Proof. Apply the fixed-\(q\) FK clause of OpenAI (2026, Theorem 1.1) and forget the degree measure from its Gromov–Hausdorff–Prokhorov conclusion. The finite law, all-edge distance, ordinary unit-area sphere law and fixed metric normalization are exactly those above. ◻

Proof. A primal vertex \(v\) is incident to \(2\deg(v)\) flag occurrences; a loop contributes twice to \(\deg(v)\) and all four of its occurrences are counted. Thus its mass is \(2\deg(v)/(4n)\), proving (5). There are \(2n\) oriented edge occurrences, and exactly \(\deg(v)\) have tail \(v\). Uniform re-rooting preserves the weight \(q^{\ell/2}\); equivalently, count rooted occurrences before quotienting by the automorphism group. This also proves the assertion in the presence of automorphisms. Couple \(x\sim m_n\) to \(\pi_n(x)\). Their \(H_n\) images lie in the same flag image, so their distance is at most \(\max_T\mathop{\mathrm{diam}}H_n(T)\). Testing against uniformly continuous functions on the compact sphere proves the last assertion. ◻

Proof. The inventory bijection is the rooted map bijection of (Sheffield 2015, secs. 4.1–4.2). The conversion from a spanning subgraph to the auxiliary spanning tree is the construction of Bernardi (2008, sec. 4), as explained in Sheffield (2015, sec. 4.1). Its elementary sewing objects are incidence occurrences, so loops, bridges, and repeated incidences are included. A matched burger/order pair forms the two exploration triangles in one quadrangle. The \(L\)-pairs give the primal spanning-tree edges; the \(R\)-pairs give the dual spanning-tree edges, whose quadrangles carry the remaining primal edges. Resolving a flexible order selects the auxiliary tree/cotree diagonal. Restoring its FK diagonal changes the decoration, not the quadrangle or the underlying all-edge primal map.

A fully matched word has \(n\) burgers and \(n\) orders. Its unconditioned probability is \[\left(\frac{1-b}{16}\right)^n \left(\frac{2b}{1-b}\right)^{N_{\mathsf F}}.\] The bijection’s loop-opening operation gives one initial interface and one additional interface for each flexible order. Since \(2b/(1-b)=\sqrt q\), the probability is a constant depending only on \(n,q\) times \(q^{\#\Gamma/2}\), which is the specified finite map weight. The rooted bijection is bijective, rather than a quotient weighted by automorphism counts. Uniform rerooting at an incidence preserves the law because every map under consideration has the same number of root incidences. These observations also show why deleting cotree-associated primal edges would change the metric in the theorem. ◻

Proof. After resolution this is the ordinary tree/cotree contour exploration in (Sheffield 2015, sec. 4.1, Figures 6–7). A burger creates an edge to a child in its tree and its order returns along that edge. Letters of the other type hold this tree position fixed. The depth of the common ancestor of the positions at \(k,l\) is the minimum contour height between them, proving the two distance identities. The exploration crosses each tree-edge side and each quadrangle side occurrence, which proves coverage of the corners, including repeated occurrences. A letter triangle is bounded by the two successive exploration sides and the created or traversed tree edge. Pairing it with its matching letter triangle gives exactly the associated quadrangle. Subdivision into flags introduces an edge midpoint but no additional incidence choice. This proves the last assertions without any bound on vertex or face degrees. ◻

Proof. The bilateral assertion is (Sheffield 2015, Theorem 2.5). In the original units the covariance rates of the two net burger counts are \((1-b)/2,(1-b)/2,b/2\); multiplying heights by \(\sqrt{2/(1-b)}\) gives the stated convention. The conditioned assertion is (Gwynne and Sun 2015, Theorem 1.8), followed by the deterministic change from duration two to duration one. The unidentified-order estimate is (Sheffield 2015, Remark 3.17); see also the reduced-word estimates in (Gwynne, Mao, et al. 2019). Each changed resolution changes a contour coordinate by at most one, so their total number bounds the supremum error. This last assertion concerns a raw unconditioned block. The transfer of discrete observations to conditioned extractions will be proved in Section 6. ◻

Proof. For the cone, the surface and contour identification are the mating-of-trees and welding theorems (Duplantier et al. 2021, Theorems 1.9 and 1.11). For the ordinary area-one sphere, (Miller and Sheffield 2019, Theorem 1.1) supplies the Brownian-loop law and determination of the surface explored by an independent unclocked whole-plane space-filling SLE. In our range \(\gamma^2\in(2,4)\) the covariance is the one in these theorems. The marked-point sampling rule is (Duplantier et al. 2021, Proposition A.13). We use the common deterministic boundary-length convention specified above, with area-clock duration one.

Call a scalar chord strict if the contour is strictly above its endpoint height between the endpoints. The finite-chain assertion uses a further geometric part of the DMS construction. There are finitely many visits to each spatial point; ordering their left and right flow-line sectors links consecutive visits by scalar chords. This is the smallest generated equivalence relation in (Duplantier et al. 2021, Lemmas 8.9, 8.13–8.15 and Theorem 8.18), without taking its closure. A non-strict scalar chord can be split at its possible intermediate minimum visit into adjacent strict chords; scalar fibers have at most three visits. The same finite-visit and adjacent-sector argument applies to the independent whole-plane exploration of the sphere away from its exploration root: changing to quantum-area time preserves the visits, their order and their frontier incidences. At the root one additionally identifies \(0\) and \(1\). Thus the sphere statement combines its Brownian-loop realization with this geometric argument; it does not invoke a separately stated sphere finite-chain theorem. The quotient is topological because the resulting continuous bijection from the compact time quotient to the sphere is a homeomorphism. Running-minimum frontier coordinates are the quantum lengths supplied by the same welding construction. ◻

Proof. This fixed-area field law originates in the sphere construction of David, Kupiainen, Rhodes and Vargas (David et al. 2016, sec. 3.3, Proposition 3.8 and Section 3.4). We use its form in Borga et al. (2026, Definition 2.2); Aru et al. (2017, Theorem 1.1 and Remark 1.2) identify it with the ordinary DMS sphere. The mark resampling statement and local comparison, including the \(\gamma\)-insertion, are (Borga et al. 2026, Lemmas 2.3–2.4). Subtracting the constant in (9) gives area one by the identity \(\mu_{h+c}=e^{\gamma c}\mu_h\). ◻

Proof. The displayed inequality follows by bounding the two endpoint errors and the error of the minimum. A vertex \(u\) on a tree geodesic from \(a\) to \(b\) satisfies \(d(a,u)\le d(a,b)\). The exact contour-tree identity therefore bounds all the asserted distances by \(d_{c_n}(s_n,t_n)\), up to one rescaled edge if a cut lies inside a step. This tends to zero. If the assertion about \(\eta\) were false, choose a representative \(u_n\) violating it and then a convergent subsequence of \(u_n\). Its limit \(u\) satisfies \(d_c(s,u)=0\), so \(\eta(u)=\eta(s)\) by the scalar chord rule, contradicting continuity of \(\eta\).

Consecutive faces of a dual-tree path share the primal edge crossed by that tree edge. Their entire boundary walks therefore have connected union, including when a boundary edge or vertex is visited repeatedly. A corner of any of these faces represents its dual-tree vertex, so the same distance bound applies to all its representatives. This proves the last assertion. No bound on a face degree is involved. ◻

Proof. First choose one representative for every primal vertex and every dual vertex. Representatives of a given primal vertex have zero \(L_n\)-tree distance; representatives of a given dual vertex have zero \(R_n\)-tree distance. Uniform contour convergence, compactness of the time interval, and the chord rule show that the maximum disagreement between their \(\eta\)-images tends to zero. This is a uniform statement: a failing sequence of vertices and two of their occurrences would have convergent time representatives and would contradict the limiting chord rule.

At a tour cut the primal and dual corner states have that same time representative. In one quadrangle its four corner states are linked successively by equality of a primal or a dual vertex, together with the two consecutive-cut errors of its exploration triangles. The preceding uniform assertion and continuity of \(\eta\) show that all these representatives agree up to a uniform error tending to zero. It follows that the chosen representatives of the primal vertex and face center of every flag agree to this accuracy. Assign its edge midpoint the image of one fixed corner occurrence of that edge. The preceding quadrangle argument gives the same conclusion for this assignment.

Identify \(S\) with the unit sphere in \(\mathbb R^3\). On every flag send a point with barycentric coordinates \((a_1,a_2,a_3)\) to \[\frac{a_1 y_1+a_2 y_2+a_3 y_3} {|a_1 y_1+a_2 y_2+a_3 y_3|},\] where \(y_1,y_2,y_3\) are the assigned images of its vertices. For all large \(n\) the three images lie in a common ball of radius less than \(\pi/4\), so the denominator is nonzero. These formulas agree on glued sides and give a continuous map on the flag quotient. Their images stay in the same small spherical ball. This proves (i) and the flag assertion in (ii). Every flag in a primal or dual star shares the image of its central vertex, so its entire star has diameter at most twice the maximum flag diameter. The assertions for edges and faces follow as well. Finally the tour meshes become dense in \([0,1]\), \(\eta\) is onto, and every tour position has a primal corner vertex with the stated small error. This proves (iii). ◻

Proof. Choose two successive compact neighborhoods of \(K\) in \(U\). The inverse image of the smaller closed neighborhood is compact and lies inside the open inverse image of the larger one. Cover it by finitely many intervals whose closures lie in that open inverse image. This proves (11). If a sequence of omitted tour representatives belonged to stars of flags meeting \(K\), Proposition 12 would force their \(\eta\)-images into an arbitrarily small neighborhood of \(K\). They would eventually lie in \(\eta^{-1}(\overline{K'})\subset I\), a contradiction. Avoiding \(p_*\) keeps the inverse images away from \(0,1\). In the plane, transience of \(\eta\) makes the inverse image of every compact set compact; the same finite-cover proof then applies. ◻

Proof. First suppose that representatives \(s_n,t_n\) of \(u_n,v_n\) converge to \(s,t\) with \(s\sim_L t\). The primal-tree path joins these vertices and has image tending to the single point \(\eta(s)\) by Lemma 11 and Proposition 12. If instead \(s\sim_R t\), join the incident dual vertices by their dual-tree path. The union of the primal boundaries of its faces is connected, contains both primal corner vertices, and has the same vanishing image diameter. A walk in this connected subgraph supplies the required primal connection.

For arbitrary \(s,t\) with \(\eta(s)=\eta(t)\), choose a finite chain of \(L\)- and \(R\)-chords joining them, using Theorem 9. For each intermediate time choose its nearest discrete cut and its primal corner vertex. The preceding construction for each link gives finitely many walks whose union tends to \(\eta(s)\); concatenate them.

If the uniform assertion failed, there would be \(n_j\to\infty\), \(\delta_j\downarrow0\), points \(x_j\), and two prescribed vertices with images in \(B(x_j,\delta_j)\) admitting no walk in \(B(x_j,\epsilon)\). Choose time representatives and pass to convergent subsequences \(x_j\to x\), \(s_j\to s\), \(t_j\to t\). Proposition 12 gives \(\eta(s)=\eta(t)=x\). The finite-chain construction just proved produces the forbidden walks. For an arbitrary point of a flag, first join it inside that flag to its primal vertex. The extra pieces have vanishing projected diameter by Proposition 12. ◻

Proof. Apply Lemma 14 with a spatial error smaller than the clearance of the chosen finite requirements. Uniform continuity gives a finite time partition for each \(c_i\) on whose pieces its oscillation is smaller than the resulting \(\delta/3\). Choose primal representatives near the partition points using Proposition 12(iii), retaining the prescribed endpoints. Join successive representatives by the lemma. Subdivide at the intermediate ports before making these choices. Concatenation proves the ordered assertion, and the same construction works for all the finitely many paths at once. ◻

Proof. Take pairwise disjoint small closed junction disks about the vertices of \(G\). In each disk its three incident edges are three disjoint stubs except for their common endpoint. Choose three further intermediate points on the stubs, and narrow disjoint ribbons along the remaining edge portions. The ribbons of different edges are disjoint outside the junction disks. Inside a junction disk, choose the three ribbon ends disjoint, so each meets only its own stub neighborhood. Make all these neighborhoods small enough for the desired error.

Assign one primal lift to every graph vertex. By Corollary 15, lift its three initial stubs from that common lift to the three intermediate points. Their union is a connected branch set \(B_v\) within the chosen junction neighborhood. The branch sets belonging to distinct vertices are disjoint, since their projected neighborhoods are disjoint. Lift each remaining edge portion between the corresponding stub endpoints. The error may be chosen so small that different connecting walks can intersect only within branch sets at a common endpoint, and no connecting walk meets any other branch set. In particular, near a junction each connector meets only the end of its own stub; all such contacts occur before the prescribed marks delimiting the middle of that connector.

For an edge \(vw\), take the portion from its last visit to \(B_v\) before its first subsequent visit to \(B_w\). Erase loops from that finite walk. The resulting simple paths have interiors disjoint from every branch set and from each other. At a branch set retain a minimal finite tree joining its three attachment points. After the three outgoing paths are attached, this tree has exactly one vertex of degree three and every other vertex of degree two; this also covers coincident attachment points and the case where one attachment lies between the other two. Removing unused branches therefore gives an actual subdivision of \(G\). The use of three attachments is the reason for choosing a cubic skeleton.

Here is the order control needed to define \(h_n\). Give each edge ribbon a continuous longitudinal coordinate, equal to its edge parameter on the central arc. In the path-lifting construction use successive coordinate intervals of length less than \(\sigma\), and choose the spatial errors so that each joining piece stays within \(\sigma\) of its prescribed interval. If a loop-erasure removes a portion between two occurrences of one vertex, the two original parameters differ by at most \(3\sigma\): their common projected point has a single longitudinal coordinate within the required error of both. Truncation at the branch sets changes only the initial or final stub. Assign increasing central-arc parameters to the surviving vertices, and interpolate strictly increasingly along the surviving edges. The skipped parameter intervals have oscillation tending to zero as \(\sigma\downarrow0\). Within a junction neighborhood map the three remaining tree arms to the three initial edge pieces. This gives a graph homeomorphism whose error is bounded by the ribbon widths, junction diameters, and those vanishing oscillations. Choose them in that order to make the total error smaller than \(\epsilon\).

Finally a subdivision induces the same embedding of the suppressed graph. Whitney’s theorem says that two embeddings of a finite simple \(3\)-connected planar graph in the sphere are equivalent up to a sphere homeomorphism (Whitney 1932). Its hypotheses hold for \(G\), and hence the identified facial cycles agree, with at most one global orientation reversal. ◻

Proof. Fix a small \(\epsilon>0\). Triangulate the reference sphere with simplicial triangles of diameter less than \(\epsilon/10\), and take its embedded dual graph \(G\). Draw dual edges through adjacent triangle interiors. Its faces lie in stars of the original vertices, so their diameters are less than \(\epsilon\). This graph is cubic and \(3\)-connected. For the latter assertion, delete any two dual vertices, or equivalently the corresponding two closed triangles. Their union has connected complement in the sphere, whether the triangles are disjoint, meet at a vertex, or share an edge. A path in that complement between two remaining triangle interiors can be perturbed away from triangulation vertices; the triangles it successively crosses give a dual path avoiding the two deleted vertices. Thus deleting fewer than three dual vertices does not disconnect the dual graph. The simplicial condition also ensures that the dual has neither loops nor multiple edges.

Apply Lemma 16 with an error much smaller than \(\epsilon\). Denote by \(F_n\) the face of \(G_n\) corresponding to a face \(F\) of \(G\). Choose \(z_F\) so that \(\overline F\subset B(z_F,\epsilon)\). We claim that, for all large \(n\), \[ P_n(\overline{F_n})\subset B(z_F,3\epsilon). \tag{14}\] The boundaries already satisfy this assertion by the graph homeomorphism estimate. If the claim fails in the interiors, pass to a sequence \(x_n\in F_n\) with \(P_n(x_n)\to y\notin B(z_F,3\epsilon)\). Choose a vertex \(v\) of \(G\) outside \(B(z_F,3\epsilon)\); such a vertex exists for sufficiently small \(\epsilon\), because the dual graph has small faces and covers the sphere. The complement of \(\overline B(z_F,2\epsilon)\) is connected, so join \(y\) to \(v\) by a path with clearance from \(B(z_F,3\epsilon/2)\). Lift this path from the prescribed point \(x_n\) to the junction lift of \(v\), using Corollary 15. Its projection avoids \(B(z_F,3\epsilon/2)\) for all large \(n\), whereas the projection of \(\partial F_n\) is contained in that ball. The lifted path therefore avoids \(\partial F_n\). Its last point is outside \(\overline{F_n}\), since \(v\) is not incident to the corresponding face. This contradicts the Jordan separation of \(\partial F_n\). There are finitely many faces, so (14) holds for all of them simultaneously.

Each face of a \(3\)-connected sphere embedding is a Jordan disk. The graph homeomorphism \(h_n\) therefore extends across every face to a homeomorphism \(H_{n,\epsilon}:X_n\to S\). For completeness, a homeomorphism between the boundary circles of two closed disks extends by identifying each disk with the standard closed disk and extending the resulting circle homeomorphism radially. The Jordan–Schönflies theorem gives these disk identifications; here all boundaries are finite polygonal curves in triangulated surfaces. The facial correspondence makes the extensions agree on common edges. For \(x\in\overline{F_n}\), both \(H_{n,\epsilon}(x)\in\overline F\) and \(P_n(x)\in B(z_F,3\epsilon)\), and hence \[\sup_{x\in X_n}\rho(H_{n,\epsilon}(x),P_n(x)) \le 4\epsilon.\]

Take \(\epsilon_k\downarrow0\). At each \(k\) the construction works eventually; increase its threshold \(n_k\) so that the thresholds are strictly increasing, and use \(H_{n,\epsilon_k}\) for \(n_k\le n<n_{k+1}\). This proves (12). Equation (13) follows from Proposition 12 and the triangle inequality.

All finite walk choices can be made first in a fixed enumeration. The graph and its disk extensions can likewise be chosen among piecewise linear maps after finite subdivisions, with rational coordinates in fixed spherical charts. These form a countable collection; the polygonal disk construction and an arbitrarily small perturbation give a member satisfying any strictly weakened error bound above. Choose the first such member. Its uniform error is measurable by taking a supremum over a countable dense set in each flag. Choosing the largest available accuracy among the first \(n\) accuracies gives a measurable version of the diagonal construction. Finally, the whole argument is deterministic on the event of contour convergence and the peanosphere assertions, so retaining other observations does not alter it. ◻

Proof. Transience makes the full time preimage of a spatial compact bounded. Choose a large disk containing the tests and all their buffers, then a larger disk for the finite joining routes below. Both Brownian coordinates are simultaneously below any prescribed pair of levels at arbitrarily distant times in either direction. Indeed, after an invertible linear change of coordinates this follows from planar Brownian recurrence applied to a disk contained in the corresponding open cone. Choose cuts \(a<b\) beyond the complete time preimages in use, at which both coordinates are strictly below all their heights. Rounding the cuts to lattice times preserves these strict inequalities for all large \(n\).

No scalar chord from a protected time can cross either cut: it would have to pass below its own endpoint height. Thus every protected primal vertex, dual vertex, matched quadrangle, and their full stars are determined inside \((a,b)\). Translate the resolved axial walk on \([a,b]\) into the strictly positive quadrant, precede it by a monotone axial path from the origin to its initial point, and follow it by a monotone axial path from its terminal point to the origin. This is a finite nonnegative resolved excursion and hence a finite spherical tree/cotree sewing. The appended steps and all incidence classes which they affect are below the protected heights. The exact tree/cotree incidence rule therefore leaves the protected full stars unchanged. This proves (i).

The projection and scalar-path proofs apply in any compact time interval containing the complete preimages involved. Strict lower cuts confine all the relevant discrete representatives to that interval. A finite chain of limiting chords uses only finitely many times, and its discrete tree paths can be read in a bounded enlargement. The sequential compactness proof of local joining consequently gives (ii), uniformly on each compact spatial buffer. There is no need to bound the length of one of these paths.

We give the exterior argument for (iii), since it prevents the cap from entering a tested region. Choose a finite simple cubic \(3\)-connected planar skeleton \(G\) whose bounded faces have diameter less than \(\epsilon\) throughout the large disk, and whose outer face misses that disk. Such a graph is the dual of a fine triangulation of a disk completed on the sphere by a vertex at infinity; the dual edges adjacent to that vertex can be drawn in a collar of the disk. The proof of \(3\)-connectivity in Theorem 17 applies without change. Lift its stubs and edge ribbons using (ii). All these finite walks and their complete incidence stars are in a bounded time protection.

Take \(T\) larger than every time in this protection and so large that both limiting tour tails \((-\infty,-T]\cup[T,\infty)\) avoid a disk containing \(G\) and its ribbons. Retain two tour positions near \(-T,T\) as exterior sentinels. For each bounded face of \(G\), join each sentinel to a graph vertex not incident to that face by a spatial path avoiding a slightly enlarged disk about that face. Retain also a path between the two sentinels outside the disk containing \(G\). There are finitely many such routes. Enlarge the time protection to contain their lifted routes before choosing the final lower cuts \(a<-T<T<b\).

The tour strip from \(T\) to \(b\) is connected to the appended terminal piece, and the strip from \(a\) to \(-T\) is connected to the appended initial piece. These strips avoid the lifted skeleton: an intersection would be an incidence in one of its full stars, whose representatives have already been retained between \(-T\) and \(T\). All cap-affected flags are connected to these strips through the appended tour, including any old exploration triangle whose partner is supplied by an appended step. Thus the cap and both strips lie in one face of the lifted skeleton. The sentinel routes show that this face is not any of the bounded corresponding faces: a route to a vertex not incident to such a face avoids its boundary, by its projection clearance. Whitney’s facial correspondence therefore identifies it as the outer face. Equivalently, the cap changes only that exterior component; every other complementary disk and all its incidence data are the original ones.

