemma. For a random initial component one first conditions on it and subtracts its initial constant, then uses rational balls with positive clearance. The conclusion is still a probability-zero statement, so no bound uniform over all such domains is needed. ◻

Proof. For \(x\notin C\), choose a small ball \(B_x\Subset O\setminus C\). Its closure misses \(C_n^{\eta_n}\) for all large \(n\). Membership in \(U_n\) is then constant throughout \(B_x\); otherwise the connected ball would meet its actual relative boundary. A dense test point in \(B_x\) determines this constant eventually. The indicator of \(U\) is also constant there, since \(B_x\) lies in one component of \(O\setminus C\). Thus the indicators converge at every point outside \(C\). Dominated convergence and the zero area of \(C\) prove the claim. ◻

Proof. Work first with \(Q\subset\operatorname{int}Q^+\) and a positive clearance as above, on an outcome where the list length has stabilized. Every \(C_{*,j}\) is contained in \(A_\tau\). By 48, it has zero area in \(\operatorname{int}Q^+\), and the actual relative boundary of \(\mathcal O_\delta(C_{\delta,j})\) lies within \(O(\delta)\) of \(C_{\delta,j}\).

We identify membership at a dense mark \(q\in\mathcal Q\cap (\operatorname{int}Q^+\setminus C_{*,j})\). If \(q\) belongs to the odd-hole union along infinitely many meshes, its positive distance from \(C_{*,j}\) gives a fixed disk about \(q\) inside that union on those meshes. The relevant odd depth is bounded by \(L\). Indeed every enclosing odd ancestor is a distinct cluster: after entering its odd hole, later clusters lie strictly inside that hole. The exterior-connected seed must discover each such ancestor before \(C_{\delta,j}\), and its odd-hole union contains \(q\). It is therefore another entry of the same list for \(Q^+\). After passing to a further subsequence, the depth is one fixed \(r\leq L\), and the representative of \(q\) lies in the fixed disk. Thus \(C_{\delta,j}\) supplies the marked address \((\iota,q,r)\). The stabilized equality bit makes the entry visible, and 58 identifies its canonical cluster and odd hole.

Conversely, suppose the entry is visible and \(q\notin C_{*,j}\) lies in an odd hole of \(C_j^{\rm can}\). That hole has the marked address \((\iota,q,r)\) at a finite depth. The partition equality makes its supplier equal to the list entry for all sufficiently small mesh, so \(q\) eventually belongs to \(\mathcal O_\delta(C_{\delta,j})\). These two implications show that the membership indicators converge at every dense mark off \(C_{*,j}\). In the visible case their limit is \[\mathbf 1_{\mathcal O(C_j^{\rm can})\setminus C_{*,j}},\] and in the invisible case it is zero. This also rules out oscillation between membership and nonmembership: any infinite membership subsequence supplies a fixed visible address, after which the converse gives eventual membership.

The displayed visible set is a union of components of \(\operatorname{int}Q^+\setminus C_{*,j}\). In fact 58 gives \(C_j^{\rm can}\subset C_{*,j}\), and the boundary of every canonical odd hole is contained in \(C_j^{\rm can}\); a component of the smaller complement cannot cross such a boundary. Removing \(C_{*,j}\) changes no area on this region. We may therefore apply 61 to each list entry. Its odd-hole indicators converge in \(L^1(Q)\) to the canonical odd-hole indicator, or to zero for an invisible entry. Thus even a collection of long or numerous collapsing holes has vanishing area unless it supplies an open marked witness.

The signs have stabilized and the list is finite, so the discrete signed sum in [eq:discrete-discovered-mean] converges in \(L^1(Q)\) to the sum in the statement. For a visible entry its sign is the canonical sign because it eventually supplies a fixed marked address. In the wired case the connected rectangle \(Q^+\), being disjoint from \(W_0\), lies in one pointed initial wired hole. Compact containment puts it in one discrete initial hole for fine mesh, even if that hole pinches elsewhere. Its initial label is therefore one constant \(b_0\in\{-a,a\}\) on \(Q^+\). This supplies the additional baseline term. Remove a deterministic truncation \(L\leq M\) using [eq:contributor-domination] and the initial count tail. The uniform higher logarithmic moments in 32 give uniform integrability when passing tested conditional means.