For a bounded face, the connected-exterior proof of (14) now applies. Its points have time representatives within the fixed window, so a violating sequence has a convergent representative. The required path away from its small boundary can be lifted in the original map by (ii); if that path reaches a cut-affected region, it has already reached the outer component, which gives the same separation contradiction. Hence the entire projected face lies in a \(3\epsilon\) disk. Extend the graph homeomorphism over all bounded faces and extend arbitrarily over the outer face. Since the tested compact sets avoid the outer face and its collar, their inverse images are in the unchanged part, with error at most \(4\epsilon\). Let \(\epsilon\downarrow0\).

All windows, skeletons, and routes just used are finite at each stage. Enumerate the desired compact tests and accuracies, retain the first \(k\) at stage \(k\), and enlarge their windows before choosing the cuts. Choosing the discrete index after that finite stage gives the asserted simultaneous diagonal version. ◻

Proof. Use \(H_n\) to transfer the two walks to the reference sphere. Its uniform error is smaller than all the fixed port clearances for large \(n\). Each walk then has a subpath crossing the indicated opposite sides of a common quadrilateral. Two such transverse paths intersect: a simple subarc of the first joining its two sides separates the other two sides, by the Jordan theorem. Their inverse images intersect as well. The interiors of distinct edges in an embedded primal graph are disjoint, so an intersection of primal walks gives a common vertex, also when one of their edges is a loop or they share an edge. Apply Proposition 18 in the bilateral case. ◻

Proof. For large \(n\), the \(H_n\)-image of the walk remains in a slightly enlarged annulus between \(B_0\) and \(B_3\). The short pointwise homotopy from \(P_n\) to \(H_n\) avoids \(B_0\) and an exterior point, so the winding number is unchanged. Decompose the finite closed walk at repeated vertices into simple circuits, deleting immediate backtracks. Winding numbers add in this decomposition. At least one simple circuit has nonzero winding about \(B_0\); its embedded image is a Jordan curve, and therefore it separates all of \(B_0\) from the exterior of \(B_3\). Its bounded side lies inside \(B_3\), because the connected exterior of \(B_3\) avoids the circuit. Keeping a smaller fixed error than the clearances between the four disks gives precisely the assertions about all \(P_n\)-lifts and about \(P_n(\Omega_n)\). The edge and cost assertion follows from the circuit decomposition. The protected-window homeomorphism proves the local version. ◻

Proof. Choose a cubic \(3\)-connected skeleton with face diameter much smaller than \(\epsilon\), perturbing it to avoid the excluded points. Put disjoint small junction disks at its vertices and narrow disjoint edge ribbons between them. In each junction disk prescribe four paths following the sides of a small square, extended past its corners so that adjacent paths have strict transverse crossings there. Prescribe edge paths which run from the central hole of one junction square through its edge ribbon to the central hole of the other.

Corollary 19 stitches the four square paths into a closed walk of nonzero winding about its central hole. To see the winding directly, cut consecutive paths at their forced intersections in the disjoint corner boxes and retain the intervening side portions. Their concatenation follows the square once, and a homotopy within its four thin side corridors preserves that winding. Corollary 20 then supplies a simple junction circuit. Every incident edge walk starts inside its central hole and exits its outer disk, so it meets that circuit. Thus all paths at a junction belong to one connected branch set, without specifying their lattice endpoints in advance.

Truncate each edge walk between its last visit to the first branch set and its first visit to the second, and erase loops. Different edge corridors are disjoint away from their assigned junctions. As in Lemma 16, minimal trees joining the three attachments produce a subdivision of the cubic skeleton. Its edges follow the reference edges in order. Under the actual homeomorphism \(H_n\) it is an embedded graph on the reference sphere with arbitrarily small tracking error. Whitney’s facial correspondence and the connected exterior of each small face, exactly as in the proof of (14), imply that every complementary component has diameter at most \(\epsilon\). Keeping the entire connected union of the supplied walks only subdivides these components. No additional edges have been used, which proves the cost assertion.

Finally a continuous path in an open set can be followed through any fixed finite ordered ports by a finite-length \(D\)-path. To verify this, cover its compact image by finitely many \(D\)-balls whose doubled balls lie in the open set, subdivide the parameter interval so successive selected points lie in one such ball, and join them by short \(D\)-geodesics. Those geodesics remain in the doubled balls. Applying this to the finite path list, with smaller tubes and the same strict ports, gives finite \(D\)-length reference paths with the identical topological requirements. ◻

Proof. Use a positive buffer between \(U\) and \(\partial V\) in (Gwynne, Miller, et al. 2019, Lemma 2.4). It determines the two frontier flow lines from a countable dense set, stopped on exiting \(U\), and hence each directed traversal, with its elapsed Lebesgue-area clock. Quantum time is obtained by measuring successive images with \(\mu_h|_U\). For an incoming or outgoing frontier arc compactly contained in \(V\), restrict the field to its indicated side and use the LQG boundary measure in the welding construction of (Duplantier et al. 2021). To make the locality explicit, truncate that side by crosscuts inside \(V\) and map it to a half-disk near a retained subarc. Apply boundary Gaussian multiplicative chaos to the transformed field there. Two such charts give the same measure by boundary coordinate covariance: their transition fixes a real boundary interval and extends by reflection across it. Exhaust the retained arc by these subarcs. Thus restriction to a smaller arc commutes with the measure construction, and lengths and outgoing minus incoming length increments are local once the sides and arcs are specified. This statement supplies no height shift between different traversals.

Properness of the plane exploration puts all visits to \(\overline U\) in a compact time interval; on the sphere the time interval is already compact. Each traversal reaching \(K\) crosses the positive Euclidean distance \(\mathop{\mathrm{dist}}(K,U^c)\). Uniform continuity gives a positive lower bound on the duration of such a crossing, so only finitely many disjoint traversals can reach \(K\). The marked-cell assertion is precisely (Contreras Hip and Gwynne 2026, Lemma 3.7), or follows by applying the traversal determination to consecutive local marks. Its extension to the field laws used here is made on a fixed buffer by the local field comparisons above. Relative heights will be recovered from aligned frontiers below. The remaining traversal order and the conditional laws of discrete observations are treated in Section 6. ◻

Proof. The deterministic-point assertion follows from the imaginary-geometry construction; see (Borga et al. 2026, sec. 2.4, property (vi)). Independence and integration against the random area measure give the second assertion. For the last, otherwise choose \(z_j\to z\) visited at \(t_j\notin I\). Properness, or compactness of the spherical time interval, gives a subsequence \(t_j\to t'\notin I\). Continuity gives \(\eta(t')=z\), contradicting the unique visit. ◻

Proof. Whitening the two-dimensional Brownian motion sends the quadrant to a wedge of opening \(\theta\). Its positive homogeneous harmonic function vanishing on the boundary is \(r^{\pi/\theta} \sin(\pi\varphi/\theta)\). Brownian scaling and the killed wedge kernel give the survival power \(\varepsilon^{\pi/\theta}\); in a half-plane it is \(\varepsilon\). One can equivalently use (Gwynne, Mao, et al. 2019, Lemma 3.2) for this survival estimate. For the endpoint assertion split the duration in two. Bound the first transition density by the free Gaussian density, reverse the second killed transition using kernel symmetry, and integrate the survival bound over the \(k\) wall coordinates, of total volume \(O(\varepsilon^k)\). Gaussian tails make the unrestricted coordinate integral finite.

For completeness, the image-dimension conclusion needs a counting argument, not merely the dimension of the record-time set. Fix \(\nu>0\), a compact window, and a witness \(r>0\). On the almost-sure Brownian modulus event, the oscillation over every interval of length \(2\delta\) is at most \(C\delta^{1/2-\nu}\) for all small dyadic \(\delta\). If a grid interval contains a joint record, Brownian motion starting at its appropriate grid endpoint stays in a quadrant enlarged by \(C\delta^{1/2-\nu}\) for at least \(r/2\). On truncating the random modulus constant by a deterministic bound, the survival estimate bounds the expected number of such grid intervals by \(C\delta^{-1+d_2-2\nu d_2}\). Markov’s inequality and Borel–Cantelli give, for every \(\zeta>0\), the eventual almost-sure bound \(\delta^{-1+d_2-2\nu d_2-\zeta}\) for their number. Each image lies in a ball of radius \(C\delta^{1/2-\nu}\). Letting \(\nu,\zeta\downarrow0\) through rational values gives (15). Coordinate projection cannot increase the number or size of covering balls. ◻

Proof. We first work with bilateral Brownian motion and use countably many compact windows and rational positive witnesses throughout. Record conditions can be shortened, so any two distinct times can be placed in disjoint rational windows with a positive gap, retaining their respective record witnesses inside those windows.

For two such windows, condition on their Brownian increments relative to their own starting heights. The independent increment across their gap gives their relative starting height a nondegenerate Gaussian density in \(\mathbb R^2\). Thus a random set determined by the first window and one determined by the second are translated relative to each other by a vector with a density. If both sets have upper box dimension at most \(a<1\), their difference set has upper box dimension at most \(2a<2\) and hence zero planar Lebesgue measure. Lemma 24 applies to the image of forward joint records in the first window and backward joint records in the second. A common chord would give equal two-dimensional heights in these two images, an event of probability zero. This proves (i).

We give both parts of the branch-level argument. First, a scalar local minimum at a time \(t\) cannot also be a one-sided record of the other coordinate. Restrict to witnesses of fixed positive length on both sides of \(t\). On a grid interval of length \(\delta\) containing such a time, the increments in one direction must stay above \(-C\delta^{1/2-\nu}\) in one coordinate, and those in the other direction in both coordinates. The past and future increments at the two grid endpoints are independent. The survival bounds therefore give \(C\delta^{(1-2\nu)(d_1+d_2)}\) for a specified interval. Since \(d_1+d_2>1\), choose \(\nu>0\) small enough that this exponent exceeds one; a union bound over \(O(\delta^{-1})\) intervals then tends to zero. Remove the modulus truncation and then the rational witness restriction. This proves the assertion when \(t\) itself is the local minimum.

Second, put a distinct scalar local minimum in a rational window separated from the joint record. The local-minimum heights of a scalar Brownian path in this window form a countable set: every strict local minimum is the unique minimum of some rational interval. The relevant coordinate projection of the joint-record image in the other window has upper box dimension less than one, hence Lebesgue measure zero. Condition on the increments in both windows. The relevant coordinate of the intervening Gaussian increment has a density; it almost surely does not translate this null set to any of the countably many local-minimum heights. Countable exhaustion proves (ii).

We recall why the scalar facts just used hold simultaneously. Brownian motion has a unique minimum on every fixed compact interval, and minima on disjoint rational intervals have distinct heights almost surely, by the transition density and independent increments. These countably many statements imply that every local minimum is strict, that different local minima have different heights, and that a scalar chord class has at most three visits. Indeed, four visits at the same minimal height would give two distinct local minima at that height.

At a time in (iii), both coordinates are backward records. A branch triple would put the respective coordinate at a local-minimum level, contrary to (ii). In particular two earlier suppliers for one colour, together with the order time, would give that excluded triple. A supplier shared by both colours would contradict (i). The chord relation is closed on a compact time window, so the admissible tuples with spans at least \(a\) form a compact set. Their distinct supplier times consequently have a positive minimum separation. For the perturbation assertion, extract a convergent subsequence of any sequence of admissible tuples. Uniform contour convergence preserves the chord equalities and inequalities, and the lower span bound remains \(a\). Uniqueness identifies the limiting suppliers; their positive separation preserves chronological order. A finite cover of the compact tuple set gives the asserted uniformity.

Finally, on every compact subinterval of \((0,1)\) the quadrant excursion law is absolutely continuous with respect to a Brownian path with a starting-point density, after localization of its endpoint heights. This follows from its positive killed transition and entrance densities; see (Gwynne and Sun 2015, sec. 2). The events proved above are local on a finite union of compact time windows. Countable exhaustion therefore transfers them to the excursion interior. Its common chord between the two global endpoints is the stipulated root convention. ◻

Proof. We first distinguish a scalar class from its possibly larger spatial fibre. For a coordinate \(c\), let \(E_c\) be the set of heights of visits which have a positive-span one-sided record in \(c\) and also a positive-span one-sided record in the other coordinate. This is a single pathwise Lebesgue-null set of heights. For witnesses in the same direction, the height-image estimate in Lemma 24 gives dimension at most \(2(1-2/\gamma^2)<1\). Here and below take a countable union over compact time windows and positive rational lower bounds for the witness spans.

For witnesses in opposite directions, cover a fixed window by intervals of length \(\delta\). On the Brownian modulus event with oscillation at most \(\delta^{1/2-\varepsilon}\), an interval containing such a visit forces scalar survival for a fixed positive duration backward from its left endpoint and forward from its right endpoint, each with initial allowance at most \(2\delta^{1/2-\varepsilon}\). These two Brownian increments lie on disjoint time intervals. The reflection principle therefore bounds the probability by \(C\delta^{1-2\varepsilon}\). The expected number of occupied intervals is at most \(C\delta^{-2\varepsilon}\). On a dyadic sequence, Markov’s inequality and Borel–Cantelli bound the count eventually by \(\delta^{-4\varepsilon}\); the excluded modulus events are summable. Letting \(\varepsilon\downarrow0\) and using any Brownian Hölder exponent below \(1/2\) shows that the height image has dimension zero. Thus both directions contribute only a null set to \(E_c\). Interior excursion absolute continuity gives the same assertion for the finite bridge.

An interior point of a genuine scalar ancestral segment belongs to a nontrivial scalar class. Indeed its level is below the endpoint height: the last visit to that level before the endpoint and the first visit afterward exist by recurrence, or by the lower excursion endpoints, and the contour stays above the level between them. Every visit in that class has a positive-span one-sided \(c\)-record witness: join it to any different visit of the same class. If an opposite-colour chord is incident to one of its visits, its height consequently belongs to \(E_c\). Outside \(E_c\) no such chord can leave the class. By the finite-chain identification rule of Theorem 9, its entire spatial fibre is then that one scalar class.

Consider an aligned overlap with local parameters \(r\) and \(r+\Delta\). Almost every \(r\) in its nondegenerate interval has an actual height outside \(E_c\). Equality of its spatial points therefore gives equality in the scalar tree and forces \(\Delta=x_i-x_j\), where \(x_i,x_j\) are the actual used-start heights. The same statement holds on a dense set of parameters. The ancestral segments are continuous in the scalar-tree metric, so equality extends to the whole overlap and its closure. This also shows why an isolated coincidence arising from the other colour cannot give an accepted overlap.

For the converse, write the blocks as \([a_i,b_i]\), \([a_j,b_j]\), with \(b_i<a_j\), and let their scalar minima be \(m_i,m_j\). Their endpoint ancestral arcs intersect in the scalar tree on the height interval \[[\max\{m_i,m_j\},M],\qquad M=\min_{[b_i,a_j]}c,\] when this interval is nonempty. A chord at level \(h\) between used-interior visits has \(m_i,m_j\le h\le M\), by its suffix and prefix tests. The interval cannot be a singleton. If, for example, \(m_i=M=h\), the visit in the first block is an interior local minimum at level \(h\), and a further visit at that level lies in \((b_i,a_j)\). There is a preceding visit before the first block, since the bilateral contour visits lower heights in its past, or since the interior excursion starts below the tested level. Its last return to level \(h\) before this block, the local minimum, the intervening visit, and the visit in the second block would be four visits of one scalar class. This is impossible: the middle two would be distinct local minima at the same height. The case \(m_j=M\) is the time reversal. Cut endpoints are almost surely not scalar record times, since the atlas uses deterministic or independently marked cuts. They therefore do not create an endpoint exception in this argument.

The nondegenerate common ancestral interval projects to an aligned same-colour overlap, with its local boundary-length parameter. It contains the chord level in its closure. For three visits, the middle visit is the strict scalar minimum between the outer ones. The two adjacent pairs share the incident segment above that level; the outer pair shares the segment below it. All three visits of a protected spatial point are retained. Their branch boundary is thus a retained block minimum; if two visits are in one block, its exact internal minimum accounts for their identification. This proves the converse and the branch assertion. ◻

Proof. Suppose there are candidate triples \((s_j,t_j,h_j)\) and minimizing times \(u_j\in[s_j,t_j]\cap O\) for which the difference in (16) tends to zero. Pass to a limit \((s,t,h)\in\mathcal C\) and \(u\in[s,t]\cap O\). We have \(c(u)=h\). Every \(v\in(s,t)\cap O\) belongs to \([s_j,t_j]\) eventually, and hence \(c(v)\ge h\); the same inequality at the endpoints follows from \(c(s)=c(t)=h\). By the candidate condition it also holds on \([s,t]\cap I\). Thus \[c(s)=c(u)=c(t)=\min_{[s,t]}c=h.\] The scalar chord rule gives \(\eta(u)=\eta(s)\in K\), contradicting \(u\notin I\). The same argument applies when \(s=t\).

The strict gap is stable under sufficiently small uniform contour perturbations and endpoint perturbations within compact instruction cells. In a finite block cover, suffix minima, prefix minima, and the minima of the intervening retained pieces are included in the tested condition. The remaining minimum is separated as above. One can therefore put rational height bands strictly between the protected candidate heights and that remaining obstruction. Lemma 29 uses these bands together with the exact discrete retained minima; no approximation of an exact lattice equality is inferred from uniform convergence alone. ◻

Proof. Choose \(K\Subset K_1^\circ\Subset K_1\Subset U\). The compact set \(\eta^{-1}(K_1)\) has a finite interval cover whose closure lies in \(\eta^{-1}(U)\). Enlarge and merge its intervals to place \(\eta^{-1}(K)\) in used interiors. By Lemma 13, all incidences of the required stars lie there eventually. Every visit to a supplier vertex is included, not only a selected path endpoint.

Within a block, scalar equalities and minimum tests are exact word operations. For each scalar colour, form the finite graph of blocks having a positive aligned frontier overlap. Lemma 27 proves that each edge gives the actual height difference, read from the two local length parameters. Choose a spanning forest by the fixed block enumeration; every height difference within a component is then a forest sum. Pairs in different components cannot have a protected scalar chord, by the converse part of that Lemma, so they are rejected locally.

Here is the local choice of the remaining bands. Fix an ordered pair of blocks in one component. In component height coordinates, let \(\mathcal C\) be the compact family of candidate endpoints and heights after testing the suffix, prefix, and every intervening retained minimum in that component. One endpoint has its image in \(K\). The closures of the positive aligned overlaps define a local subset \(\mathcal G\subset\mathcal C\), and let \(\mathcal B=\mathcal C\setminus\mathcal G\). By Lemma 27, \(\mathcal G\) is exactly the set of true scalar chords; \(\mathcal B\) consists of the rejected candidates. In particular this definition reads no omitted contour minimum.

We use the omitted minimum only to prove that these local sets admit a finite code. Its value \(M_{ij}\), on the portions between the two blocks omitted from this component, is one fixed threshold: \[(s,t,h)\in\mathcal G\quad\Longleftrightarrow\quad h\le M_{ij} \quad\text{for }(s,t,h)\in\mathcal C.\] It has a strict gap from the candidate heights. Indeed, a convergent sequence of omitted minimizing visits approaching such a height would give an omitted visit in the same scalar class as a protected endpoint, as in Lemma 28. The full-preimage cover places that visit in a retained block interior. The converse of Lemma 27 then places this block in the same component, contradicting its omission. This also covers a retained block belonging provisionally to another component.

The gap separates the compact candidate family from the threshold \(M_{ij}\). Thus its accepted and rejected parts are both closed in \(\mathcal C\); compactness of \(\mathcal B\) is a consequence of this gap, not merely of its being a complement. If \(\mathcal G\) and \(\mathcal B\) are both nonempty, their height projections are compact and satisfy \[\max h(\mathcal G)<\min h(\mathcal B).\] Choose the first rational threshold and positive rational clearance strictly separating these two locally defined sets, and allow exact candidate matches only below that threshold. If one set is empty, choose the corresponding all-or-none instruction. If there are no omitted portions, all candidates pass. These choices are measurable: extrema of the compact local candidate and overlap relations are measurable, and the rational search is countable. For the rejected set, the extrema can be obtained by first imposing positive rational distance from the closed overlap relation and then taking monotone limits. Aligned overlap itself is a local measurable test: on each rational parameter subinterval, with rational domain margins and a bounded offset range, test whether a constant translation makes the uniform distance between the two labelled arcs zero. The admissible offsets form a compact set, so its nonemptiness and least element are measurable. A positive overlap has only one translation, by Lemma 27. Thus even the rejection decisions use only local data. The unknown value \(M_{ij}\) is never an instruction input or a selection criterion.

A scalar branch boundary uses the exact minimum in its retained middle block, with no rational approximation to that equality. Additional supplier neighborhoods retain the adjacent overlap segments. To specify the discrete decision, write \(c_{i,n}\) for the contour of block \(i\) relative to its used start and \(x_{i,n}\) for that start height in its forest component. All heights here use the same rescaled units; equality is still tested exactly. For endpoints \(s,t\) in ordered blocks \(i,j\), first require \[c_{i,n}(s)+x_{i,n}=c_{j,n}(t)+x_{j,n}=h.\] Test that the suffix after \(s\), the prefix before \(t\), and each intervening retained minimum in this component are at least \(h\). These are raw-word tests. If there are omitted portions, also require that \(h\) lie below the selected rational threshold, with its prescribed clearance; use the all-or-none rule when appropriate. Endpoints in different forest components are rejected. This is the complete acceptance predicate for a scalar link.

There are finitely many block pairs and colours. Were their decisions wrong along arbitrarily close converging inputs, pass to limiting candidate endpoints and heights. A retained equality or minimum is decided by the exact tests just specified. Any remaining discrepancy must come from an omitted minimum crossing its rational band, contrary to the strict gap. Retained branch minima are never rounded to that band. This proves inclusion of genuine protected links and rejection of false ones without enumerating the Brownian branch points or inferring lattice equalities from uniform convergence.

Reduce each extended raw word internally. An unresolved flexible order in the used interior has both candidate suppliers before the left extension start. Otherwise it would have been internally resolved. Limits of those candidates give two positive-span backward chords. Their earlier endpoints are distinct by Lemma 25. A branch ambiguity would place the joint record at a scalar middle minimum or at the height of a separated scalar branch visit; the same genericity assertion excludes both. Their relative order is therefore stable. The positive left extension supplies the span lower bound \(a\) in Lemma 25(iii), and the protected supplier window supplies its compact interval \(J\). A finite rational cover of the admissible tuples consequently specifies all needed external types, simultaneously for every unresolved order in the used interior.

Unresolved decisions in unused extension portions may be arbitrary: they do not change internal matches, and used-start heights are free inputs. A portion supplying a tested incidence is promoted to a used supplier portion first. All comparisons other than the retained equalities and minimum tests have strict margins and persist in a small continuum neighborhood; those excluded tests are computed exactly from the raw words. The neighborhood’s boundaries can avoid the countably many atoms of the relevant distance-to-boundary variables. This gives the countable atlas.

Disjoint closed spatial buffers have disjoint compact time preimages, hence disjoint finite covers. Within one buffer, taking unions and merging overlapping intervals realizes all original readouts as restrictions of one incidence record. ◻

Proof. For a coordinate \(c\), a used start \(s\), and a later local time \(t\), the incoming and outgoing frontier lengths for that segment are \[Z_c(s)-\min_{[s,t]}Z_c, \qquad Z_c(t)-\min_{[s,t]}Z_c.\] Their difference is \(Z_c(t)-Z_c(s)\). For a point in the left extension, reverse the interval and subtract in the opposite order. The local length construction of Lemma 22 provides these functions on the indicated sides. Elapsed quantum time identifies their arguments. Rational local times and contour continuity therefore recover the whole relative path, not only its endpoint displacement. All cuts used here are within-block offsets; the recovery requires no elapsed time between distinct blocks. Lemma 27 recovers the forest differences from aligned boundary lengths. If several extended blocks belong to one spatial buffer, retain all of these data together, with their relative traversal order. No independence between blocks in this one aggregate input is asserted. ◻

Proof. All quantities are functions of the exact incidence record. In particular, extremal length uses the actual triangle shapes and oriented side identifications. It is a numerical variational quantity in \([0,\infty]\); it need not be continuous in a combinatorial presentation to be retained. Nonnegative path costs and extremal lengths are compactified, while the hyperspace of closed subsets of a compact metric space is compact. A diagonal extraction over the countable instructions and numbers of marks gives joint tightness. The two assertions about finite records are the two directions of Hausdorff convergence. The continuous forgetting maps take the discrete refined record sets onto their coarser sets; continuity on compact spaces gives the same identities after Hausdorff convergence. To retain a particular finite refinement of an approximating path, mark that path first and extract its refined record. Its coarse projection is the original limiting record. The same argument retains any finite collection of marked subpaths, since all are read from that one path before taking limits. A diagonal extraction can retain countably many such finite records, but by itself does not turn their spatial ranges into a parameterized continuous path. Later metric arguments supply the needed cost-based modulus. We make no converse inference about a maximum from a path hyperspace; its numerical maximum–minimum value is a separate coordinate.

For the first-exit test, stop a path on its first vertex outside \(V\). Its initial part lies in \(V\); its final edge and endpoint lie in \(W\) for large \(n\), by the full-star protection and vanishing projected mesh. The minimum in the displayed formula can therefore be computed in that retained incidence record. This includes every inner vertex, since every occurrence and full star has been retained.

Finally, for a compact set inside an open spatial confinement, \(\eta^{-1}\) of the compact set has a finite rational cover with closure inside the preimage of the confinement. Intermediate compact protections give both inner and outer such covers. Lemma 29 makes their discrete cell tests correct eventually. Apply this to ports, ranges, and target complements, with one more margin for stars. Countable refinement supplies arbitrarily small open losses. Every assertion used below is made with these margins fixed first. ◻

Proof. Bounded joint convergence gives convergence of the normalizer and each bounded continuous test multiplied by the weight. Division by the positive normalizer gives the limit \[\frac{w(e,I,A)}{\int w(e,I,A)R_e(\mathrm dI,\mathrm dA)} R_e(\mathrm dI,\mathrm dA)K(I,\mathrm dO).\] A monotone-class argument identifies the conditional kernel. No continuity of \(K\) in its input is required. Independent complete packages converge jointly to the product of their joint laws. A subsequent input-dependent weight changes the input law but preserves the conditional product of output kernels. An unused auxiliary variable may be integrated first. ◻

Proof. The iid ladder limit follows from the stopping rule and the contour invariance principle, as in (Gwynne and Sun 2017, sec. 2.1). Let \(\sigma^2=\operatorname{Var}(L_1^\circ)=(1-b)/2\), \(\beta=\operatorname{Cov}(L_1^\circ,R_1^\circ)/\sigma^2=b/(1-b)\), and \(\nu^2=\operatorname{Var}(R_1^\circ)-\beta^2\sigma^2>0\). Writing \(R^\circ=\beta L^\circ+\nu B\) with independent standard \(B\), the reflection principle gives \[ g(t,y)=\frac{1}{2\pi\sigma\nu t^2} \exp\!\left\{-\frac{1}{2\sigma^2t} -\frac{(y+\beta)^2}{2\nu^2t}\right\}. \tag{22}\] It is bounded and extends continuously by zero for \(t\le0\). If \(\widehat g_{s_b}\) denotes the hitting density at depth \(s_b\) for the variance-one process \(Z\), then \[ \widehat g_{s_b}(t,z)=s_b^{-1}g(t,z/s_b). \tag{23}\] Indeed the hitting time is the same and the second endpoint is multiplied by \(s_b\). This is the explicit change from the raw lattice density to normalized boundary-length coordinates.

We specify the lattice factor rather than suppress it. For \(\varphi(s,u)=\mathbb Ee^{i(sT_1+uY_1)}\), domain-of-attraction convergence gives uniform compact convergence of \(\varphi(s/m^2,u/m)^m\). On the compact set \[\mathcal A=\{(s,u):\tfrac12\le\sqrt{|s|}+|u|\le2\},\] the limiting modulus has a maximum \(\rho<1\), since the limiting law has the density \(g\). Uniform convergence bounds \(\sup_{\mathcal A}|\varphi(s/m^2,u/m)|^m\) by some \(\rho'<1\) for all sufficiently large \(m\). For small \(r=\sqrt{|s|}+|u|>0\), choose \(m=\lfloor r^{-1}\rfloor\). Then \((m^2s,mu)\in\mathcal A\), so \(|\varphi(s,u)|^m\le\rho'\). Taking the \(m\)-th root gives \(1-|\varphi(s,u)|\ge c/m\ge cr\), the required bound near zero. The forward raw words \(\mathsf h,\mathsf h\mathsf C,\mathsf h\mathsf c, \mathsf h\mathsf h\mathsf H\) have pairs \((1,0),(2,1),(2,-1),(3,0)\), respectively, in the backward displacement convention. In each word the leftmost \(\mathsf h\) is the first surviving type-1 burger found by the backward search. Their differences generate the parity lattice. Thus the only unit-modulus points on the Fourier torus are zero and \((\pi,\pi)\); away from them powers decay exponentially. Near them, the scaled Fourier integrands are dominated by \(e^{-c(\sqrt{|s|}+|u|)}\), an integrable function. Fourier inversion gives uniform convergence. On the feasible coset the two neighborhoods contribute equally, producing the factor two; on the other coset they cancel. This proves (21).

Fix \(0<\varepsilon<1/2\), put \(r=\lfloor\varepsilon m\rfloor\), and retain the first \(m-r\) marked blocks. Their density under the specified total endpoint is \[ \frac{p_r(k-J_{m-r},\ell-S_{m-r})}{p_m(k,\ell)}. \tag{24}\] On compact target sets it is at most \(C\varepsilon^{-3}\) and has the continuous limit given by (21). This proves convergence of the retained prefix. Under the total endpoint, complete marked blocks are exchangeable. The last \(r\) blocks have the same ordered marked law as the first \(r\), whose conditional density is bounded by \[\frac{p_{m-r}(k-J_r,\ell-S_r)}{p_m(k,\ell)}\le C.\] Their duration and full relative contour oscillation therefore vanish in the \(m^2,m\) units when first \(m\to\infty\), then \(\varepsilon\downarrow0\). This uses unconditioned stopped-contour convergence; no internal path is reversed. Converging together proves the full bridge limit.

For parameter continuity, the first coordinate above its lower wall is a three-dimensional Bessel bridge from \(a\) to zero. Independent standard bridges \(b^3,b^1\) on \([0,1]\) give \[A_u=\|(1-u)ae_1+\sigma\sqrt t\,b^3_u\|,\quad L^\circ_{tu}=A_u-a, \qquad R^\circ_{tu}=\beta L^\circ_{tu}+u(y+\beta a)+\nu\sqrt t\,b^1_u.\] The killed Brownian transition density verifies this representation. It is continuous for \(a,t>0\); applying the preceding convergence to each convergent parameter sequence yields compact uniformity. Positive killed transition densities and Brownian path support give the stated tube positivity. ◻

Proof. Fix a chart instruction and a good continuity neighborhood. We give the finite identity before taking limits.

Proof of 32. Use the aggregate kernels \(K^0\) constructed in (17). By 36, their augmented reference law has form (29), and its limiting change of density depends only on the retained inputs. Applying 33 therefore preserves these kernels under the actual word law. The recovery \(\mathsf I=\mathsf I(I)\) and the spatial relabelling described above give \(K(I,\mathrm dO)\). Multiplying a reference law by a positive smooth function of its input and normalizing leaves its conditional output law unchanged. Thus changing the auxiliary densities, and integrating parameters unused by the readout, identifies the kernels on overlapping input laws.

For a finite family of disjoint buffers, the aggregate reference packages, including their full inputs and output arrays, are independent. Their joint limits are products. Apply Lemma 36 to their union: the limiting resampling density depends only on the continuum input, so Lemma 33 preserves that conditional product. The same calculation applies in the finite bridge. Tests of the whole contour can be kept in the calculation; a monotone class of bounded continuous tests and the determination in Theorem 9 identify conditioning on \(\mathcal C\). No additional information in \(\mathcal C\) changes the kernel. Finite-dimensional products determine the assertion for countable arrays.

Compatibility is obtained before limiting. When two charts describe the same finite test, their readouts are restrictions of the same union incidence record and coincide on its good event for all large indices. The good-event exhaustion in Lemma 36 transfers this equality to the limits. Countably many such equalities hold simultaneously, and remain so under an absolutely continuous change of input law. This constructs one compatible array, not a collection of unrelated conditional versions.

For bounded changes of field density, multiply joint continuous tests by the density, first approximating that density in probability by bounded continuous functions of the input. Truncate an integrable density to remove boundedness. This proves the third assertion. Local marks can be retained first and then integrated with their conditional laws. Independent marks in disjoint regions preserve the conditional product.

For fixed offsets include their scaled versions in the same aggregate reference experiment before extraction. Its finite joint laws retain the actual correlations between these offsets; independence is used only between disjoint spatial buffers.

A measurable continuum functional has its fixed joint law with the input. On a set of input probability at least \(1-\varepsilon\), Lusin approximation replaces any bounded test of that functional by a continuous one. Include these approximations in the extraction and let \(\varepsilon\) tend to zero. Finally, variables sampled conditionally on \(\mathcal C\), independently of the discrete outputs, can be appended by disintegration. This proves the final assertion. ◻

Proof. Properness places all visits to \(K\) in a compact time interval. Uniform continuity there supplies \(\delta>0\) such that moving between \(K\) and \(V^c\) takes at least \(\delta\) units of time. Each distinct traversal meeting \(K\) contains such a move, so only finitely many occur. Lemma 22 reads their directed pieces in any strict imaginary enlargement. They can be enumerated measurably by taking the first representative in the fixed dense-point enumeration defining the local flow-line construction. The global curve order is determined by the imaginary field before it is parameterized by quantum area. Hence its conditional probabilities given \(J\) have the stated independence of real-field and clock choices. For a finite conditional distribution, \(\mathbb P[w_O(J)=0]=0\), which proves feasibility.

For completeness, order determination can be witnessed in a finite region. For the finitely many representatives, trace their left and right frontier flow lines until the corresponding same-angle branches have merged. The whole-plane flow-line tree construction of space-filling SLE gives these mergers at finite times almost surely; the resulting finite trees and their cyclic sides determine the relative Peano order (Miller and Sheffield 2017). Their compact traces lie in a finite disk. Until exit from a slightly larger disk they are local sets determined by the imaginary field there, so the witnessed order is determined there as well. For increasing deterministic disks, the probability that all of these finitely many witnesses are contained tends to one. Intersecting with deterministic bounds on the number of representatives proves the exhaustion assertion.

By Theorem 32, conditional on the whole fields and order \(o\), the actual local output array has a kernel \(K_o(I,\mathrm dO)\), where \(I\) now omits the order. This is defined for almost every local input with \(w_o(J)>0\). Keep the actual component and, conditionally on the fields and that component, sample each of the other feasible components independently from its kernel. The entire vector then has conditional law \(\prod_{o:w_o>0}K_o(I,\mathrm dO_o)\), which is independent of the actual order and depends only on local restrictions. The same construction uses the joint array at all retained fixed scale offsets, so it does not alter their within-order consistency. Conditional products between disjoint regions follow from Theorem 32, using independent auxiliary fills. This proves the last assertion. ◻

Proof. Let \(t\) be the unique preimage, and let \((a,b)\) be the component of \(\eta^{-1}(V)\) containing \(t\). If every neighborhood of \(z\) had a visit outside \((a,b)\), properness would give a convergent sequence of such times. Its limit would be a preimage of \(z\) outside \((a,b)\), contradicting uniqueness. Thus all visits to some \(U\) lie in that component. Within a single directed traversal, order is its own direction; labelled length differences are intrinsic to its local frontiers. The local measurability assertion follows from Lemma 22 and Theorem 32. ◻

Proof. Take larger disjoint disks around the compact buffers, with a pin outside them. Conditional on the exterior GFF, the fields in the disks are independent zero-boundary GFFs plus harmonic functions. Each harmonic function on a strict inner buffer has a smooth finite-energy extension supported in its disk. Conditional Cameron–Martin equivalence and then integration show that the joint restrictions are equivalent to independent zero-boundary restrictions. For the cone comparison, choose the normalization tests strictly inside its normalization circle and their buffers away from its marked center. There the field is an ordinary GFF plus the logarithmic drift, which has a finite-energy cutoff on the buffers. The same argument applies there and to its coordinate images. Finite smooth shifts are directly Cameron–Martin changes. For the imaginary field use its Markov decomposition modulo the angular period, with an independent uniform phase (Miller and Sheffield 2017, Propositions 2.8, 2.9 and 2.11). The argument applies to its harmonic part too. Densities can be truncated and integrated as in Theorem 32.

For affine covariance, express one intrinsic chart in its two coordinates. The quantum coordinate rules identify its area and labelled boundary lengths, and imaginary geometry identifies its directed frontiers. The retained raw incidence record and triangle shapes are literally the same; only their spatial labels change. Consequently their finite observations agree before extraction. Taking joint limits proves covariance of the kernels. The uniform phase convention makes translations and positive dilations exact symmetries of the unclocked whole-plane imaginary input. Absolute continuity alone would not prove this covariance; here it follows from the intrinsic finite readouts.

Finally, dividing quantum area units by \(v\) subtracts \(\gamma^{-1}\log v\) from the field and divides boundary-length units by \(\sqrt v\). Changing the denominator by \(v^p\) gives the last identity in (30) exactly on the same finite map. For integer time units use \(v_n=\lfloor vn\rfloor/n\) throughout this identity and retain the same raw cuts. Then \(v_n\to v\). If Poisson cuts are used, transform their intensity by the same clock change; include all required intensities in the countable atlas before extraction. Thus there is no endpoint-rounding error measured in an arbitrary graph-cost unit. Passing to the joint indexed limit proves the formula. No relation between \(c_n\) and \(c_{\lfloor vn\rfloor}\), or scaling rule for an unindexed passage limit under addition to \(h\), is used. ◻

Proof. Fix the ordinary normalized reference input \(h_*\) and its imaginary input with uniform phase. Choose common smooth auxiliary forest and free-cut laws for the reference experiments, keeping each instruction’s deterministic cuts fixed. The density invariance in 32(3) allows this choice without changing the resulting actual-law kernels or prescribed observations. The observation marks themselves use the fixed normalized convention in the statement. Independent reference restrictions are used in the disjoint enlarged buffers. At label \(v_r\), sample \(h_*\) in that label’s unit convention. In the original convention this is \(h=h_*+\gamma^{-1}\log v_r\); the indexed unit identity (30) then gives \(h_{v_r}=h_*\). Thus the label changes the effective microscopic time and cost units, while the base normalized input has a fixed law. Write \(P_r\) for these unconditioned row laws and \(\nu\) for their common marginal of continuum inputs and once-drawn marks. Thus \(\pi_{\rm input}P_r=\nu\) for every \(r\).

At stage \(r\) retain the first \(r\) coordinates and relative offsets \(w_1,\ldots,w_r\). First hold \(r,v_r\) fixed. The fixed-label construction supplies microscopic approximants for that row’s unconditioned joint law, including its protected reference experiment and reference metric observations. Choose an approximant along the sequence defining that row, with base integer scale \(n_r\), so that the first \(r\) bounded joint tests have error at most \(2^{-r}\) and \[N_r=n_rv_r\longrightarrow\infty,\qquad n_rv_r\min_{j\le r}w_j\ge r.\] Include the corresponding protected-density tests among those approximated. For integer-time conventions keep each raw cut and readout exactly and round only the unit used to scale its labels. Since the smallest unit tends to infinity, this changes the continuum labels by \(o(1)\); it changes no edge or path cost.

Re-expressed in the effective unit \(N_r\), these approximants form one raw-word sequence with a fixed continuum reference input law and arbitrary deterministic output denominators \(c_{n_r}v_r^p\). For any fixed finite list of offsets, its joint field inputs are \(h_*-\gamma^{-1}\log w_j\), a law independent of \(r\). Apply Lemma 36 directly to this sequence. Its durations, forest variables, guards, noise depths, and ladder limits are all expressed in \(N_r\) units; in particular its raw depth window is proportional to \(\sqrt{N_r}\). Its reference densities are fixed, and its compact good-chart normalizers have the positive limits proved there. The diagonal macro-input marginals converge to \(\nu\). Its actual-law good-chart exhaustion is therefore uniform along this sequence: first fix finitely many \(\nu\)-continuity good cells of total mass at least \(1-\varepsilon\), then let \(r\) tend to infinity, and finally let \(\varepsilon\) decrease to zero. A finite union handles any fixed finite list of buffers and offsets. The exact resampling identity and independent complete reference packages therefore prove the conditional product for this diagonal directly. Summable joint test approximation identifies the result with the desired limit of the unconditioned row laws. Subsequent fixed locally absolutely continuous field changes use 32(3). No assertion is made here for arbitrary varying field densities.

The fixed measurable functionals are retained by the Lusin argument in 32, now under the common input marginal \(\nu\). The finite-order augmentation uses that same fixed imaginary marginal and the joint retained array. Thus it also applies to the diagonal, without changing any actual-order coordinate. ◻

Proof. By Proposition 10, before normalization write the field as \(h_0+d\), where \(h_0\) is an ordinary whole-plane GFF with a pin, \(d\) is the deterministic marked drift, and the law has a strictly positive integrable tilt. Normalization gives \[h=h_0+d-\gamma^{-1}\log\mu_{h_0+d}(\widehat{\mathbb C}).\] Repin outside \(K\) first. The change is an additive random constant, which cancels in this normalization; its effect on the original tilt is another positive integrable density. We may therefore use an exterior pin throughout the following Gaussian calculation.

Choose a smooth finite-energy \(g\) with \(0\le g\le1\), equal to one precisely on \(K\), less than one off \(K\), and zero on an open set near the pin. It can be compactly supported away from the marks. Orthogonal Gaussian decomposition in the direction \(g\) gives \(h_0=h_\perp+Xg\), with \(X\) a nondegenerate Gaussian independent of \(h_\perp\). Conditional on \(h_\perp\), set \(\nu=\mu_{h_\perp+d}\). This is a finite measure of full support, and its boundary mass on the chosen circles is zero. On \(K\) the normalized field is \(h_\perp+d+c(X)\), where \[c(x)=x-\gamma^{-1}\log\int e^{\gamma xg}\,\mathrm d\nu.\] Differentiation under the integral gives \[c'(x)=1- \frac{\int g e^{\gamma xg}\,\mathrm d\nu} {\int e^{\gamma xg}\,\mathrm d\nu}>0.\] The inequality is strict because the exterior set where \(g<1\) has positive mass. Since \(\nu(\{g=0\})>0\), \(c(x)\to-\infty\) as \(x\to-\infty\). Also \[\int e^{\gamma x(g-1)}\,\mathrm d\nu\longrightarrow\nu(K) \quad(x\to\infty),\] by dominated convergence, so \(c(x)\to-\gamma^{-1}\log\nu(K)\). Thus \(c\) is a continuously differentiable increasing bijection onto that interval, with strictly positive derivative.