Finally, 48 identifies the full conditional mean on these tests, including across \(A_\tau\), by its neighborhood removal argument. Hence the ordinary function just obtained is \(M_\tau\), with no extra distribution on the explored set. On a connected ball avoiding \(C_j^{\rm can}\), its odd-hole indicator is constant. The baseline is also constant on a ball avoiding \(W_0\). The localized list contains only finitely many entries, regardless of the diameters of their associated canonical clusters. This proves the last assertion. ◻

Proof. Every point is on a canonical support or the initial wall. Let \(C_\delta\) be lattice free clusters with a positive-diameter Hausdorff limit \(C_*\). The limit is compact and connected. Take \(x\in C_*\setminus W_0\) and a distinct point \(y\in C_*\). Choose an axial direction \(e\) and a rational \(t\) with \(e\cdot y<t<e\cdot x\). At the fixed column stop \(\tau(e,t)\), the cluster \(C_\delta\) has been discovered for all sufficiently small mesh, because it has a point near \(y\) in an earlier column. A small rational ball about \(x\) lies ahead of the seed and has positive clearance from \(W_0\). The same clearances hold at fine mesh. Choose \(Q\subset\operatorname{int}Q^+\) around that ball and a rational lower clearance \(r\). The target cluster is an entry in the finite localized list at this already specified stop.

Let \(\mathcal U_\tau\) be the finite union of the canonical supports associated with that list’s visible entries. If \(x\notin\mathcal U_\tau\), a smaller rational ball \(B\) around \(x\) misses \(\mathcal U_\tau\), \(W_0\), and the moving seed. By 62, \(M_\tau\) is a constant distribution on \(B\). Let \(\Lambda_{Q^+}=\{\mathop{\mathrm{dist}}(Q^+,W_0\cup S)>0\}\), the stopped-measurable clearance event for this localization. In the present initial domains it is the event in [eq:stopped-full-characteristic]. That identity gives, for every real smooth test \(\varphi\) supported in \(Q^+\), \[ \mathbf 1_{\Lambda_{Q^+}} \mathbb E\!\left[e^{i(h-M_\tau)(\varphi)}\mid\mathscr E_\tau\right] =\mathbf 1_{\Lambda_{Q^+}}\exp\!\left\{-\tfrac12 G_{D\setminus A_\tau}(\varphi,\varphi)\right\}. \tag{113}\] It holds for all linear combinations of finitely many tests, with the killed GFF extended by zero across \(A_\tau\). It therefore supplies the local residual law required by 60, including the mollifiers whose supports cross the explored set. Since \(A_\tau\) is boundary-connected and the complementary components are simply connected, that lemma says \(A_\tau\cap B=\varnothing\). But \(C_\delta\) was revealed at this stop, so \(C_*\subset A_\tau\), a contradiction.

The choice of \(B\) may depend on the later canonical data. Its existence implies one of the countably many \(\mathscr E_\tau\)-measurable zero events that \(M_\tau\) is constant on a fixed rational ball and \(A_\tau\) meets it. The conditional law used in each of these events is the one at the fixed stopped state. All directions, column thresholds, balls, and localizations come from countable families, and each localized list has finitely many entries. The preceding conclusion therefore holds simultaneously for every possible subsequential \(C_*\) and every \(x\in C_*\setminus W_0\). We have proved \[ C_*\subset W_0\ \cup\!\bigcup_{C\in\mathcal C_{\rm can}}C, \tag{114}\] where \(\mathcal C_{\rm can}\) is the complete canonical free family under consideration. For wired descendants the argument used the single composite exploration of the original \(h\). A ball away from \(W_0\) lies in one pointed wired hole and has a constant initial label, even when its discrete initial chamber pinches into several limiting holes.