Compare the joint variables \((h_\perp,c(X))\) with \((h_\perp,X)\). Conditional change of variable gives a density which is finite and strictly positive exactly when \(e^{\gamma y}\nu(K)<1\). This event is the measurable condition \(\mu_{h_\perp+d+y}(K)<1\) on the resulting patch field. Pushing forward therefore proves equivalence of the normalized patch law with the unnormalized \((h_0+d)|_K\) law restricted to that event. The initial positive tilt preserves its null sets and positivity. A finite-energy cutoff of \(d\) away from the marks, and Proposition 39, replace the unnormalized restrictions by ordinary ones, jointly on all patches. This proves the real-field statement at actual heights.

The unclocked imaginary field is independent of the sphere real field in this chart, by Proposition 10. Its separated restrictions are equivalent to independent ordinary restrictions by Proposition 39; enlarged imaginary buffers may be chosen before this comparison. Finally, Theorem 32 attaches precisely the same conditional output kernels on both sides. Multiplying a joint input law by a strictly positive density multiplies the input–output law by that same density. This proves the final assertion for the actual sphere array, including its products and all-vertex escape readouts. ◻

Proof. Condition on the whole fields. On \(V_j\), the auxiliary output has exactly the kernel required by \(h'=h+f_j\); outside \(W_j\), the original exterior has exactly the unchanged kernel. Their product is the corresponding marginal of the common output array, by Theorem 32. Countable exterior restrictions are obtained by exhausting strict buffers outside \(W_j\). Disintegrate the full common array given these retained restrictions and sample the remaining coordinates from that conditional law. This preserves the auxiliary interior and old exterior exactly. Local Poisson marks may be resampled with the correct conditional area law in the changed region and retained outside. The order is unchanged, since the imaginary field is unchanged.

Before restriction or selection, the resulting pair has law \[\mathop{\mathrm{Law}}(h',O')=L_j(h')\mathbb P_{\rm out}(\mathrm dh',\mathrm dO').\] The Cameron–Martin formula for this fixed pinned law gives \[L_j(h')=\exp\{(h',f_j)_\nabla-\tfrac12\|f_j\|_\nabla^2\}, \qquad L_j(h+f_j)=\exp\{(h,f_j)_\nabla+\tfrac12\|f_j\|_\nabla^2\}.\] Here the pairing is the Gaussian linear functional. The second formula expresses the likelihood cutoff directly in the source field. Clipping this likelihood to \(L\) bounds the entire image measure by \(L\mathbb P_{\rm out}\). Restricting the source to any further event can only decrease its image measure. This remains true when that event depends on the original interior, auxiliary draws for other recipes, a minimizing path, or a selected hit. The discarded source coordinates are not required to be independent of the retained ones. Summing these measure inequalities proves the finite-list bound. For compactly supported smooth \(f_j\), the Gaussian pairing is measurable in its support buffer and has variance \(\|f_j\|_\nabla^2\). The clipping events can consequently be included among the fixed continuum tests before selection. ◻

Proof. Choose a smooth cutoff supported in \(E^+\) and equal to one near \(E\). For a harmonic function \(g\), its product with this cutoff is an extension in \(H^1_0(U)\), equal to \(g\) on \(E\), whose Dirichlet norm is at most \(C\|g\|_{L^2(E^+)}\). Indeed, multiply \(\Delta g=0\) by \(\chi^2g\), integrate by parts, and use \(2ab\le a^2/2+2b^2\) to bound \(\int\chi^2|\nabla g|^2\) by \(4\int g^2|\nabla\chi|^2\); use nested cutoffs for the product extension. Constants have extensions of bounded norm by the same cutoff. Apply this to both fields in the product Dirichlet space.

If \(P_f\) is the fresh law shifted by such an extension \(f\), and \(P_0\) the unshifted fresh law, Cameron–Martin and Cauchy–Schwarz give, for every event \(A\), \[ P_f(A)\le e^{\|f\|_\nabla^2/2}P_0(A)^{1/2}, \qquad P_f(A)\ge e^{-\|f\|_\nabla^2}P_0(A)^2. \tag{32}\] These inequalities still hold after adjoining the same conditional kernel: the likelihood ratio is a function of the fields alone.

Decompose the reference pair on \(U\) by the domain Markov property. Its harmonic parts, after separating the bounded imaginary phase, have almost surely finite \(L^2(E^+)\) norms. Their extensions are independent of the two fresh parts. Choose a bound \(H_0\) whose extension event has probability at least \(1/2\). Integrating the second inequality in (32) gives \[P_{\rm ref}(A)\ge\tfrac12e^{-H_0^2}P_0(A)^2.\] The first inequality then bounds the target probability by a constant times \(P_{\rm ref}(A)^{1/4}\).

For the other direction, if \(P_{\rm ref}(A)\ge\eta\), choose \(H_\eta\) so that the reference extension exceeds \(H_\eta\) with probability at most \(\eta/2\). Integrating the first inequality gives \[P_0(A)\ge e^{-H_\eta^2}(\eta/2)^2.\] The second inequality gives the asserted positive lower bound, for example \(e^{-H_t^2-2H_\eta^2}(\eta/2)^4\), where \(H_t\) bounds the target extension norm. All bounds concern the field laws and therefore are independent of the output label. ◻

Proof. Let \(J\) be the local unclocked imaginary input. For each feasible order \(o\) let \(w_o(J)>0\) be its conditional probability and \(K_o\) its local output kernel. Lemma 37 identifies these weights and kernels: the weights and finite feasible set depend on \(J\), independently of the real field and volume label. Adjoin the other order coordinates as in that lemma, retaining the actual coordinate, so that the conditional vector law is \(\prod_oK_o\).

For any \(\zeta>0\), choose \(J_0<\infty\) and \(\delta>0\) so that, with reference probability at least \(1-\zeta\), there are at most \(J_0\) feasible orders and all their weights are at least \(\delta\). This is an exhaustion of the full traversal list for the fixed protection, under one imaginary marginal law common to all labels. Neither the bound nor the minimum weight is asserted uniformly over all imaginary inputs. On this event, the conditional probability that some coordinate fails is at most \[\sum_oK_o(\text{failure at }o) \le\delta^{-1}\sum_ow_o(J)K_o(\text{failure at }o).\] After integration, the all-order failure probability is at most \(\zeta+\varepsilon/\delta\). Choose \(\zeta\) first and then \(\varepsilon\). The product assertion is the augmented product assertion of Lemma 37. The same argument works for a finite joint array of readouts in one buffer. ◻

Proof. We verify the covariance assertion, including the effect of moving centers. Evaluation of \(\mathcal H_j\) at \(c_j+r_ju\) is the boundary pairing against harmonic measure \(\nu_{j,u}\) on \(\partial U_j\). For \(u\in E^+\), its density on the two boundary circles, after scaling, is bounded uniformly, as are its derivatives on any slightly smaller fixed shell containing \(E\). The charge representing \(G_j(u)\) is \(\nu_{j,u}-\sigma_j\), where \(\sigma_j\) is uniform measure on the circle of radius \(32r_j\). It has total mass zero, bounded total variation, and support at distances comparable to \(r_j\) from \(y\). The increment \(H_j-H_{j-1}\) is represented by \(\sigma_j-\sigma_{j-1}\), with the same properties at these two adjacent scales. Smooth circle densities make their logarithmic self-energies finite and uniformly bounded after scaling. The additive \(\log r_j^{-1}\) cancels because the charges have mass zero. The ratio of the adjacent radii is fixed.

For two nonadjacent scales, the smaller charge has support of diameter \(O(r_k)\), at distance comparable to \(r_j\) from the larger charge. Subtract the logarithmic kernel at one point of the smaller support. Its gradient there is \(O(r_j^{-1})\); the total-mass cancellation gives a covariance bound \(C r_k/r_j\). For increments, associate the charge to its larger of the two radii. Enlarging the constant to cover adjacent scales, every covariance block therefore has norm at most \[C e^{-s(|j-k|-2)_+}.\] For the \(L^2\) blocks, the same estimate follows from the uniform pointwise covariance bound and the fixed area of \(E^+\). For the smooth part of the Green function, cancellation in both variables bounds the block by \(C_Dr_jr_k\), with the larger adjacent radius used for an increment. Its operator norm is at most \(C_D\sum_jr_j^2\); it is uniformly bounded once the largest radius is small. Pin terms for the imaginary whole-plane field are sums of a function of either variable and disappear against the zero-mass charges.

The row sums of the exponential block bound are uniformly finite. Cauchy–Schwarz, or the block Schur bound, gives the asserted operator norm. Integrating the diagonal covariance bound gives the trace estimate. This also constructs the profiles as Gaussian \(L^2\)-valued variables.

If \(\lambda_\ell\) are the covariance eigenvalues, for \(0<t<(4\Lambda)^{-1}\), \[\mathbb Ee^{t\|Z\|^2} =\prod_\ell(1-2t\lambda_\ell)^{-1/2} \le \exp\!\left(2t\sum_\ell\lambda_\ell\right) \le e^{C'(m+1)}.\] Finite-dimensional approximation and monotone convergence justify the formula. Markov’s inequality, followed by increasing \(A\), proves (34). ◻

Proof. There are two different grid counts in the proof. We control coarse circle averages at the first and last radii on their own grids. The harmonic-profile and fresh-shell failures are then made to have arbitrarily large exponential rates, so that they can be summed over the much finer grid indexing every displaced scale chain.

Choose \(2<\beta<Q\) and set \[a=\frac{\log4}{4\gamma(Q+\beta)},\qquad B=\frac{4\log4}{\gamma(Q-\beta)},\qquad t_j=aN+js,\quad m=\left\lfloor\frac{(B-a)N}{s}\right\rfloor .\] All buffers are contained in a fixed compact neighborhood of \(K\) for large \(N\). Use the grids and nearest-center chains above. The number of finest-grid points needed to cover \(K\) is at most \(C_Ke^{2BN}\).

First control the two endpoints on their own grids. The circle averages there have variance \(t_j+O_K(1)\). There are at most \(C_Ke^{2t_j}\) centers on the grid at scale \(j\). The Gaussian tail bound therefore gives, for \(j=0,m\), \[ \mathbb P[\text{some such center has }|H_j|>\beta t_j] \le C_K e^{-(\beta^2/2-2)t_j} \le C_K e^{-c_KN}. \tag{36}\] Increasing \(C_K\) absorbs the bounded variance correction. On their complement, for every chain the exterior-measurable process \[Y_j=\gamma(Qt_j-H_j)+\log M\] satisfies \(Y_0<N\log4\) and \(Y_m>2N\log4\) for sufficiently large \(N\). The fixed multiplier \(M\) changes only this last threshold.

Choose \(R>2B+3\), and then the budget \(A\) in Lemma 45. On its good event, \[ \sum_{j=1}^m(Y_j-Y_{j-1})^2\le A_1N \tag{37}\] for a deterministic \(A_1\), since \(t_j-t_{j-1}=s\) and \(m=O(N)\).

Assign each \(i\in I_N\) to the first \(j\) for which \(Y_j\ge i\log4\). Discard the assignment if \(Y_j-i\log4>L\), where \(L\ge\log4\) will be fixed shortly. For one upward jump \(\Delta=Y_j-Y_{j-1}\), the number of discarded lattice levels is zero unless \(\Delta>L\), and in that case is at most \[\Delta/\log4+1\le 2\Delta/\log4 \le\frac{2\Delta^2}{L\log4}.\] The sum in (37) therefore bounds the number discarded by \(2A_1N/(L\log4)\). Choose \(L\) so large that this is at most \(\theta N/8\). Every remaining assignment satisfies \[ |Y_j-i\log4|\le L, \tag{38}\] and a single \(j\) has at most \(K_L=\lceil2L/\log4\rceil+2\) such assignments.

Now choose \(H\) so large that \(K_LA/H^2\le\theta/8\). There are at most \(AN/H^2\) indices at which \(\|G_j\|_2^2+\|\widetilde G_j\|_2^2>H^2\). Deleting their labels leaves at least \(\theta N/2\) labels at at least \[\kappa_0N,\qquad \kappa_0=\theta/(2K_L),\] distinct radii. Select one label at each such radius, for example the smallest. Endpoint tests, budget tests, label assignments, and this selection are all \(\mathcal F\)-measurable. This is why the coarse circles in (33) were placed outside every fresh annulus.

For a selected pair \((j,i)\), the conditional real field in (35) is a fresh field on \(U\) plus \[G_j+\left(H_j-Qt_j-\gamma^{-1}\log M +i\gamma^{-1}\log4\right).\] The constant in parentheses has absolute value at most \(L/\gamma\). The imaginary conditional field is a fresh field plus \(\widetilde G_j\) and a phase in a bounded interval. Lemma 44 first makes the reference all-order failure probability small. Lemma 43 then makes its conditional target failure probability at most any prescribed \(p_*>0\), by choosing \(\varepsilon_*\) small enough. These choices are made after \(A,L,H\) and the geometric constants have been fixed.

The selected augmented tests are conditionally independent: integrate the fresh-field product law against the conditional product of local kernels in the disjoint buffers. The random selection causes no change, since it is \(\mathcal F\)-measurable and takes values in a finite deterministic list. With \(\ell\ge\kappa_0N\) selected radii, a union bound over subsets shows \[ \mathbb P[\text{fewer than }\ell/2\text{ tests succeed}\mid\mathcal F] \le 2^\ell p_*^{\ell/2} \le 2^{m+1}p_*^{\kappa_0N/2}. \tag{39}\] Choose \(p_*\) to make this at most \(e^{-RN}\). Together with the square-budget bound this has an arbitrarily large exponential rate. Taking the union over the at most \(C_Ke^{2BN}\) finest-grid chains leaves a summable error. The endpoint errors are added using (36); they were union-bounded on their own grids and do not incur the finest-grid entropy.

For any \(x\in K\), choose a finest-grid point \(y\) within \(\epsilon_0r_m\). At every scale of its chain, \[|x-c_j(y)|\le\epsilon_0r_m+\epsilon_0r_j \le 2\epsilon_0r_j<r_j/256.\] Its at least \(\kappa_0N/2\) successes give the required distinct radii. Take the union of the successful tests produced by all chains. At a fixed center and radius, the alignment interval allows at most \(K_L\) labels. If several succeed, retaining any one preserves coverage, because their central holes are identical. The radius-\(r\) buffers lie in disks of radius \(16r\) about grid points spaced by \(\epsilon_0r\), so their overlap is at most a constant depending only on \(\epsilon_0\).

Borel–Cantelli proves eventual coverage. This event belongs to the restriction of the fields and kernels to any fixed neighborhood of \(K\); finitely many large initial radii can be discarded. A fixed absolutely continuous field-law change preserves this probability-one event by Proposition 39. It need not preserve the displayed summable numerical probability bound. Finally intersect the probability-one events for a countable compact exhaustion and countably many prescribed test families. ◻

Proof. At each center and radius retain one good shell density. Sum them over the centers whose buffers meet \(U\), and then sum over the retained radii and divide by \(N\). At radius \(r\), bounded overlap and grid counting give total energy at most \(C\mathop{\mathrm{Area}}(U^{+CR_N})/\beta_m\). There are at most \(CN\) radii. Cauchy–Schwarz for their sum proves (46).

For the charge, choose a coordinate in which the path interval has oscillation at least its diameter divided by \(\sqrt2\), and restrict to a subinterval between a minimum and a maximum of that coordinate. Discard coordinate values within \(CR_N\) of either endpoint. At each remaining value choose a visit of the path. This visit belongs to at least \(cN\) good central holes. A hole at radius \(r\) can account for a coordinate interval of length at most \(Cr\), and the path must make a full crossing of its buffered shell. Integrating the number of covering holes over the coordinate interval gives \[cN\bigl(\operatorname{osc}-CR_N\bigr) \le C\sum_{\text{shells crossed}}r.\] The integral of the summed shell densities is at least the sum on the right before the harmless fixed factor. Divide by \(N\). Strict inner and outer margins absorb the uniform projection errors, proving (47). Integrals over disjoint parameter intervals add even when their traces overlap. No choice of the density was made using the path. ◻

Proof. Fix \(h>0\), and choose \(N\) with \(CR_N<ch/4\). A collection of disjoint path intervals each of projected diameter at least \(h\) then pays at least \(ch/2\) per interval for all sufficiently large \(m\). By (43) and (46), its expected cardinality is bounded by \(C/h\). The same bound holds for the maximal cardinality, by monotone convergence over finite collections.

Here is a measurable reparameterization implementing compactness. For a continuous projected path, mark its successive first moves by distance \(2^{-k}\), and denote their finite number by \(J_k\). Give each such mark clock mass \(2^{-k}/(1+J_k)\), for every \(k\ge1\), and add normalized Lebesgue mass in its original parameter. The total clock mass lies in \([1,2]\). Use its inverse, inserting a pause at each atom, and normalize the new parameter to \([0,1]\). This is a representative of the same path up to zero uniform reparameterization distance. On a set where \(J_k\le M_k\) for every \(k\), a new parameter interval shorter than \(2^{-k-1}/(1+M_k)\) contains no complete move at scale \(2^{-k}\), and the image oscillation there is at most \(2^{2-k}\).

The expected crossing-count bounds allow \(M_k<\infty\) to be chosen with total exceptional probability as small as desired. For each fixed \(k\), the finitely many indices preceding its eventual bound can be included by increasing \(M_k\): their crossing counts are finite almost surely. Thus the reparameterized laws are tight in \(C([0,1],K)\), by Arzelà–Ascoli. Prokhorov’s theorem gives \(\nu\), and endpoint separation passes to the limit.

To prove the length assertion without exchanging infinite suprema, fix \(J\). Consider at most \(J\) disjoint closed parameter intervals with rational endpoints whose images have positive clearance inside \(U\), and sum the distances between their endpoint images. The supremum of these sums is a nonnegative lower semicontinuous path functional: each strict confinement test is open, and each finite sum is continuous on that test. For every such collection, (47) and Cauchy–Schwarz bound its flow expectation by \[C\sqrt{\mathop{\mathrm{Area}}(U^{+CR_N})}+CJR_N+o_m(1).\] The bound also holds for the supremum, because the density charge holds simultaneously for all collections. Pass first to the weak flow limit, then let \(N\to\infty\) with \(J\) fixed, and finally let \(J\to\infty\). These finite interval sums exhaust variation in the open set \(U\). Taking one larger confinement first proves finite total variation for almost every path; the same argument in each \(U\) then proves (48) for its arclength measure, with multiplicity.

Every limit path has length at least \(d_0\), so \(\Lambda\ne0\). The constant in (48) is uniform over finite unions of boxes. Cover a planar null set by countably many rational boxes of arbitrarily small total area, apply the estimate to finite subunions, and increase to the full union. This proves \(\Lambda\ll\mathrm dz\). ◻

Proof. First take a bounded real readout and set \(p(z)=\mathbb E[Y(z)\mid\mathcal H]\), with jointly measurable versions. Conditional independence at distinct points and Fubini give, for every rational disk \(V\), \[\mathbb E\left[\left(\int_V(Y(z)-p(z))\,\mathrm dz\right)^2 \middle|\mathcal H\right]=0.\] The diagonal in \(V\times V\) has area zero. A countable disk exhaustion and Lebesgue differentiation imply \(Y(z)=p(z)\) for almost every \(z\), almost surely. A second application of Fubini gives this identity almost surely at almost every deterministic point.

For completeness, the point germ of a Dirichlet GFF is trivial. Realize the field as an isonormal Gaussian process over \(H^1_0(V)\), with its Dirichlet inner product. The Gaussian subspace generated by its restriction to \(B(z,r)\) is the closed span of the Green potentials of test functions supported there. An element in the intersection of these subspaces is orthogonal to every smooth function vanishing near \(z\). Such functions are dense in \(H^1_0(V)\): a logarithmic cutoff around \(z\) has Dirichlet energy tending to zero, so a point has zero capacity. The subspace intersection is therefore zero. Decreasing orthogonal projections, first on each finite Gaussian chaos and then by \(L^2\) approximation, show that the intersection of the completed Gaussian sigma fields is trivial. The same argument applies to the direct sum of two independent fields. Taking the imaginary field modulo its period only decreases the observable sigma fields. Local absolute continuity in Proposition 39 transfers triviality to the actual field restrictions and their phase conventions.

For almost every deterministic \(z\), the conditional expectation \(p(z)\) is measurable with respect to each shrinking field restriction, hence to this trivial germ. It is deterministic, and so is \(Y(z)\). A countable family of bounded continuous functions separating points proves the assertion for a separable target space.

Finally rotate a local coordinate about \(z\), making the corresponding imaginary-field phase adjustment. The intrinsic time/height ports and flag-surface variational outputs are unchanged; only their spatial coordinate descriptions transform. This is the affine covariance in Proposition 39. The original and transformed field laws are locally equivalent. Thus every probability-zero or probability-one assertion about a germ transports to the transformed germ. Since its value is deterministic, that value must obey the claimed rotation rule. Taking countable intersections completes the proof. ◻

Proof. Suppose otherwise, and take the record-low extraction described in Section 8.2: successive record lows \(j_m\) with \(\beta_{j_m}\to0\). For every fixed backward offset \(i\), \(\beta_{j_m-i}\ge\beta_{j_m}\) eventually. Retain all fixed offset laws and all local EL arrays in units \(\beta_m=\beta_{j_m}\), using the diagonal extraction in Theorem 32. The second inequality of (45), through (44), gives the wider-band shell barriers in these units at every fixed backward label \(v_i=4^{-i}\). The aligned-annulus result and Lemma 47 therefore apply.

The first inequality of (45) retains a positive-probability small-EL event. Retain the trimmed cylinder flows from (44) as witnesses in the joint extraction. On a fixed buffer containing their confinement and the local tests, Proposition 39 gives a strictly positive Radon–Nikodym derivative of the ordinary field-pair restrictions relative to the cone restrictions. Reweight the entire coupled extraction by this field-dependent derivative. The local field and array marginal then has the ordinary law with the same kernels; the positive-probability flow event and its pathwise inequalities persist under this equivalent change of law. No kernel is conditioned on the flows. Their paths have fixed positive advance in a compact set and traffic energy \(O(\beta_m)\). Lemma 48 gives, on that event, a probability flow \(\nu\) of nonconstant rectifiable paths with nonzero absolutely continuous average arclength \(\Lambda\).

We describe the local event witnessed by these paths. Take a finite sufficiently fine set of unoriented directions, closed under a quarter-turn. For a direction \(v\), a center \(z\), and \(s>0\), put \[R_v(z,s)=\{z+t v+u v^\perp: |t|<3s/2,\ |u|<s/4\}.\] A strict crossing runs from \(t<-s\) to \(t>s\), has compact confinement in this rectangle, and is represented by intermediate port covers. Let \(E_v(z)\) be the event that at every sufficiently small dyadic \(s\) some such cover has finite limiting EL in units \(\beta_m\). Explicitly, this is a countable union over the first dyadic scale, followed by a countable intersection over smaller scales, and a union over admissible finite covers and finite integer EL bounds. Whether a cover fits strictly inside the indicated rectangle is a measurable open geometric test. Thus \(E_v(z)\) is jointly measurable in \(z\) and the local arrays. Discarding finitely many scales leaves the event unchanged, so it is a local germ readout. Its finite EL bound may depend on the scale.

At arclength-almost every interior point of almost every path of \(\nu\), arclength parameterization has a unit tangent. Choose \(v\) within the angular slack of this tangent. Differentiability at that parameter value gives strict crossings of \(R_v(z,s)\) at every sufficiently small \(s\). Almost every path belongs to the support of \(\nu\). Each fixed strict tube crossing is an open path event containing that path, so has positive \(\nu\)-mass. Weak convergence gives positive mass in the discrete flows for all sufficiently large \(m\). Restrict and normalize that mass in (43). Its energy is still \(O(\beta_m)\), with a finite constant depending on this radius and tube. A finite intermediate cover of positive mass supplies the required local EL witness. This proves \(E_v(z)\) at \(\Lambda\)-almost every such point for some direction \(v\).

Since \(\Lambda\ne0\) and \(\Lambda\ll\mathrm dz\), on the positive-probability flow event at least one of the finitely many events \(E_v\) holds on a set of positive planar area. Lemma 49 makes its indicator deterministic at almost every deterministic point and transports it under a quarter-turn. By Fubini, on a probability-one event these equalities hold outside one planar null set, simultaneously for the finite set of directions. The positive-area flow witness therefore meets that full-area set, where \(E_v(z)\) and \(E_{v^\perp}(z)\) have the same truth value. Choose a point satisfying both, and then one fixed dyadic radius below their two starting scales.

Let \(\mathcal F_{v,m}\) and \(\mathcal F_{v^\perp,m}\) be the two discrete curve families specified by the selected covers. Their extremal lengths are at most \(C_1\beta_m\) and \(C_2\beta_m\), respectively, for all sufficiently large \(m\). Choose a common central quadrilateral \(Q_m\), with projected sides near those of \(\{|t|<s/2,|u|<s/2\}\), and smooth it on the discrete surface within the available margins. As shown in Figure 4, the narrower strips have lateral clearance from the inappropriate sides of this quadrilateral. Every member of either family therefore contains a subpath crossing the corresponding pair of sides of \(Q_m\). For any density \(\sigma\) on \(Q_m\), its extension by zero to the surface has the same energy, and \[\inf_{\gamma\in\mathcal F_{v,m}}\int_\gamma\sigma\,\mathrm ds_m \ge \inf_{\gamma\in\mathcal C_{\leftrightarrow}(Q_m)} \int_\gamma\sigma\,\mathrm ds_m.\] Taking the supremum in (41), and arguing identically in the perpendicular direction, yields \[\mathop{\mathrm{EL}}(\mathcal C_{\leftrightarrow}(Q_m)) \le\mathop{\mathrm{EL}}(\mathcal F_{v,m})\le C_1\beta_m, \qquad \mathop{\mathrm{EL}}(\mathcal C_{\updownarrow}(Q_m)) \le\mathop{\mathrm{EL}}(\mathcal F_{v^\perp,m})\le C_2\beta_m.\] The constants are finite because the radius and the two covers have now been fixed. Their product tends to zero, contradicting (42). This excludes the record-low extraction. ◻

Proof. Proposition 50 makes the high-probability barriers uniformly positive in raw EL units. To pass from powers of four to an arbitrary diverging integer sequence, write \(n=4^{j_n}v_n\), where \(j_n=\lfloor\log_4 n\rfloor\) and \(1\le v_n<4\), and extract \(v_n\to v\in[1,4]\) jointly with every fixed offset array. Reexpress the same raw cuts and incidence readouts in the new units: the area clock is divided by \(v_n\), frontier heights by \(\sqrt{v_n}\), and the real field is shifted by \(-\gamma^{-1}\log v_n\). The actual flag-surface extremal length is unchanged. Strict chart margins allow the converging factors to be retained through this indexed-array identity. On each fixed buffered domain, the constant shifts lie in a bounded family and admit cutoff extensions of uniformly bounded Dirichlet norm. The two-sided Cameron–Martin comparison therefore chooses a common high-probability barrier threshold before annular alignment. This argument does not identify differently indexed kernels at an unshifted field. Apply the aligned-annulus result and Lemma 47 with \(\beta_m=1\).

For a curve making advance \(d\) compactly in \(U\), choose \(N\) with \(CR_N<cd/4\). The density charges the entire family by at least \(cd/2\), for large \(m\), and has bounded energy. Its extremal length is therefore bounded below by a constant times \(d^2/\mathop{\mathrm{Area}}(U^{+CR_N})\). A finite collection of ordinary patches catches a fixed positive subadvance of every curve of spherical diameter at least \(d\). To obtain such a collection, first remove disjoint sufficiently small neighborhoods of the finitely many exceptional marks; a curve of diameter \(d\) either makes an advance away from them or crosses one of their surrounding compact annuli. Cover the resulting compact regions by finitely many ordinary boxes and sum the densities. This proves the first claim for the entire family at once.

For a through crossing of \(A(z,r,2r)\), take a slightly larger annular union of boxes of area at most \(Cr^2\). Each crossing makes projected advance at least a fixed multiple of \(r\) there. The preceding quotient is bounded below by a universal constant. The union can avoid the center and all other exceptional marks; in particular this proves the assertion for fixed shells surrounding an exceptional center. Changes of ordinary field law preserve the eventual statement and its constant. Countable extraction handles all rational tests and margins simultaneously. Lemma 41 transfers these probability-one conclusions to the finite sphere at its actual area-normalized heights; Theorem 32 retains the same incidence and extremal-length readouts. ◻

Proof. The two ring estimates below give equicontinuity and exclude collapsed continua. Annuli with shifted centers will then bound the linear distortion of each limit.

Proof. Let \(f_n=\phi_n\circ H_n^{-1}\to f\) uniformly. The image of a local quadrilateral \(R\) has the same modulus as its actual flag-surface preimage. Strict inner and outer port versions therefore determine the modulus of \(f(R)\): uniform convergence of maps and inverses places the four sides between arbitrarily small collars, with their marked sides in the correct order. Conformal rectangle modulus is continuous under these Jordan collar approximations. The two transverse comparisons in (42) supply the two sides of this determination. Thus all these local moduli are readouts of the local EL arrays. Global three-point normalization does not enter them, since Möbius postcomposition preserves modulus.

The distortion bound (53) implies that \(f\) is differentiable almost everywhere (Cristea 1989, Lemma 2). Its quasiconformality also gives a nonzero Jacobian almost everywhere (Astala et al. 2009, Theorem 3.1.2). At a point \(z\) satisfying both conclusions put \[ T(z)=\frac{Df(z)^{\mathsf T}Df(z)}{|\det Df(z)|}. \tag{54}\] This positive symmetric tensor has determinant one and bounded eccentricity. It is a germ readout of the local moduli. Here is an explicit way to recover it. For a unit vector \(v\) and \(t>0\), let \(R_{v,t}\) have length one in direction \(v\) and width \(t\). Differentiability implies uniform convergence of the rescaled images of \(z+rR_{v,t}\) to \(Df(z)R_{v,t}\), for fixed \(t\). If \(L\) is a nonsingular linear map, constant density and straight-line flows give \[\frac{(|Lv|-t|Lv^\perp|)_+^2}{t|\det L|} \le \mathop{\mathrm{EL}}_{\leftrightarrow}(LR_{v,t}) \le \frac{|Lv|^2}{t|\det L|}.\] Multiplying by \(t\) and letting \(t\downarrow0\) recovers \(v^{\mathsf T}T(z)v\). Three fixed directions, with countable sequences of \(r,t\) and strict covers, determine the tensor. This proves both local readability and spatial measurability.

Lemma 49 makes \(T(z)\) deterministic for almost every deterministic \(z\). A coordinate rotation conjugates it by the rotation matrix; the possible orientation sign of \(f\) has disappeared from (54). Applying the rotation part of that lemma to a quarter-turn gives \(T=R^{\mathsf T}TR\). For a symmetric two-by-two matrix this sets the off-diagonal entries to zero and equates the diagonal entries. Since \(\det T=1\), we obtain \(T=I\) almost everywhere. The quasiconformal differential is therefore a similarity almost everywhere, of one fixed orientation. The analytic characterization of quasiconformal maps (Astala et al. 2009, Theorem 2.5.4) makes \(f\) conformal or anticonformal. Isolated marked points are removable. ◻

Proof. For continuous functions \(f_1,f_2,f_3\) on \(\widehat{\mathbb C}\), conditional expectation of their product is \[\mathbb E\!\left[\prod_{j=1}^3 f_j(H_n(x_n^j))\, \middle|\,\text{surface data}\right] =\prod_{j=1}^3\int f_j\,\mathrm d(H_n)_*m_n.\] Weak convergence passes the right side to \(\prod_j\int f_j\,\mathrm d\mu\). Products of continuous test functions determine the law on the compact product sphere, proving the joint conditional assertion. Diffuseness gives distinctness. In a chart avoiding the limiting points at infinity, the normalized Möbius map is the usual cross-ratio expression; its coefficients depend continuously on the distinct triple. The resulting maps and their inverses converge uniformly in spherical distance by compactness. Other chart choices give the same conclusion when one point is infinite. ◻

Proof. An original edge contributes four equal-mass flags and one quadrangle. Choosing a uniform quadrangle and then one of its two matched tour steps gives a uniform step of the exploration. By Proposition 12 and Theorem 17, every point in any of its four flags and either step representative have vanishing discrepancy in the reference coordinate, uniformly over the map. Hence, for every continuous test function \(g\), \[\int g\,\mathrm d(H_n)_*m_n -\frac1{2n}\sum_{j=1}^{2n}g(\eta(j/(2n))) \longrightarrow0.\] The continuity of \(\eta\) and its quantum-area parameterization identify the Riemann-sum limit as \(\int g\,\mathrm d\mu_h\). Repeated incidence labels cause no change: the four flags are counted as occurrences. Thus \((H_n)_*m_n\Rightarrow\mu_h\).

The maps \(H_n\) were chosen before the fresh samples. Apply Lemma 54 to this reference area convergence. The limiting marks have conditional law \(\mu_h^{\otimes3}\), jointly with the retained map and local observations, and are almost surely distinct. Thus both the source marks and their prescribed target images satisfy the normalization hypothesis of the ring criterion.

Conformal invariance identifies the through EL of \((\phi_n\circ H_n^{-1})(A)\) with the actual surface EL for the corresponding annulus. Proposition 51, with strict collars, gives the hypothesis of Lemma 52 simultaneously for the countable annulus library. That lemma gives uniform subsequential convergence to a quasiconformal homeomorphism. Lemma 53 makes it conformal or anticonformal, hence Möbius or anti-Möbius on the sphere. Uniform convergence and convergence of the moving marks preserve their prescribed images. These three images specify the map once its orientation is fixed. All extractions retained the same local arrays and topological correspondence, so no observation acquires a different coordinate. ◻

Proof. Write \(f_n=\phi_n\circ H_n^{-1}\). Uniform convergence and the reference area convergence prove the measure assertion. If \(\omega_f\) is a modulus of continuity of \(f\) on the sphere, then \[\max_T\mathop{\mathrm{diam}}_\rho\phi_n(T) \le2\|f_n-f\|_\infty+ \omega_f\!\left(\max_T\mathop{\mathrm{diam}}_\rho H_n(T)\right) \longrightarrow0\] by (40). Every sequence has a further joint extraction with these conclusions, which proves the stated in-probability assertion about mesh. The three samples and \(\phi_n\) throughout are the prescribed finite ones. ◻

Proof. Use the exact conditional rerooting statement in Lemma 4. A uniform oriented-edge tail can be coupled to a uniform flag by choosing one of its two sides, and then to an independent uniform point of that flag. The two canonical positions differ by at most the maximum flag diameter. Conditional on the root-free map data and the normalization samples, this gives the same limiting continuous-test integrals as sampling from \((\phi_n)_*m_n\). Apply Corollary 56. The resulting area measure is nonatomic. ◻

Proof. If flexible matches interlaced, say \(i<i'<j<j'\), the burger at \(i'\) would still be available when the flexible order at \(j\) takes the older burger at \(i\). This contradicts freshness. Thus the discrete chords are noncrossing, and the same holds for their limits.

We use the full statement of Gwynne–Sun (Gwynne and Sun 2015, Theorem 1.11), with its time interval \([0,2]\) rescaled to \([0,1]\). The theorem is for the empty-reduced word at each positive integer \(n\) and each fixed inventory parameter in \((0,1/2)\). Its second assertion identifies all subsequential limits of flexible orders with nonvanishing interval span; its third assertion gives convergence of both entrance quantities and eventual agreement of the wall label. Its fourth assertion simultaneously approximates maximal cone intervals in every rational-endpoint open interval containing a specified rational interior mark. These statements are stronger than uniform contour convergence.

Apply these assertions first at the adjacent-cut times of the cited theorem. The deterministic one-mesh bound just noted gives the same endpoint limits and wall labels at our occurrence times. In particular, it gives the upper endpoint-pair limit. To check the lower limit explicitly, fix \([a,b]=[v(b),b]\Subset(0,1)\). Choose open intervals \(I_k\) with rational endpoints, containing \([a,b]\), whose closures decrease to \([a,b]\), and fix a rational point of \((a,b)\). The maximal cone interval in \(I_k\) containing that point contains \([a,b]\): cone intervals containing the same interior point are ordered by inclusion. Its endpoints therefore tend to \(a,b\). For each \(k\) the distinguished discrete maximal intervals converge by the cited theorem. A diagonal choice approximates \([a,b]\). Equivalently, first apply the theorem simultaneously to the countable family of rational tests and then take their closure. Compactness of the endpoint-pair space turns the upper and lower conditions into Hausdorff convergence.

There is no additional nondegenerate circular chord at an excursion end. For example, a sequence with one endpoint tending to \(0\) and the other tending to an interior time would force an interior coordinate value to be zero. A sequence ending at \(1\) with its entrance bounded away from \(0\) would do the same, using equality in the entrance-wall coordinate. Both contradict strict interior positivity. The sole possible full-duration interval has equal endpoints on \(\mathbb T\) and is already included among the diagonal pairs. The resulting closed chord system is determined by the excursion. Hence retaining it causes no ambiguity in the joint subsequential couplings: its limiting conditional law given the excursion is a point mass. ◻

Proof. In an unoccupied flag triangle the retained corner is cut off by \(v=2/3\). In an occupied triangle that corner is already contained in the strip \(f\le1/3\), and the boundary is \(f=1/3\). On a common vertex–face-center side these two rules meet at the same point. On the other two side types the occurrences are paired within the same edge quadrangle and have the same occupation status. Consequently the four ports of an edge quadrangle have exactly the usual two FK pairings. They are shown in Figure 2. This verification is occurrence by occurrence, so loops and bridges do not identify distinct ports accidentally.

Starting from \(C_n\), restore the FK pairing at each flexible quadrangle. Cut its two marked visits and reconnect the four cut ends by the other noncrossing pairing. In the abstract disk this is a cut along \([t_i,t_j]_{\rm d}\). Noninterlacing ensures that the chord lies in one current face, and its distinct endpoints divide that face into two. Induction therefore identifies the final cyclic routes with all disk faces, giving \(1+N_{\mathsf F}\) of each. Each reconnection stays in its own quadrangle. The polygonal flag drawing has the same four-port pairings, so it has exactly these cyclic routes in the exposed time-circle order. This argument retains even the smallest finite loops; it does not replace a chord between consecutive occurrences by a diagonal pair.

For the side assertion, realize each switch by a narrow bridge in the disk on the side of its resolved-tree edge. The two burger types use the two opposite disks of \(C_n\). Within either disk the bridges may be chosen disjoint because their endpoints do not interlace. Fix an outer bridge with interval \([a,b]\), its child bridge \([c,d]\), and a reference point on \(C_n(b,a)\). If the bridges use the same disk, the face route bounds the strip between them. The skipped arc \(C_n(c,d)\) and the reference arc \(C_n(b,a)\) lie on the same side of this route. If the child bridge uses the other disk, the skipped arc \(C_n(c,d)\) lies between that bridge and the outer bridge, while the reference arc remains on the other side. Thus the two test points are separated exactly when the bridge types differ; see Figure 5. Subsequent reconnections are disjoint from this face route and cannot change either test point’s component. This proves the criterion. The same conclusion can be checked directly from the detailed component identification in (Gwynne, Mao, et al. 2019, Lemmas 2.7 and 2.9). ◻

Proof. Chord endpoint convergence implies Hausdorff convergence of the underlying laminations. A compact connected subset of \(F\) has positive distance from \(\mathcal L\) and therefore lies in one discrete face eventually. Exhausting \(F\) by such compact subsets gives \(\overline F\subset\liminf\overline F_n\). Every boundary chord of \(F\) is approximated by discrete chords, and \(F_n\) lies on the side of each approximating chord selected by \(x\). The intersection of these closed half-disks is \(\overline F\). This gives the reverse inclusion and proves the first cell assertion.

For a varying sequence, compactness of the space of compact convex subsets of \(\overline{\mathbb D}\) gives a limit \(K\). If \(K\) has interior, no limiting chord can cut that interior: an approximating chord would cut the corresponding discrete face. Choose an interior point of \(K\) outside the lamination. The first part then identifies \(K\) with a face closure. Such a point exists, since otherwise the lamination would contain an open disk in \(K\); a chord through that disk would again cut \(K\), which is impossible.

For the curve assertion in this case, parameterize \(\partial F_n\) and \(\partial F\) by radial bearings from a fixed interior mark. Their radial functions converge uniformly. Indeed an interior disk about the mark is contained in all sufficiently large \(F_n\), and convexity then bounds the radial modulus uniformly; pointwise convergence follows from the cell convergence. These boundary parameterizations preserve cyclic order. On exposed circle arcs, the spatial routes converge to \(\eta\) uniformly. On a chord side, Lemma 59 confines the route to a matched quadrangle, whose image diameter tends to zero. Every subsequential limit of such a route is therefore the common endpoint value of \(Q\). The compactness argument used above to prove continuity of \(Q\) makes this convergence uniform on the parameter circle. Pauses inserted at chord endpoints are harmless: continuous nondecreasing changes of time are uniform limits of strictly increasing changes of time and give the same zero-distance class in the stated loop topology.

Suppose instead that \(K\) has no interior. It is a point or a line segment. A closed convex cell is the convex hull of its contacts with \(\partial\mathbb D\). Extracting those compact contact sets shows that \(K\) is the convex hull of its limiting circle contacts. There are at most two such contacts, since three distinct points of the circle are noncollinear. In the two-contact case, a boundary chord of a discrete cell converges to their connecting segment. For clarity, take a point strictly between the two contacts and boundary points on either side of it tending to the segment; these points are on chord sides, and any limiting side through an interior point of the segment has exactly those two circle endpoints. The endpoints consequently have equal \(Q\) values. All exposed-circle route values and all chord-side route values have that same subsequential limit. Thus the spatial diameters tend to zero.

To prove local finiteness, suppose distinct limiting faces had loop diameters bounded below. Their closures have a subsequential convex limit. The preceding collapsed-cell argument, applied with the fixed lamination, rules out an empty interior. If the limit has interior, an interior disk is eventually contained in every face in that subsequence, contradicting disjointness of distinct faces. Thus only finitely many limiting face loops have diameter above any positive threshold.

Fix \(\varepsilon>0\). Match all limiting face loops of diameter greater than \(\varepsilon/2\) to the discrete faces selected by their interior marks. There are finitely many, their selected faces are distinct for large \(n\), and their cyclic errors tend to zero. If unmatched discrete loops of diameter greater than \(\varepsilon\) remained along a subsequence, Part 2 would produce a noncollapsed limiting face loop of diameter at least \(\varepsilon\). Its interior mark would eventually belong to the unmatched discrete face, contrary to the selected matching. This proves the final assertion, and a diagonal sequence of \(\varepsilon\) gives the claimed loop-ensemble topology. ◻

Proof. First consider a radial branching tree in a proper simply connected domain \(D\), with force point immediately on either side of its boundary root. The alternating construction is the nested-domain construction of (Aru et al. 2022, sec. 2.1.6). Its first full crossing, completed through the separated component, gives an outermost loop. The component of its complement containing a target is exactly the radial target component at the crossing time. Conditional on the explored branches and completions, the unexplored trees in the components are independent radial trees. At the next generation the force side is reversed, as follows from \(\theta\mapsto2\pi-\theta\) and the strong Markov property (Aru et al. 2022, Remark 2.3). Reflection preserves the unmarked domain CLE law. Induction for finitely many targets and generations, followed by the countable union, therefore gives the independent-domain iteration defining nested \(\mathrm{CLE}_{\kappa}\). This also gives consistency at a discovered interior bubble: its descendant loops are exactly those of the continued tree in that bubble, and other loops of the domain ensemble cannot enter its interior. Thus the two crossing directions give consecutive generations in this construction.