The sets in the cover are compact and pairwise disjoint, including \(W_0\): this follows from 50 and the fact that each retained cluster lies strictly inside its parent. They form a countable family. Their intersections with \(C_*\) are therefore a disjoint closed cover of the compact connected set \(C_*\). Sierpiński’s continuum partition theorem (Sierpiński 1918), with a complete proof in (Freiwald 2014, V, Theorem 5.10), says that at most one member of such a cover is nonempty. Consequently \(C_*\) is contained in one canonical cluster or entirely in \(W_0\).

We exclude the latter case using the actual chamber wall. Let \(\Gamma_\delta\) be the component-side open \(W\) perimeter of the initial chamber containing \(C_\delta\), with the medial resolution at pinches. In the wired case \(W_{0,\delta}\) is a current support, whereas the spatial estimate uses the attached matching wall. 10 gives \[\sup_{z\in C_\delta}\mathop{\mathrm{dist}}(z,\Gamma_\delta) \leq \sup_{z\in C_\delta}\mathop{\mathrm{dist}}(z,W_{0,\delta})+O(\delta).\] Its first-exit proof uses the resolved chamber face: a segment from an adjacent chamber point toward the revealed support first meets this chamber’s own medial rim, including at a pinch. The free case uses the polygonal boundary. If \(C_*\subset W_0\), Hausdorff convergence of \(C_\delta\) and \(W_{0,\delta}\) makes the right side tend to zero.

Choose one fixed buffered square \(Q_{\rm all}\) containing \(X\) inside a larger regular ambient box, and use the event \(\Xi_\delta(\tau_0;b,\eta)\) of 11 in this square, at the initial stop \(\tau_0\). Its connected cell region contains the initial revealed support up to \(O(\delta)\). If the limit above has diameter greater than \(2b\), then for every fixed \(\eta>0\), at all sufficiently fine meshes \(C_\delta\) has a connected passage of diameter at least \(b\) in \(Q_{\rm all}\) inside that \(\eta\)-collar. It is still unsearched at \(\tau_0\). Thus the existence of such a limit forces membership in \(\liminf_n\Xi_{\delta_n}(\tau_0;b,\eta)\) for every fixed \(\eta\). For any deterministic \(\eta_m\downarrow0\), Fatou’s inequality and the bound in [eq:spatial-frozen-shadow], integrated over the initial cut data, give \[\mathbb P\!\left[\bigcap_m\liminf_n \Xi_{\delta_n}(\tau_0;b,\eta_m)\right] \leq \inf_m\limsup_{\delta\downarrow0} \omega_{Q_{\rm all},b}(\eta_m,\delta)=0.\] The collar width is fixed before each mesh limit; its possible dependence on the future Hausdorff limit is used only for the eventual inclusion. Countably many rational \(b>0\) exclude every positive-diameter limit contained in \(W_0\). We conclude that \(C_*\) is contained in one canonical cluster.

Visible precursors. For the visible precursor of a canonical cluster \(C\), 58 puts \(C\) in every Hausdorff limit. The preceding containment puts that limit in a single canonical cluster. Pairwise disjointness forces it to be \(C\) itself. Compactness of \(\mathscr H\) then shows that the entire visible precursor sequence converges to \(C\). In particular every canonical cluster is obtained.

Invisible precursors. Return to any lattice sequence with positive-diameter limit \(C_*\subset C\). Choose an axial direction \(e\) in which \[w=\max_{z\in C_*}e\cdot z-\min_{z\in C_*}e\cdot z>0,\] and fix a rational \(b\in(0,w/4)\). This sequence is eventually among the \((e,b)\) records. Their length has stabilized to a finite number, so one fixed rank \(k\) supplies it along infinitely many meshes. The full sequence \(\widehat C_{\delta,k}^{e,b}\) already has a Hausdorff limit, which must equal \(C_*\) on that subsequence. Replace the possibly alternating original sequence by this fixed recorded sequence. This replacement classifies its support limit; it makes no assertion that visibility transfers from the original choices.