We pass to the whole plane using stopping domains. Choose a deterministic target, put it at \(0\), and use the bi-infinite Loewner time of its branch from \(\infty\). Define \[S_m=\inf\{t\ge -m:\theta(t)\in\{0,2\pi\}\},\qquad T_m=\inf\{t\ge S_m:\theta(t)=2\pi-\theta(S_m)\}.\] These are stopping times. Full angular crossings occur arbitrarily far toward the initial end (Gwynne, Miller, and Qian 2021, Lemma 2.7), so \(T_m\to-\infty\) almost surely. Explicitly, fix any earlier full crossing. Once \(-m\) precedes its starting endpoint visit, \(T_m\) is no later than its ending visit: whichever endpoint is hit first after \(-m\), both endpoints have been visited by then.

Let \(D_m\) be the component containing \(0\) after \(T_m\). The tip and force prime end coincide in the endpoint state \(\theta(T_m)\). The whole-plane strong Markov property and conditional independence of branches after target separation imply that the unexplored tree in \(D_m\), conditionally on the exposed branch, is a radial branching tree with the indicated endpoint force side (Miller and Sheffield 2017, Proposition 2.2 and Section 2.1.3). The preceding domain argument shows that its future ensemble \(\Gamma^{(m)}\) has conditional law \(\mathrm{CLE}_{\kappa}\) in \(D_m\). Its curves lie in \(\overline D_m\) and may touch \(\partial D_m\).

We check consistency with the global readout using only these domain ensembles. Fix a globally generated loop \(G\) with target \(w\) and a finite starting crossing. For sufficiently large \(k\), the cut \(T_k\) precedes both the separation of \(w\) from \(0\) and that crossing. Then \(w\in D_k\), its branch shares the exposed prefix, and \(G\) is a loop of \(\Gamma^{(k)}\): every subsequent full crossing and its completion are computed from the same restricted tree. Increase \(k\) so that \(T_k<T_m\) as well. In the domain construction for \(\Gamma^{(k)}\), \(T_m\) is a later full-crossing time toward \(0\), and \(D_m\) is the corresponding interior bubble. Domain consistency therefore says that \(G\), if its trace enters \(D_m\), is a descendant loop generated by the continued tree there, hence belongs to \(\Gamma^{(m)}\). Conversely every loop of \(\Gamma^{(m)}\) is a global crossing loop. There are countably many cuts and generated loops, so these statements hold simultaneously. In particular the global and future readouts agree on all loop tests in a compact subset of \(D_m\), including loops which touch \(\partial D_m\) elsewhere.

The branch tends to \(\infty\) at its initial end. Thus every fixed disk lies in \(D_m\) eventually: for sufficiently negative \(T_m\) the entire exposed past lies outside that disk, whose connected interior belongs to the \(0\)-component. The domain limit theorem (Miller et al. 2015, Theorem A.1) identifies the limit of \(\mathrm{CLE}_{\kappa}\) in such domains with the unique whole-plane law. Its conformal matching estimate is uniform over all proper simply connected domains, with error tending to zero as their boundary recedes from a fixed disk. It applies conditionally to \(D_m\): bounded continuous tests of the loop configurations in a fixed compact set converge to the fixed whole-plane expectation, and dominated convergence removes the conditioning. The local agreement just proved identifies the global alternating readout with this limit. It also transfers local finiteness and the absence of shared arcs directly from a domain ensemble, since every compact set is eventually inside some \(D_m\).

The whole-plane construction in (Gwynne, Miller, and Qian 2021, sec. 2.3.2) uses a positive-crossing subsequence. Its agreement in law with this same large-domain limit is stated in Remark 2.6 of that paper. Consequently its Möbius-invariance theorem (Gwynne, Miller, and Qian 2021, Theorem 1.1) applies to the present unmarked ensemble. ◻

Proof. We give the frontier dictionary in the cut surface, where different prime ends over one spatial point remain distinct. Color the left side of the branch red and its right side blue. The boundary of its current target component consists, in prime-end order, of a red arc and a blue arc between the tip \(W\) and the marked junction \(O\). These are the boundary colors and exploration order of (Aru et al. 2022, sec. 2.1.7 and Figure 9). The force prime end is precisely this junction \(O\): it is transported by the Loewner map until a collision, and the same boundary-color rule continues it after the collision. The two prime-end arcs have strictly positive quantum length when they are nontrivial. Therefore one color is absent exactly when \(O\) and \(W\) coincide on the corresponding side, that is, when \(\theta=0\) or \(\theta=2\pi\).

We use the precise bubble readout of (Gwynne and Miller 2021a, sec. 2.3.4, Definition 2.2 and Figure 4; Lemma 3.1): a filled bubble has interval \([v(b),b]\), its boundary length is \(|Z(b)-Z(v(b))|\), and its side is specified by the coordinate equal at the two endpoints. This statement uses backward cone intervals, with the same entrance convention as here. The following frontier verification explains its use on the sphere. When a monocolored component is entered at time \(a\), the space-filling order visits all its descendants before returning to a different component; write \(b\) for its completion time. Its interior is visited only during this block, although its boundary points may be visited again. The frontier outside this component is unchanged throughout its filling. Subtracting its terminal heights therefore gives the two remaining frontier lengths \[ \bigl(L(s)-L(b),\ R(s)-R(b)\bigr),\qquad a\le s\le b. \tag{55}\] They are nonnegative, both are zero at \(b\), and exactly one is zero at \(a\). The first entry \(a\) is the entrance \(v(b)\): in the running-minimum frontier construction of Theorem 9, extending the block farther backwards crosses one of its two terminal frontier levels.

The converse uses the same frontier construction. For a cone interval \([v(b),b]\), cut the two frontier trees at heights \(L(b)\) and \(R(b)\) and retain its filling block. The horizontal identifications inside the block stay above those cuts. Its unfilled boundary arcs are parameterized by the height intervals in (55); at its first time the arc for the entrance wall is empty. Maximality of \(v(b)\) prevents a preceding block from remaining on these same two sides of the cuts. Thus this is exactly the monocolored component entered there. If the interval contains \(u\), it is the branch’s target component, since its interior contains \(\eta(u)\); if it precedes \(u\), it is an erased off-target filling. This verifies both directions of the readout without imposing a new law on the field inside the component. Erasing earlier off-target fillings does not change these two boundary arcs in the target component. The argument concerns the restricted quotient and its length coordinates on \([a,b]\Subset(0,1)\), which are the same in the sphere realization of Theorem 9; no infinite-volume surface law is being assigned to that block.

Combining (55) with the prime-end description identifies the entrance-wall labels with the two endpoint states of \(\theta\), up to interchanging the colors. Target components shrink along a branch, and their contiguous filling blocks are consequently ordered by reverse inclusion. The exact backward-cone erasure for a target branch is also stated in (Aru et al. 2022, sec. 4.2 and Definition 4.14); the ancestor-free construction underlying it is (Duplantier et al. 2021, Proposition 1.13 and Lemma 10.4).

Starting at either state, the last visit to it before reaching the other is the beginning of a full crossing of \([0,2\pi]\). The alternating constructor therefore gives precisely the stated switches and their between-tips portions. Conditional on that portion, the remaining piece is chordal SLE in the specified prime-end component, as in (Aru et al. 2022, sec. 2.1.6) and (Gwynne, Miller, and Qian 2021, sec. 2.3.2). More precisely, let \(\sigma,\tau\) be the starting and ending branch times of the full crossing, and view the target bubble at \(\sigma\) in its prime-end disk. The tip and force point coincide at its designated boundary root \(x\). The completing branch approaches the later lift of this same \(x\) through the component separated from the target at \(\tau\). The excursion and completion therefore give a closed continuous curve \(\beta\) in the closed prime-end disk, starting and ending at \(x\). Both pieces lie in the larger cone interval’s filling.

Let \(f\) map that disk to the outer bubble. Its continuous extension exists because the boundary is locally connected. The closed curves \(f((1-\varepsilon)\beta)\) lie in the bubble and converge uniformly to the completed loop as \(\varepsilon\downarrow0\). Simple connectedness gives zero winding about every point outside the bubble for each approximant. Winding stability gives the same assertion for the completed loop at every off-trace point outside the bubble. Thus every surrounded typical target is in that bubble.

Iteration supplies exactly the successive nested loops about a target. Their domains decrease to that target. In the whole-plane version they also exhaust toward the initial point. Equivalently, the full crossings of the recurrent angular diffusion occur at all scales; the whole-plane branch is transient at its initial end and tends to its target at its terminal end. For any fixed compact subinterval of \((0,1)\), large enclosing bubbles contain its image. Their fillings therefore contain that time subinterval, and their endpoints tend to \(0,1\).

Finally, for the unconditioned Brownian contour, the two boundary processes after ancestor-free erasure, read in forward branch order, have negative jumps and stable index \(\kappa/4\in(1,2)\), one for each side (Duplantier et al. 2021, Proposition 1.13). This reverses the time direction of the positive-jump processes in that statement. Each Lévy measure has infinite mass near zero, so each coordinate has jumps in every nonempty clock interval. The jumps are precisely the erased cone intervals of the indicated wall type. The time-changed trace is continuous directly: every inverse-clock jump skips an interval \([a,b]\) with \(\eta(a)=\eta(b)\), and \(\eta\) is continuous. This is the inverse-clock argument in (Duplantier et al. 2021, Proposition 10.6). The erasure retains the full ordered branch traversal; the skipped intervals are space-filling visits to off-target bubbles, not moving pieces of that branch. Thus a portion on which the branch changes position occupies a nonempty interval of its moving clock. For the excursion, exhaust by strict interior rational windows containing the full Brownian-time span of the tested clock interval, including all its erased intervals. Local absolute continuity transfers the null event that one label is missing there. Taking the countable intersection gives the assertion for rational and independent uniform horizons. Only this almost-sure property is transferred; the excursion’s time-changed coordinates need not be Lévy processes. ◻

Proof. Two intervals with the same terminal time have the same entrance \(v(t)\), so are equal. Suppose instead that \([a,d]\) is properly contained in \([a,b]\) and has the same entrance. Interchange the two coordinates if necessary so that the outer wall is \(R\): \(R(a)=R(b)\) and \(R\ge R(b)\) on \([a,b]\). If the inner wall is \(L\), its cone inequality and the outer inequality give \(R(a)\ge R(d)\ge R(b)=R(a)\). Hence \([a,d]\) is both an \(L\)-chord and an \(R\)-chord, contrary to Lemma 25(i). If the inner wall is also \(R\), then \(R(a)=R(d)=R(b)\), and \(d\) is a scalar local minimum at that level, strict by Brownian scalar genericity. But \(b\) is a backward joint record with positive witness, so its \(R\)-height cannot be a scalar branch level by Lemma 25(ii). This rules out a shared entrance as well. ◻

Proof. The enclosing chords of Lemma 63 approach the full time interval. An interior point of a face eventually lies between one of those chords and the opposite circle arc. The maximal circle gap of that face containing time \(0\) therefore has endpoints in \((0,1)\). Its bounding chord is the asserted outer chord. Almost every time in \((a,b)\) is inside a child: typical times have arbitrarily small enclosing cone intervals, whereas no cone chord can cut the interior of \(F\). We henceforth impose this fact for the independent uniform targets as well as for almost every time.

Let \([c,d]\) be a child and take a target horizon \(r\in(c,d)\). Ancestor-free erasure toward \(r\), restricted to \([a,c]\), is exactly the face contour. Here is the elementary nesting check. Every child interval before \(c\) is a filling disconnected from that target and is erased. Conversely a deleted cone interval meeting a retained circle contact would have to cross the outer chord or a child chord, unless it were contained in one of the skipped children. Crossing is impossible, and the latter case deletes no further face contact. Endpoint differences are constant pauses. Thus the two procedures retain the same points in the same order.

For an opposite-type child, there is no intermediate containing interval between it and \([a,b]\), by maximality of the child. Lemma 64 gives \(a<c<d<b\). Their closures are consequently the two endpoints of a full-winding excursion in Lemma 63. The preceding erasure check identifies the face prefix with its between-tips branch segment. That segment is nonconstant: a full crossing of the angular interval makes a nontrivial Loewner hull. Its traversal order is the branch order.

Conversely, take the two consecutive nested intervals of a target switch. A chord from the target time on the circle to time \(0\), outside the larger interval, crosses both bounding chords. The open segment between those crossings meets no lamination chord. A chord meeting it would separate these two circle points and therefore give an intermediate interval containing the target. The endpoints of the segment are distinct, since the two bounding chords are distinct and their intersections are in their relative interiors. Closedness of the lamination puts this open segment in one complementary face. That face has the two specified chords on its boundary. This proves existence; the side of the outer chord fixes uniqueness. It also proves that the face has interior even when chords accumulate on either side of the branch.

We next make the exhaustion assertion precise. On a varying compact subarc of \(\alpha_F\) before a later varying portion, choose a child strictly after it and a rational horizon inside that child. Such a child exists because almost every intervening time is in a child, and a child cannot skip two retained contacts with distinct images. The preceding erasure check identifies the earlier subarc with an interval of this horizon’s ancestor-free trace. By Lemma 63, this is the full ordered moving trace. Its portion between distinct spatial values therefore occupies a nonempty interval of the moving clock: an interval collapsed by that clock is a skipped filling whose two endpoint images agree. On this compact earlier part of the trace, the erasure check identifies its retained set with the face contacts, up to endpoint pauses. The jumps of the inverse clock are the maximal open gaps of this retained set, hence precisely the immediate child intervals there. An interval inside an already skipped child is not another inverse-clock jump. The compact restriction before the later child’s entrance removes any cutoff at the target horizon. The last assertion of Lemma 63 therefore gives immediate children of both labels on the chosen varying portion. This use of the stable processes is simultaneous for all rational horizons and all interior windows; the face and subarc may therefore be chosen afterwards.

Set \(x_0=\alpha_F(b)\). Given a sufficiently small \(\delta>0\), let \(t_\delta\) be the last time with \(\rho(\alpha_F(t_\delta),x_0)\ge\delta\). There is a varying portion after \(t_\delta\) and before the contour becomes constant at \(x_0\). Apply the preceding paragraph to a compact part of that variation, leaving some variation after it. An opposite-type child entrance \(c_\delta>t_\delta\) occurs there, and \[\mathop{\mathrm{diam}}_\rho\alpha_F([c_\delta,b])\le2\delta.\] Choosing successively smaller \(\delta\) and later varying subarcs makes the entrances nondecreasing. If a constant terminal interval is present, the entrances approach its first point and the same bound holds. This produces nested nonconstant prefixes with vanishing remaining diameter.

All these prefixes belong to the same CLE loop: each contains the first nonconstant prefix, and distinct CLE loops share no nonconstant arc. They also follow the same oriented traversal from the same starting prime end. No self-tracing prevents switching to a different visit of that initial arc. Parameterize this traversal on \([0,1]\); the prefix endpoints are nondecreasing parameters, bounded by \(1\), and hence converge to a parameter \(s\). Continuity of that traversal and the vanishing remainder diameter identify the face contour with the traversal up to \(s\), allowing pauses. Since the face contour is closed, it is a closed initial portion, as asserted. ◻

Proof. The deterministic step is a local index comparison. Suppose that a closed curve \(C\) and \(L\) share an oriented compact segment \(A\), and write \(L=A*B\), \(C=A*E\). If the complementary arcs \(B,E\) miss a connected ball \(U\), then additivity of degree gives \[ \operatorname{Ind}(L,z)-\operatorname{Ind}(C,z) =\operatorname{Ind}(B*\overleftarrow E,z), \qquad z\in U\setminus L. \tag{56}\] The right side is constant on \(U\). Consequently, if two off-trace points of \(U\) have different indices for \(C\), they also have different indices for \(L\). If \(P\) misses \(U\), its index is constant there and cannot agree with \(L\) at both points. Local constancy off compact traces then gives the disk \(V\) in the statement.

We will apply this comparison with \(C\) a chordal SLE closed along a boundary side. To cover an initial portion chosen after seeing the loop, we first construct one simultaneous family of comparison paths. Every moving open portion of a loop contains an interior segment with law absolutely continuous with respect to a segment of ordinary chordal \(\mathrm{SLE}_{\kappa}\). Indeed each loop is its full-winding branch piece followed by its conditional chordal completion. In the first piece use countably many windows stopped while the driving and force points are separated by at least \(1/m\), in a fixed bounded time window after a strict initial cut. The force drift is bounded there, and coordinate change followed by Girsanov gives the asserted local absolute continuity (Schramm and Wilson 2005, Theorem 3 and Section 5). Force collisions have no time interval. In the completion use its conditional chordal law in the cut domain. Each moving interval contains an interior excursion: a continuous Loewner trace spends zero capacity time on the domain boundary (Yuan 2022, Proposition 1.7). Restrict to a compact subinterval of such an excursion. Enumerate the countably many branches, completions, and stopped windows, and refine them by all rational compact subintervals in these interior excursions. The resulting countable family of comparison segments contains a member inside every moving open portion, including those selected subsequently.

An ordinary chordal SLE in a Jordan disk has open pockets of both passage sides arbitrarily close to each moving interior trace point. To verify the needed two-sided assertion, decorate a quantum wedge by an independent chordal SLE. Its left and right boundary-length processes in quantum natural time are independent spectrally negative stable processes of index \(\kappa/4\) (Duplantier et al. 2021, Theorem 10.1). Both have jumps in every nonempty time interval, and each jump cuts off an open pocket on its designated side. We need this clock to describe the full ordered traversal, as well as to be continuous. The proof of (Duplantier et al. 2021, Theorem 1.16), with both force weights zero, obtains ordinary chordal SLE by welding two wedges of weight \(\gamma^2-2\) to a middle wedge of weight \(2-\gamma^2/2\); the ambient weight is \(3\gamma^2/2-2\). In the middle wedge the interface is a concatenation of \(\mathrm{SLE}_{\kappa}(\kappa/2-4;\kappa/2-4)\) traces. identify its entire ordered counterflow traversal and its continuous quantum-natural parameterization. The inverse-clock jumps skip space-filling visits to bubbles, not moving portions of that counterflow traversal.

The exterior welding in the proof of Theorem 1.16 retains this same interface and the ordered quantum bubbles it encloses. The intrinsic bubble description of quantum natural time in (Duplantier et al. 2021, Theorem 6.22, Definition 6.23 and Remark 10.5) therefore retains its clock, with the fixed deterministic normalization. This construction gives a continuous parameterization of the full ordinary chordal traversal. Every moving portion thus occupies a positive clock interval. Jumps of each stable coordinate in every rational clock interval, together with continuity, imply that closing tips of pockets of both sides approach every moving interior trace point simultaneously. Interior points of these pockets may be chosen arbitrarily close to their tips and remain off the entire chordal trace. These nearby interior points supply the two-sided test. Integrating out the auxiliary field gives the property for the unclocked chordal law.

Close the reference chordal path by one specified boundary side of the disk. Its winding indices on pockets of the two passage sides differ by one. One way to see this without a smoothness assumption is ordered approximation of a continuous Loewner trace by simple traces (Yuan 2022, Theorem 1.1). Truncate away from its terminal point, complete within a shrinking terminal neighborhood, and close on the same boundary side. The Jordan curve theorem gives the side test for each approximant, and uniform convergence preserves winding about any fixed off-trace points. For a cut domain perform this argument in its prime-end disk, using only interior test points. Hence the assertion survives conformal coordinates and the stopped local changes of measure above.

The full ordinary chordal reference law also has strong no self-tracing. For a fixed indexed full crossing, the conditional completion given the explored trunk is ordinary chordal SLE in its remaining component, and its concatenation with the trunk is a CLE loop (Gwynne, Miller, and Qian 2021, sec. 2.3.2). Normalize this component to a prime-end disk. An open portion whose image there lay in its complementary trace would give the same containment in the physical CLE loop. It has an interior moving subinterval by (Yuan 2022, Proposition 1.7); conformal injectivity there prevents its physical image from being constant. This contradicts the strong CLE property stated above. The normalized completion has the fixed ordinary chordal law conditionally on the trunk, so the deduction gives a probability-one property of that law. The same interior argument transfers it to the cut-domain reference paths used here.

Before choosing any initial portion, retain a complete reference path for every member of the countable comparison family. For one member, disintegrate its reference chordal law given the compared segment and weight by the Radon–Nikodym derivative of the actual segment law. Sampling from this conditional extension kernel couples the actual segment with a complete chordal path which agrees with it; the full path marginal is absolutely continuous with respect to chordal SLE. These kernels exist on the continuous-path spaces. Use auxiliary randomness for the countably many extensions, and intersect all their no-tracing and two-sided-pocket events, together with the corresponding events for the original loops. This gives a single probability-one event on the enlarged space.

On that event fix arbitrary \(L\) and \(P\). The omitted moving portion cannot have its entire trace in \(P\), by no self-tracing. It therefore contains a moving open interval whose trace misses the compact trace of \(P\). Choose an already enumerated comparison segment inside it, and let \(c\) be its already retained complete reference path. Thus \(c\) and \(L\) agree on an open parameter segment, and the required properties of both full curves hold before this choice.

For any continuous curve with no self-tracing, the set of repeated parameter times is meagre. To check this, for disjoint compact rational parameter intervals \(J,K\), the set of times in \(J\) whose images lie in the image of \(K\) is closed. It has empty relative interior: otherwise an open portion would have its image contained in the complementary trace, violating the strong no-tracing property. The countable union of these sets covers all repeated times, with the cyclic endpoint convention for \(L\). Apply this to both \(L\) and the complete path \(c\). On their common open segment choose, by the Baire theorem, a time visited only once by each full curve. Let its position be \(x\).

Choose a smaller common compact segment \(A\) around that time. Both complementary arcs are compact and miss \(x\). There is consequently a ball \(U\) about \(x\) which misses these two complementary arcs, \(P\), and the cut-domain boundary. Close \(c\) by its boundary-side arc to obtain \(C\). Inside \(U\), the only portions of \(L\) and \(C\) are their identical oriented segment \(A\). The two passage pockets of \(c\) supply points of \(U\) at which the indices of \(C\) differ by one. The deterministic comparison (56) now supplies the required open disk \(V\).

All subsequent choices are deterministic on the single event just constructed. The conclusion therefore holds for every closed initial portion. Since this conclusion concerns only the original loops, integrating out the auxiliary reference paths gives the asserted probability-one event on the original space. ◻

Proof. We first transfer the finite side test to a fixed face. For each fixed boundary chord of \(F\), the corresponding sides of its approximating faces have the same limiting endpoints. To verify this, take a point in the relative interior of that chord and a transverse line through it meeting \(F\). The endpoints of the resulting convex-cell sections converge. Their supporting chord sides have subsequential endpoint limits on the circle; a limiting side through the chosen interior point must be the full bounding chord, since its line has only two circle intersections. Lemma 58 then gives eventual agreement of their labels.

Fix a typical target in one child and a test time outside the outer interval, using time \(0\) as the reference point at infinity. These locations are off the limiting face contour: by Lemma 65, that contour is an initial portion of a CLE loop, and the targets avoid all CLE traces. For a target time \(u\) strictly inside a child, choose discrete tour times \(u_n\to u\). Convergence of the two child endpoints puts \(u_n\) inside the corresponding discrete child for all large \(n\), and the spatial correspondence sends its tour point to \(\eta(u)\) in the limit. The analogous assertion holds for the exterior reference point. Their positive distance from the compact limiting contour and uniform loop convergence preserve the finite winding test. Lemma 59 therefore implies \[ \operatorname{Ind}(\alpha_F,\eta(u))\ne0 \quad\Longleftrightarrow\quad u\text{ is in an opposite-type child of }F. \tag{57}\] Targets outside the outer interval have index zero. The nonzero index has the orientation sign of the prefix. The statement holds simultaneously for all faces and all the countably many targets; there are only countably many faces, since each contains a rational disk point.

By Lemma 65, a nonconstant face is a closed initial portion \(P\) of one CLE loop \(L\). Conversely every CLE loop has a surrounded open component, hence contains a typical target. Its switch supplies a face with a nonconstant prefix, so no CLE loop is absent from this assignment.

We show that targets surrounded by one fixed \(L\) give the same outer interval. Let \(I_1,I_2\) be their switch outer intervals. Every typical target surrounded by \(L\) lies in each outer bubble by Lemma 63. Typical points have unique visit times, so both intervals contain both target times and are nested. Suppose, for contradiction, that \(I_2\) is a proper subinterval of \(I_1\). Let \(J_1\) be the child of \(I_1\) used by its switch. The first target time lies in the interiors of both \(J_1\) and \(I_2\). These two intervals are therefore comparable, and maximality of \(J_1\) among proper subintervals of \(I_1\) gives \(I_2\subseteq J_1\). Thus the second target also sees the switch with outer interval \(I_1\). The between-tips prefix is identical for both targets in that child, by target invariance and the erasure description. It identifies \(L\) for both targets, since it is a nonconstant shared arc. But the second target already sees the switch with outer interval \(I_2\) producing \(L\), contradicting the once-only nested construction. Hence \(I_1=I_2\). The face on its inner side is unique, so every target surrounded by \(L\) is tested by an opposite-type child of the same face.

It follows from (57) that \(P\) and \(L\) have the same index on every typical target surrounded by \(L\). They also agree on every typical target outside \(L\). Indeed targets outside the outer interval have index zero. A target inside it is almost surely in a child; if it were in an opposite-type child, that switch would produce a loop surrounding the target and sharing the nonconstant prefix of \(L\). It would therefore be \(L\), a contradiction. A same-type child has index zero by the side test. The indices of \(P\) and \(L\) thus agree on the entire dense typical target set. Lemma 66 excludes any nonconstant omitted portion. Hence the face traverses the complete loop in its order.

Finally, two nonconstant faces assigned to the same \(L\) would, by their opposite-type children, supply two outer intervals for targets surrounded by \(L\). We just proved that these intervals coincide; their unique inner face is then the same. This proves multiplicity one and completes the identification. ◻

Proof. Combine Lemmas 58 and 61 with Theorem 67. The topological correspondence is the one of Theorem 17. All limiting chord and face data are measurable functions of the retained excursion, so the identification holds in its existing joint coupling. In particular, it is not necessary to recouple an individual loop separately from the other observations. ◻

Proof. The reference surface is decorated by an independent unclocked space-filling SLE in Theorem 9. Its exploration tree and unmarked CLE are functions of that unclocked curve; changing to the quantum-area clock changes neither. Thus the conditional loop law is fixed. Given the surface and three fresh area marks, \(T\) is deterministic and the loop law remains unchanged by its action: whole-plane \(\mathrm{CLE}_{\kappa}\) is invariant under every Möbius transformation (Gwynne, Miller, and Qian 2021, Theorem 1.1). Taking conditional expectations and then forgetting the unnormalized surface and marks proves the claimed conditional independence in the normalized embedding.

For the convergence assertion, a uniform individual-loop matching of error \(\delta\) becomes a matching of error at most \[2\|\phi_n\circ H_n^{-1}-f\|_\infty+\omega_f(\delta),\] where \(\omega_f\) is a modulus of continuity of \(f\) on the compact sphere. Uniform continuity of \(f^{-1}\) ensures that every loop of prescribed positive diameter in the new coordinate had diameter bounded below in the old one. Hence the complete matchings of Corollary 68 remain complete after this coordinate change. Theorem 55 identifies \(f\) and its single orientation sign, proving the last assertion. ◻

Proof. Existence, uniqueness, locality, and Weyl scaling are (Gwynne and Miller 2021b); conformal covariance is (Gwynne and Miller 2020, Theorem 1.3). Properness is (Dubédat et al. 2020, Lemma 3.8); a proper length space is geodesic, as also noted at the start of Section 4 of that paper. Fixed-pair uniqueness is (Miller and Qian 2020, Theorem 1.2). The precise internal estimate, including its probability and normalization assertions, is (Dubédat et al. 2020, Lemma 3.20, equations (3.66)–(3.67)). Apply it along dyadic radii and Borel–Cantelli to obtain (60). The scale constant is \(r^{\xi Q}\) for the uniquely normalized LQG metric. On a fixed compact chart, local field absolute continuity and Weyl scaling transfer the almost-sure assertion to our other real-field laws away from their insertions. Uniform probability bounds under a changed law require the density control established when that law is used. ◻

Proof. Intersect fixed-pair uniqueness over \(A\times A\) with the simultaneous strong-confluence event of (Bhatia and Kavvadias 2026, Theorem 2 and Proposition 6). In a given open geodesic subarc choose a strictly interior interval \([s,t]\) and \(0<\varepsilon<(t-s)/2\). Proposition 6 supplies metric neighborhoods of its two endpoints such that all geodesics between those neighborhoods contain the original segment \([s+\varepsilon,t-\varepsilon]\). Since \(A\) is also metric dense, choose distinct anchors in these neighborhoods. Their unique geodesic contains the required segment. The event is simultaneous over all starting geodesics; only the anchor pair is selected afterward. For later local applications, one first confines a sufficiently short interior piece below the positive cost of exiting a surrounding buffer. Locality and local absolute continuity then transfer this whole-plane statement to that piece. ◻

Proof. By Theorem 70, in each fixed buffered geometry the internal metric induces the Euclidean topology, positive separations between disjoint compact ports are positive, and the required confined connection costs are finite. Finite maxima, positive minima, and moduli of continuity therefore admit high-probability finite cutoffs and positive lower cutoffs. Only internal metrics in the stated dependency buffer are used. Under (35), the coordinate and constant identities for this reference metric give \[D_{h^{c,r,v}}(u,u') =v^{-p}D_h(c+ru,c+ru')\] with the corresponding internal domains. These are the ordinary unit-coordinate regularity events at every label.

For \(f\in C_c^\infty(E^\circ)\), the pairing \((h,f)_\nabla\) is a centered Gaussian of variance \(\|f\|_\nabla^2\) under the real reference law. Constants pair to zero; Dirichlet norms and pairings are invariant under affine coordinate rescaling. The same holds for the centered imaginary field. Gaussian tails and a finite union bound give any required common probability cutoff. On \(|(h,f)_\nabla|\le T\), the Cameron–Martin factor \(\exp((h,f)_\nabla-\|f\|_\nabla^2/2)\), and its reciprocal, have fixed finite bounds. These are field events and are adjoined to the local output tests without changing their kernels. ◻

Proof. Fix the annulus cover before taking a discrete approximation. Use the strictly buffered upper tests to choose all its simple primal circuits simultaneously on that approximation, with costs at most \(Lw_i+o_n(1)\), and denote their filled sides by \(\Omega_i\). Transfer them to the reference sphere by \(H_n\). For large \(n\), each specified central hole lies strictly inside its \(\Omega_i\), and \(\Omega_i\) lies in the corresponding outer disk. The errors of \(H_n\) are smaller than the fixed clearances. In the estimates below we may first use any \(a'<a\) to absorb those errors, and let \(a'\uparrow a\) after taking the limit.

Starting at \(P(0)\), select a hole containing that point and follow \(P\) until its first encounter with the boundary of the selected filled circuit. At that encounter select another hole containing the point, and repeat, stopping when the endpoint is reached. If an index is chosen several times, retain its occurrences separately. Write \(\Omega_1,\ldots,\Omega_m\) for this ordered list and \([s_j,s_{j+1}]\) for the corresponding path intervals. Each interval except possibly the last has \(D\)-length at least \(a'w_j\), and its image lies in \(\Omega_j\). These intervals have disjoint interiors, so \[\sum_{j<m}w_j\le (a')^{-1}\operatorname{len}_D(P).\] The finite cover has a positive minimum weight, so there are only finitely many complete intervals, bounded uniformly in the approximation by \(1+\operatorname{len}_D(P)/(a'\min_iw_i)\). Thus all summed \(o_n(1)\) cost errors vanish at this fixed cover. The point \(P(s_{j+1})\) belongs to \(\partial\Omega_j\cap\operatorname{int}(\Omega_{j+1})\). Consequently \(\Omega_{j+1}\) cannot be contained in \(\Omega_j\).

We reduce this list in its original order. Keep an ordered list of retained disks. When a new disk contains the last retained disk, delete that last disk, assigning its path interval to the new disk, and repeat until no such containment remains. The new disk cannot be contained in the preceding retained disk: otherwise the last disk just deleted would have been contained there as well, contrary to the defining property of the retained list. The new disk also overlaps the preceding disk in its interior, since it contains the deleted disk whose interior overlapped that preceding disk. Thus consecutive retained disks have intersecting interiors and neither contains the other. Their Jordan boundaries therefore intersect. These are intersections of actual primal circuits, so they contain common primal vertices. No transverse-crossing claim about limiting circuit traces is needed.

Choose a common vertex for each consecutive pair. On each retained circuit follow one arc between its two selected vertices; on the first and last circuit choose the free endpoints arbitrarily. If there is only one retained circuit, both endpoints may be chosen there. The resulting primal walk uses at most the full cost of each retained circuit, and hence at most \[L\sum_{j=1}^m w_j \le \frac{L}{a'}\operatorname{len}_D(P)+L\max_iw_i.\] Each retained disk has spatial diameter \(O(r_*)\), contains all path intervals assigned to it, and the assignments remain consecutive in path order. Thus the walk’s endpoints are within \(O(r_*)\) of the prescribed endpoints, and its circuits lie in any fixed open tube around \(P\) once \(r_*\) is sufficiently small.

This ordered reduction also retains finitely many intermediate ports. Assign a marked reference time to the retained disk containing its path interval. The entire corresponding circuit is within \(O(r_*)\) of its marked reference point. Choose each marked vertex to be the incoming intersection on its circuit occurrence, or the free initial vertex on the first one. Several marks assigned to one disk may use the same vertex; their order is then preserved without an additional traversal. For an individual subpassage, all retained circuit intervals strictly between its two end marks are charged to disjoint reference intervals within that subpassage. At most the two circuits carrying the end marks are not so charged. Its cost is therefore at most \[\frac{L}{a'}\operatorname{len}_D(P|_{[t,t']}) +2L\max_iw_i.\] The full-path bound above still has its single terminal term. In applications the number of prescribed marks is fixed and the covers are refined with \(\max_iw_i\to0\), so these boundary errors fit any assigned positive subpassage tolerances. All intersections and joins were made on one discrete surface. Pass to the retained finite records only after this construction, then let \(a'\uparrow a\) and refine the covers.

For the lower bound, apply the same stopping construction to a discrete path, stopping on the fixed circles beyond the tested bands. Thus every complete step contains a full band traversal. Mark in every complete step its traversal of the tested band. The marks occur on disjoint ordered path intervals. A budget \(T\) allows at most \(T/(\ell\min_i w_i)\) complete steps; thus they can be retained under subextraction. If \(I\) is the set of selected complete steps, then \[\operatorname{cost}\geq \ell\sum_{i\in I}w_i.\] Apply the same chronological deletion to the filled intermediate reference circles, retaining a chain of intersecting boundaries. Connect the successive circle-intersection points within their respective buffers, using the internal-diameter bounds. This gives a reference route of length at most \(A\sum_{i\in I}w_i\). The first and last reference connections have Euclidean displacement at most a fixed multiple of \(r_*\); their costs are \(o_{r_*}(1)\) by uniform continuity of \(D\) on the compact neighborhood. Hence \[D(x,y)\leq A\sum_{i\in I}w_i+o_{r_*}(1).\] The last, possibly incomplete, disk contributes only an endpoint error: its reference circle is approached at the last complete step, and its remaining Euclidean displacement is bounded by its diameter.

For discrete approximations, keep the finite cover, the strict band margins, and all stopping marks fixed first. The number of marks is bounded by the displayed budget. The subrecord property transfers the traversal lower tests, and the attained-witness property transfers the circuit upper tests. Robust separation and intersection hold before passing to a limit. Finally refine the annulus covers. This order of limits does not require any estimate for costs from arbitrary lattice endpoints. ◻

Proof. For a whole-plane field, Lemma 3.20 of (Dubédat et al. 2020), equations (3.66)–(3.67), gives the internal-ball bound with any \(0<\chi<\xi(Q-2)\), with polynomially high probability along dyadic scales. Borel–Cantelli gives (61); local field absolute continuity transfers its almost-sure version to the ordinary charts used here. Quantitative cutoffs are taken in the aligned reference units before this transfer.

For completeness, let \(R\) be a closed rectangle with bounded aspect ratio, \(z\in R\), and let \(F_t(z)\) clamp the two coordinates of \(z\) to the rectangle whose sides are shifted inward by \(t\). Put \(t_n=(31/32)^nt\). Then \[|F_{t_n}(z)-F_{t_{n+1}}(z)|\leq \frac{\sqrt2}{32}t_n, \qquad \mathop{\mathrm{dist}}(F_{t_n}(z),\partial R)\geq t_n.\] The ball used in (61) therefore lies strictly inside \(R\). Connecting successive clamped points and summing a geometric series gives an interior path from \(z\) to \(F_t(z)\) of length at most \(H_1t^\chi\), including a continuous extension at \(z\). For \(x,y\in R\), use \(t\) equal to the smaller of \(|x-y|\) and one quarter of the shorter side. The segment between \(F_t(x)\) and \(F_t(y)\) lies in the \(t\)-interior of \(R\). Divide it into steps of size at most \(t/8\) and use (61) on each step. Bounded aspect ratio bounds the number of steps when \(|x-y|\) is comparable to the side length; otherwise it is bounded absolutely. Adding the two inward paths proves the asserted modulus. Every constructed path lies in the closed rectangle, so enlarging the rectangle by an arbitrarily thin open collar cannot increase its cost. ◻

Proof. Choose a fixed sufficiently fine angular subdivision of a small Euclidean circle. Lemma 75 connects consecutive vertices inside balls contained in its annular neighborhood. The result is a rectifiable reference closed path of winding one, with length at most \(H_2r^\chi\). The same construction works uniformly for centers on fine meshes and, by strict hole margins, surrounds every point of the compact set. Follow its successive pieces using the upper comparison, with transverse overlapping ports. The topology comparison of Corollaries 19 and 20 gives a surrounding primal walk and hence a simple surrounding subcircuit, with cost at most \(CH_2r^\chi+o(1)\). Take countably many gridded covers and then let \(r\) decrease.

To join two nonconstant certificates ending at \(z\), choose a small surrounding circuit which both certificates must meet. Cut them at their encounters with the circuit and join along that circuit. The added cost tends to zero. This proves the triangle inequality; cases with a constant subpassage use \(F_U(z,z)=0\). Symmetry comes from reversing primal paths. Following internal reference paths proves \(F_U\leq CD(\cdot,\cdot;U)\). Consequently, for endpoints in a compact subdomain, \[|F_U(x,y)-F_U(x',y')| \leq C\bigl(D(x,x';U)+D(y,y';U)\bigr).\] Lemma 75 gives the local continuity asserted here.

In a tangent extraction the reference field and its internal metric are retained with their fixed joint law. Their compact moduli are therefore tight. A strict low-cost witness uses finitely many compact ports and retains its cost slack. Conversely, a strictly cheaper certificate in a proposed saturated limit has a finite attained record in a strict collar. In preceding laws, correct its endpoint errors by the circuits just constructed. Their costs tend uniformly to zero on compact reference-modulus cutoffs, contradicting the preceding saturation inequalities. First fix the strict collar and finite record, next take the extraction limit, and finally decrease the endpoint tolerances. This proves the stated stability without requiring costs to prescribed lattice endpoints. ◻

Proof. Use the ordered stopping construction of Lemma 74 and mark a crossing of a thinner middle band in each complete step. Its reference advance is at least \(a_1w_i\) for a fixed \(a_1>0\). The marked intervals are disjoint. Connecting the outer reference circles as in that lemma gives \[ D(x,y)\leq A_1\sum_iw_i+o(1) \tag{62}\] with a fixed \(A_1<\infty\).

For the upper assertion, run the construction along a reference geodesic. Replace each marked interval by its improved certificate, and retain the upper comparison \(C\) on the intervening pieces. At this fixed cover there are only finitely many gaps. The complete ordered upper comparison follows them with assigned cost tolerances whose sum is arbitrarily small; Lemma 76 supplies the joins with equally small total error. These tolerances are chosen after fixing the marked cover, so no error proportional to its possibly growing number of marks is left when the cover is refined. The saving is at least \(\delta a_1\sum_iw_i\). Use (62) and then refine the covers. For instance any \(0<\varepsilon_+<\delta a_1/A_1\) is admissible after the strict losses. For a general rectifiable path in a prescribed open tube, its compact range has positive reference distance from the complement of a slightly smaller buffered tube. Partition it into pieces shorter than that distance. Reference geodesics joining the consecutive endpoints then stay in the tube and have total length no greater than the original path. Apply the preceding construction to those pieces, keeping the specified ordered ports. This proves the complete upper-following assertion.

For the lower assertion, fix a normalized spatial length \(h>0\) much smaller than the thinner band’s remaining crossing distance and its clearance from the retained band collar. Use squares of half-side \(h\) centered on a grid of mesh \(h/4\). The starting point of each marked crossing lies within \(h/8\) in each coordinate of one such center. Every square so selected stays in the retained band collar, and the crossing must exit it. Suppose its first exit is on the right side \(x=c_x+h\). Retain the subpassage after its last crossing of \(x=c_x+h/2\) before that exit. It stays in the exact rectangle \[[c_x+h/2,c_x+h]\times[c_y-h,c_y+h]\] and joins its opposite vertical faces. The other three exit sides give the corresponding rectangles. These form a finite library of aspect-four rectangles; corner exits are allowed. On discrete approximations, round the stopping locations to nearby vertices and use the strict face-port and collar margins, then retain their limits. The ordered subpassages remain disjoint. The stipulated finite-library cutoffs give reference endpoint distance at least \(a_2w_i\) in the retained collar, with fixed \(a_2>0\). Apply the lower comparison \(c\) on every gap and the improved comparison on each marked rectangle. The reference triangle inequality for these finitely many ordered endpoints yields \[\operatorname{cost}\geq cD(x,y)+(c_1-c)a_2\sum_iw_i.\] If \(c=0\), the strictly positive rectangle charge itself bounds the number of complete steps at a fixed mesh. If \(c>0\), the same bound holds and the lower comparison can also be subdivided intrinsically. Thus the required marked subextractions exist in either case. Equation (62) gives any \(0<\varepsilon_-<(c_1-c)a_2/A_1\) after the strict losses.