If this record is visible, 58 puts its associated canonical cluster inside \(C_*\subset C\). Pairwise disjointness makes that cluster \(C\), and the stabilized supplier bits identify the record with the visible precursor of \(C\) for all fine meshes. It was already treated. Suppose therefore that the record is invisible. Write \(C_\delta^{\rm vis}\) for the visible precursor of \(C\). It too is eventually a record because it converges to \(C\) and \(\mathop{\mathrm{diam}}C\ge w>4b\). Among the finitely many stabilized ranks, exactly one has its supplier bit equal to one for the fixed chosen address of \(C\). Thus \(C_\delta^{\rm vis}\) has one fixed rank \(j\), and \(j\ne k\). The two ranks give one eventual discovery order. These complete-list coordinates remain outside the conditioning sigma-field of every fixed recorded stop. Smaller discoveries remain in each explored set; their local contributions are controlled by [eq:contributor-domination].

Suppose first that the visible precursor is discovered earlier, at the fixed recorded stop \(\tau\). Its limit is \(C\). For each fixed \(\eta>0\), the convergences \(\widehat C_{\delta,k}^{e,b}\to C_*\subset C\) and \(C_\delta^{\rm vis}\to C\) imply that eventually the later cluster lies inside the \(\eta\)-neighborhood of the frozen cell region at \(\tau\), using the \(O(\delta)\) displacement from \(A_{\delta,\tau}\). It is still unsearched at \(\tau\) and has a passage of diameter at least \(b\) inside \(Q_{\rm all}\). Let \[\mathcal X_\delta(b,\eta)= \{\exists\,0\le i\le D_\delta(e,b): \Xi_\delta(\tau_i;b,\eta)\}\] for the fixed \((e,b)\) sweep in that square. The event under consideration is contained in \(\liminf_n\mathcal X_{\delta_n}(b,\eta)\) for every fixed \(\eta\). For every integer \(K\), [eq:spatial-indexed-shadow] and Fatou’s inequality yield \[\mathbb P\!\left[\bigcap_m\liminf_n \mathcal X_{\delta_n}(b,\eta_m)\right] \le Ce^{-cK}+(K+1)\inf_m\limsup_{\delta\downarrow0} \omega_{Q_{\rm all},b}(\eta_m,\delta) =Ce^{-cK}.\] Letting \(K\to\infty\) gives probability zero. The spatial bound is applied to the existential future passage under its allowed history \(\mathcal H_\tau\), for finitely many fixed indices before the count tail is removed. Later visibility and limiting containment only give the eventual inclusion in the fixed-\(\eta\) events; they are not added to that conditioning.

Suppose instead that the invisible cluster is discovered first, and stop immediately after that recorded discovery. Let \(t_\delta\) be the projection of its discovery column. Every earlier column has already been visited, so the first occupied column of this cluster satisfies \[\left|t_\delta- \min_{z\in\widehat C_{\delta,k}^{e,b}}e\cdot z\right|=O(\delta).\] Hausdorff convergence preserves the minimum and maximum of the projection. Choose a point \(x\in C_*\) whose projection is more than \(w/2\) above the minimum. A rational ball about \(x\) lies a fixed positive distance ahead of the moving seed for all fine meshes. It also has positive clearance from \(W_0\), since \(C_*\subset C\) and \(C\) is compactly inside its initial parent. The stopped explored set meets that ball.

Use its localized contributor list at this fixed recorded stop. The newly discovered record meets the ball, so it is the last entry of that list and has the fixed index \(L\) after the length stabilizes. That entry is invisible: a stabilized marked supplier bit equal to one would also make the record visible. It has zero mean contribution by 62. Every earlier relevant contributor, whether its support meets the ball or its odd hole surrounds a point there, is included in the same finite list. Its associated canonical support cannot be \(C\): by 58 that would make it the already chosen visible precursor of \(C\), contrary to the discovery order. The earlier visible supports are therefore a finite family of compact sets disjoint from \(C\). Shrink to a rational ball around \(x\) missing this union. Its baseline and all remaining odd-hole indicators are constant, so \(M_\tau\) is constant there. The full residual law [eq:full-residual-use] applies at this stop, and 60 contradicts the hit of the explored set. As before, the later choice of a rational ball only implies a stopped-data zero event and adds no conditioning. This excludes the second order and hence every distinct invisible record with a positive-diameter limit. Since every positive limit of an arbitrary lattice sequence was also the limit of a recurrent fixed record, the classification is exhaustive. There is only one visible rank for a given canonical cluster. A sequence distinct from that precursor along infinitely many meshes has a recurrent rank \(k\ne j\) on those meshes, by finiteness of the record list. If that rank were visible, its limiting support inside \(C\) would make it the same chosen supplier and hence rank \(j\) eventually. Thus it is an invisible recurrent rank, which has just been excluded.

The collection topology. For clarity, outer Hausdorff convergence has the following direct link with the individual limits just classified. If \(C\in F_*^\iota\) has positive diameter, then \(\mathop{\mathrm{dist}}_{\mathscr H}(C,Z)>0\). A nearest member of \(F(\mathcal C_\delta^\iota)\) approaching \(C\) therefore cannot be a singleton for fine mesh; it is an actual lattice cluster. Conversely, every positive-diameter Hausdorff limit of lattice members belongs to \(F_*^\iota\), because these encoded families converge and \(F_*^\iota\) is closed. The fixed sweeps and their finite localized lists made the preceding classification simultaneous, so it applies to these members even when they are selected from the outer limit.

It follows that the positive members of \(F_*^{\mathrm f}\) are exactly \(\mathcal C^{\rm f}\). For \(F_*^{\mathrm w}\), a selected lattice member has a subsequence of one recurrent type: either the distinguished boundary cluster, whose limit is \(W_0=C_\partial\), or a free descendant, which has just been classified. The visible precursors give the reverse inclusion in both cases. Thus the two encoded limits equal those in the statement. 2 converts this outer Hausdorff convergence to the stated matching topology. The exclusion of duplicate invisible precursors is a stronger conclusion retained in the auxiliary lists; the set-valued metric itself forgets multiplicity. ◻

Proof of 3. By 30, \(K_\delta\) converges to \(h/a\) in every local Sobolev space \(H^s_{\rm loc}\), \(s<-1\). The arguments above can be performed on any simultaneous subsequence of the field, the wired current, and the free dual current. By 47, the distinguished wired boundary cluster converges in ordinary Hausdorff distance to \(A_{-a,a}(h)\). Its remaining pointed domains have the conditional zero-boundary GFF remainders of that same field.

Apply 63 to the free dual family in the original domain and to the free primal descendants in the composite wired exploration. The positive outer \(\mathrm{CLE}_4\) label is \(2\lambda\), and its retained split has absolute values \(0,2a\); the negative one is its reflection. The identified recursion is therefore exactly 1, with its sign-sensitive choice of split, and with every generation retained. Both families have been identified as canonical functions of the same limiting \(h\).

For completeness, these are measurable collection-valued functions. Enumerate the recursive clusters by generation and by the first mark of \(\mathcal Q\) in each generating hole, using a cemetery value for missing entries. The cluster coordinates in this countable enumeration are measurable by 50. Append \(C_\partial\) as one additional slot for the wired family. 59 makes their singleton-augmented families closed. For every open set \(\mathcal V\subset\mathscr H\), the event that \(F(\mathcal C^\iota)\) meets \(\mathcal V\) is the union of the corresponding countably many cluster hit events, together with the deterministic alternative \(Z\cap\mathcal V\ne\varnothing\). These hyperspace hit events prove measurability of both encoded factors. The boundary marker \(A_{-a,a}(h)\) is measurable in the same field.

The height and collection tightness give simultaneous joint subsequences, and every one has the pushforward of the GFF law by this single measurable construction. Hence the full mesh limit exists in the product topology, including the distinguished boundary set. Finally \(H_\delta=K_\delta/2\) and \(2a=\pi\sqrt g\), giving precisely \(h/(\pi\sqrt g)\). The parameter inequalities used throughout include \(a=2\lambda\), so the Potts endpoint is part of the conclusion. ◻