Finally, the simultaneous annulus theorem makes these constructions available about every path location with probability tending to one along the specified covers. Countable exhaustion of paths by ports, then of chart compacts and tolerances, makes the improved comparison an almost-sure statement. No rate of discrete convergence is used: the finite cover is always fixed before its discrete approximation. ◻

Proof. Fix \(\varepsilon\in(0,p)\) and a sufficiently high-probability lower traversal test in a buffered band. Start at \(j_0\) with cost unit one and, at level \(j\geq j_0\), use the denominator \[t_{j,j_0}=4^{(p-\varepsilon)(j-j_0)}.\] At each fixed offset \(j-j_0\), a bounded number of primal edges makes vanishing spatial advance as \(j_0\to\infty\). Thus the traversal lower test passes, with any fixed cost and probability slack, on arbitrarily long initial runs.

Suppose first failures existed for arbitrarily large starts. Their distance from their starts tends to infinity. Extract the joint local observations at those failures, retaining every fixed preceding offset, with denominator \(t_{j,j_0}\). At offset \(i\) the passing lower test gives a barrier proportional to \(4^{-(p-\varepsilon)i}\) in the current units. The reference metric unit there is \(4^{-pi}\). Their ratio is therefore \(4^{\varepsilon i}\). On labels \(i\in[N,2N]\), the simultaneous annulus theorem and the lower part of Lemma 74 give a lower comparison at least \(c_0 4^{\varepsilon N}D\), with \(c_0>0\) independent of \(N\). Letting \(N\) tend to infinity says that every fixed separated traversal has infinite cost in the extracted law. It strictly passes the original lower test. The continuous buffered-test convention then contradicts that these scales were failures.

Consequently a sufficiently late start has no later failure. A circuit in the normalization band contains a traversal of one of the strict lower test bands. The lower probability cutoff was chosen stronger than needed for the quantile level in (63); hence, for all sufficiently large \(j\), \[A_j\geq c_\varepsilon 4^{(p-\varepsilon)j}\] with some \(c_\varepsilon>0\). This eventual lower bound will also ensure that a single edge has vanishing cost in \(A_j\) units. Letting \(\varepsilon\) decrease to zero proves the asserted limsup. ◻

Proof. The quantile test at offset \(i\), transported to its ordinary field chart, supplies a high-probability circuit bound proportional to \(A_{j-i}/A_j\leq K4^{-pi}\). Take the quantile sufficiently high that the simultaneous annulus theorem applies after the fixed field density changes and reference regularity cutoffs. Its weights are \(w_i=4^{-pi}\). The upper part of Lemma 74 gives a finite upper constant. If the optimal upper constant were zero, follow a reference circuit in the narrower normalization band at arbitrarily small cost and stitch its finitely many transverse pieces. The reference circuit length is finite with arbitrarily high probability. The middle normalization test would then cost less than, say, \(1/2\) in \(A_j\) units with probability greater than \(\tau\), contradicting (63). Thus \(C>0\).

Define \(C\) as the infimum of deterministic almost-sure upper constants and \(c\) as the supremum of deterministic almost-sure lower constants. Use countably many rational ports, chart compacts and tolerances in these definitions. Intersect the probability-one events for constants decreasing to \(C\) and increasing to \(c\); the compact certificate extraction and arbitrary positive cost losses show that the limiting constants themselves hold. Symmetry and the existence of finite nonconstant upper passages imply \(c\leq C\).

For a volume factor \(v\), the same discrete map with time unit \(v4^j\) and denominator \(A_jv^p\) has field input \(h-\gamma^{-1}\log v\) and costs \(v^{-p}\) times the original costs. This is the joint array re-expression of Proposition 39. It does not identify two differently indexed kernels. The ordinary field laws on a strict patch are equivalent under the fixed shift, so the probability-one comparisons retain the same constants in these units. Further tangent extractions retain their inequalities by Lemma 76 and the ordered lower subrecords. ◻

Proof. Choose the band geometry and high-probability reference regularity cutoffs first. Choose \(\eta\) below the failure threshold in the simultaneous annulus theorem for a label set of density one quarter. If the first assertion failed along arbitrarily large windows, at least one quarter of their labels would have no nearly saturated witness with probability greater than \(1-\eta\). On these labels every marked reference geodesic traversal has an alternative at ratio below \(C-\delta\), with the retained strict open loss. The first part of Lemma 77 decreases the upper constant by a positive amount, contradicting its optimality. The probability threshold used in this argument depends on the annular geometry and its reference regularity, not on \(\delta\); only the required fineness and cost losses depend on \(\delta\).

If the shortcut assertion failed, a quarter of the labels would, with the requisite high probability, have no low-ratio rectangle witness. Every marked rectangular subpassage would then cost at least \(c_1D(\cdot,\cdot;\Omega)\). The second part of Lemma 77 strictly increases the lower constant. This is a contradiction also when \(c=0\). Start with an intermediate ratio between \(c\) and \(c_1\) if a strict collar or cost loss must be absorbed. The finite rectangle construction in that lemma supplies the fixed positive spatial separation. Reference diameter and modulus cutoffs, taken with probability loss less than \(\eta/2\), make all retained lengths finite and separated from zero in the normalized units. ◻

Proof. Let \(\delta_k\downarrow0\). The two sets of labels in Lemma 80 intersect. Choose indices \(i_k\to\infty\) in these intersections and put \(v_k=4^{-i_k}\). This choice is deterministic, since it uses probabilities under the common normalized input law. Extract the unconditioned joint laws, including every fixed finer offset and the reference internal metrics, by the restricted diagonal construction of Proposition 40. The discrete indices are chosen so that the effective time unit \(n_kv_k\), and every retained fixed multiple of it, tend to infinity. Only after this joint extraction do we retain the positive-probability witness events. On a fixed reference-modulus and length cutoff, the nearly saturated geodesics have equicontinuous arclength parametrizations, lie in the fixed compact band, and have separated endpoints. Extract their paths and the internal metric on their fixed open collar. A strictly cheaper limiting certificate would be attained with strict ports in the preceding laws, and its endpoint errors could be corrected by Lemma 76. This contradicts the near-saturation inequality. The limiting arc is therefore saturated. The same extraction keeps the fixed-gap shortcut witness by its strict cost and spatial margins. Its upper optimal constant is \(C\), since the inherited upper bound holds and saturation forbids a smaller one. Its lower optimal constant is strictly below \(C\); applying Lemma 80 in this law supplies a fixed positive density of still finer labels with shortcut ratio at most \(C-\delta_0\), for one \(\delta_0>0\).

Take a short strictly interior subarc of the saturated arc. It is an ambient reference geodesic: when an internal collar was used to take the extraction, shorten the arc until its length is below the positive cost of leaving a still larger collar. We are in the exterior-pinned whole-plane experiment fixed above. Proposition 6 of (Bhatia and Kavvadias 2026) applies simultaneously to every reference geodesic. It gives endpoint neighborhoods such that geodesics joining points in those neighborhoods contain a nondegenerate interior segment of our arc. Shrink these endpoint neighborhoods so that short internal connections to the arc, followed by the arc, cost less than the positive cost of leaving the larger collar. Every joining geodesic then stays in that collar. Thus the anchor selection uses only the protected internal metric, as required for the local field-law transfer. Choose the endpoints from a fixed countable Euclidean dense set. They are metric dense as well. Fixed-pair uniqueness holds simultaneously for the countably many pairs by (Miller and Qian 2020, Theorem 1.2). Thus the saturated arc contains a nondegenerate segment of one of these unique anchor geodesics.

Saturation passes to its subarcs: otherwise splice a cheaper subpassage with the upper comparison on the two remaining pieces and contradict saturation of the larger arc. Choose rational fractional times inside this segment and a rational open neighborhood with compact clearance. Countably many anchor pairs, rational intervals, neighborhoods and length cutoffs cover the positive-probability saturation event. One fixed choice has positive probability, giving (65). ◻

Proof. First restrict the positive-probability unit shortcut test so that its traversal order is determined inside a fixed finite imaginary buffer. The finite-order localization and whole-plane flow-tree determination in Lemma 37 allow the omitted probability to be less than half the original success probability. Fix this buffer before packing its copies. The resulting unit test has probability at least some \(\eta_0>0\).

Inside the chosen shell, pack \(k\) smaller copies with mutually disjoint full buffers. Their relative shape is fixed; their spatial scale decreases when \(k\) increases. Decompose both fields into fresh Dirichlet parts and exterior harmonic parts on those buffers. After subtracting the coarse real value and the imaginary phase, the harmonic oscillation on a strict inner buffer has a scale-uniform Gaussian bound. Choose a bound \(H\) so that, with probability at least \(1-\varepsilon/4\), at least \(k/2\) compartments have the required harmonic bound. Indeed the expected fraction failing any one bound is uniformly small, so Markov’s inequality gives this conclusion; independence of the harmonic parts is not used.

Here is the exact coordinate and volume compensation. If the parent annulus has center \(c\), radius \(r\), and label \(Mv\), its normalized real input is \[g(x)=h(c+rx)+Q\log r-\gamma^{-1}\log(Mv).\] For a compartment \(x=d+az\), with \(a>0\), its input at label \(v\) after the preservation shift \(b\) is \[ g(d+az)+Q\log a+\gamma^{-1}\log M+b. \tag{68}\] Write \(H_c\) for its exterior-measurable coarse value in \(g(d+a\cdot)\). Round \(-H_c-Q\log a-\gamma^{-1}\log M\) to the fixed mesh for \(b\). The residual is bounded uniformly; the absolute label \(v\) has canceled. This chooses the field of a new auxiliary draw. By (30), \(O_v=v^{-p}O\) on the same incidence array, and the reference metric has the identical scalar re-expression. Thus a unit-index shortcut has actual cost \(u\) times its unit cost, with \(u=v^p\), and its preservation kernel is exactly the kernel for \(h+b\). No change-of-field rule for a fixed passage cost is used.

On a compartment whose harmonic profile is bounded by \(H\), bounded Cameron–Martin extensions and the bounded residual shift give a uniform conditional lower probability \(p_0>0\) of success, as in Lemma 43. To see the direction of this estimate, restrict the reference harmonic profile to a fixed bounded set still carrying positive success probability. Between that restriction and any allowed target profile the Gaussian likelihood ratio has a uniformly bounded second moment. If \(L\) is the likelihood ratio and \(E\) the retained unit event, \[\mathbb P_{\rm ref}(E)^2 \leq \mathbb P_{\rm target}(E)\, \mathbb E_{\rm target}[L^2].\] This bounds the target probability below uniformly. The same calculation applies to the imaginary harmonic part modulo its phase. Conditional passage kernels do not change the likelihood ratio. The constants depend on \(\eta_0,H\), the fixed buffer shapes and the residual bound, and not on the absolute size of the compensating shift or on \(k\).

Now choose \(k\) so large that \((1-p_0)^{k/2}<\varepsilon/4\). Conditional on the exterior fields, the fresh fields and the passage draws in these disjoint buffers have product laws. Thus all good compartments fail with probability at most this amount. With this finite geometry fixed, truncate the coarse compensation values to a bounded range, losing probability less than \(\varepsilon/4\). The possible mesh values now form the finite set \(\mathcal B\). For each \(b\), sample its full joint kernel coordinate, retaining the passage records, reference metrics, all required volume offsets, and feasible-order outputs under the shifted local input. Given the fields, different prospective \(b\) coordinates may be sampled independently; their within-coordinate consistency is preserved. The disjoint-compartment arrays have product conditional laws. All these draws precede selection of a successful coordinate. The finite-order data and their directed local traversals are retained at the same time. Additional reference regularity cutoffs on these finitely many fixed geometries can be chosen to lose the remaining \(\varepsilon/4\).

This selection order is essential: the profile bound fixes \(p_0\) before the number of compartments is chosen; the finite shift menu is truncated afterward. Uniformity comes from the bounded residual and the exact array re-expression, not from continuity of passage costs in a real-field shift. ◻

Proof. The proof concerns the reference metric and a retained passage certificate. A Weyl identity is used only for \(D^f\). Choose \[0<e<\min\left\{\frac{A'u}{100}, \frac{\delta_0a'u}{16(C+1)}\right\}, \qquad B_0=2A'u.\] All individual approximate-path and gluing errors below are allocated smaller fixed fractions of \(e\).

Landing and terminal caps. Let \(x_-,x_+\in\partial B_2(0)\) minimize the two exterior landing costs. One can equivalently use approximate minimizers within the reserved error. For landing sets within \(\tau\) of those minima, the reference triangle inequality and (69) give mutual distance at least \(aw-2\tau\). The reference upper modulus therefore separates these sets by a uniform positive Euclidean amount on the cutoff event. Choose two disjoint outer caps around \(x_-,x_+\), with all original connection errors less than \(e\). Around the shortcut endpoints \(z_0,z_1\) choose disjoint terminal caps whose internal \(D_{h+b}\) connection errors to the respective endpoints are less than \(e\) in \(\Omega\). Choose outer stops just inside \(\partial B_2(0)\) and terminal stops just outside \(V\). Decrease the margin of \(V\) below the terminal-cap scale. Lemma 75 provides the uniform connection estimates needed for this operation, including when an endpoint lies at a corner.

The allowed regions; see Figure 6. Join the outer stops to their corresponding terminal stops by two disjoint simple polygonal arcs outside \(V\), avoiding the central omitted disk and the opposite caps. To construct them, first join the outer boundary component to the rectangular boundary component. Cutting along this arc leaves the planar domain connected, so the second prescribed joining arc can be drawn there. Fix their short end approaches before thickening the arcs.

Use two nested thickenings to obtain a smaller allowed set \(E_0\) and a larger set \[E_1=O\cup T_0\cup S\cup T_1.\] Here \(O\) contains the exterior and a thin inner collar of \(\partial B_2(0)\); the chamber \(S\) contains \(V\) with \(\overline S\subset\Omega\); the tubes \(T_0,T_1\) are disjoint; \(O\cap S=\varnothing\); and the only overlaps are \[O\cap T_j\subset\mathcal A_j, \qquad S\cap T_j\subset\mathcal B_j, \qquad j=0,1,\] where \(\mathcal A_j\) and \(\mathcal B_j\) are the outer and terminal caps. The relevant overlaps have closures in slightly larger caps with the same error estimates. The smaller set has positive clearance from the complement of the larger set along every interior boundary. An outer wall layer lies between them, compactly in the permitted support, so exiting \(E_1\) from the intended route requires crossing a fixed positive-width wall.

Inside each tube take a closed core \(K_j\), a finite union of overlapping rectangles of positive width, connecting its two stops. It is separated from \(O\), \(V\), and the wall layer. Its old internal diameter is bounded by \(Kw\) on a fixed-domain cutoff. A transverse marker slab spans the whole tube away from the caps. Its two longitudinal faces have a fixed positive separation. Every complete tube transit crosses this slab.

A smooth function with the required plateaux. After fixing this geometry, let \(m_*w>0\) be a lower cutoff for a crossing of any wall or marker slab. Choose \(H\geq\max\{0,b\}\) and \(J\geq\max\{0,-b\}\) such that \[e^{\xi H}m_*w>2B_0, \qquad e^{-\xi J}Kw<e/3.\] Next choose \(\rho>0\) so small that a point within \(\rho\) of a core can access that core at old cost less than \(e e^{-\xi H}/3\) in a slightly larger local collar. Keep these core collars disjoint from \(O,V\), and the wall. The choice order is \(H\), then \(J\), then \(\rho\); the lower marker cutoff uses its fixed longitudinal faces and does not depend on \(\rho\).

Choose a smooth baseline \(g\) which is nonnegative on \(O\), at least \(b\) on \(S\), exactly \(b\) on \(V\) and the smaller terminal caps, zero on the smaller outer caps and near both support boundaries, and exactly \(H\) on the walls and markers. It can be required to lie between \(\min\{0,b\}\) and \(H\). These closed plateau requirements have positive clearance, so smooth cutoff functions construct \(g\). Let \(\chi_j\) be smooth, equal to one on \(K_j\), supported in its \(\rho\)-collar, with the two supports disjoint. Set \[ f=(1-\chi_0-\chi_1)g-J(\chi_0+\chi_1). \tag{70}\] Then \(f=b\) on \(V\), \(f=0\) outside the permitted shell, and \(f=-J\) on each core. It is nonnegative on \(O\), at least \(b\) on \(S\) outside the tubes, and equals \(H\) on the wall. On a marker it equals \(H\) except within the \(\rho\)-collar of the core. Outer cap hops have multiplier at most one, and terminal cap hops have multiplier at most \(e^{\xi b}\).

Reference forcing. Follow the original exterior minimizing paths, the cheap cores and their cap hops, and an internal connection across \(V\). This gives a modified competitor of length at most \[ l_-+l_++A'u+2e<l_-+l_++B_0. \tag{71}\] For any modified anchor geodesic, the portion between its first and last visits to \(\partial B_2(0)\) has length at most \(B_0\), since the two exterior portions cost at least \(l_-,l_+\). This local portion cannot cross the high wall, and hence stays in \(E_1\). Every complete tube transit must approach its core within \(\rho\) while crossing the marker; otherwise it pays more than \(B_0\) at height \(H\). At such a marker point \(p_j\), use the access to the core, its cheap internal connection, and the terminal hop. These give \[ D^f(p_j,z_j;\hbox{local collar})<e. \tag{72}\]

The first outer cap visited must be \(\mathcal A_0\). If it were \(\mathcal A_1\), the prefix lies in \(O\), where its old length is no larger than its modified length. Joining its last point to \(x_+\) inside that cap and following the old exterior path to \(z_+\) would give an old anchor connection of length at most \(l_-+l_++B_0+e\), contrary to (69). The last outer cap is similarly \(\mathcal A_1\).

Take the first visit to \(\mathcal A_1\) after the first visit to \(\mathcal A_0\), and then the last visit to \(\mathcal A_0\) before it. The intervening open path avoids both outer caps. The allowed-set overlaps imply that it lies wholly in \(O\) or wholly in \(T_0\cup S\cup T_1\). The first possibility again gives an old competitor, now with at most two cap errors, of length \(l_-+l_++B_0+2e<l_-+l_++aw\). Thus there is an inner transition from \(T_0\) through \(S\) to \(T_1\).

On that transition, take the first terminal-cap-\(1\) visit and the last terminal-cap-\(0\) visit before it. Between them the path lies in \(S\) outside both tubes. Its shift is at least \(b\), and the cap endpoints are each within internal \(D_{h+b}\) distance \(e\) of their respective \(z_j\). The central segment therefore has modified length at least \(d-2e\). Select marker points \(p_0,p_1\) on the complete tube transits before and after this segment. As they belong to a geodesic, \[D^f(p_0,p_1)\geq d-2e.\] This last/first selection incurs at most one error per cap regardless of repeated excursions.

Passage gain and finiteness. Use (72), the output upper comparison, and the retained shortcut certificate. Concatenating with its strict margins gives \[F^f(p_0,p_1) \leq (C-\delta_0)d+2Ce \leq C D^f(p_0,p_1)-\bigl(\delta_0d-4Ce\bigr).\] The final parenthesis is at least \(\delta_0a'u/2\), allowing the reserved open losses.

Here is the order that makes the recipe family deterministic. Fixing \(M\) and the amplification geometry fixes \(e/w\) and \(B_0/w\). The initial reference-modulus cutoffs give deterministic cap radii and positive endpoint separations. All endpoint and rectangle parameters therefore lie in a deterministic compact family with fixed aspect and boundary-clearance bounds. Each valid polygonal construction persists on a neighborhood of its parameters. Choose a finite subcover of these parameter families; equivalently use sufficiently fine deterministic endpoint and rectangle bins. First add uniform reference-modulus cutoffs for this finite library, then its wall and core cutoffs, and finally choose \(H,J,\rho\) uniformly over it. The resulting finite list of smooth functions has bounded amplitude and bounded Dirichlet norm. These later wall and core cutoffs change the plateau amplitudes; they do not change the earlier spatial parameter family, gap or budget. Translation and Euclidean dilation preserve the Dirichlet energy in two dimensions. All cutoff losses can be made arbitrarily small without changing the earlier gap \(aw\) or the budget \(B_0\): the new wall and core constants are absorbed by \(H,J\). This proves the asserted finite, strict construction. ◻

Proof. Fix the recipe before imposing a hit or saturation condition. Condition on both fields. On its preservation patch retain the auxiliary draw for the exact field \(h+b\), and outside a strictly larger modification buffer retain the original exterior observations. Their joint law is the product-kernel marginal in Lemma 42. Under \(h'=h+f\), these are exactly the two required output kernels, since \(f=b\) on the first patch and \(f=0\) on the second. Complete all other output observations with their correct conditional law. The imaginary field is unchanged.

The only density change is now that of the real field. The Cameron–Martin density, evaluated on the source field, is \[L_f(h)=\exp\left((h,f)_\nabla+\frac12\|f\|_\nabla^2\right).\] On the imposed likelihood cutoff it is at most \(K\). Therefore the unselected image restricted to that cutoff is bounded by \(K\mathbb P\). Restricting further to any event determined by the old observations, the auxiliary shortcut, the hit or the chosen recipe gives a submeasure and preserves this domination. Thus no conditional independence after selecting a hit is required. The maximum cutoff over the fixed finite recipe list is uniform under spatial dilation and translation, giving the claimed single \(K\). ◻

Proof. All supports have radius at most a deterministic number tending to zero, their centers range over a fixed compact set, and their amplitudes are bounded by a fixed constant. The old \(D\)-diameters of these supports tend uniformly to zero. On either an old or a modified geodesic, the part between the first and last support visits can be replaced by a reference connection between two points of that support, at cost bounded by a fixed amplitude factor times its old diameter. Outside the support the two length rules agree. Consequently all modified anchor paths have old length tending to the old anchor distance. Global comparison by the bounded multiplier keeps them in a fixed proper metric ball. Their arclength parametrizations are equicontinuous; every subsequential limit is an old geodesic. Uniqueness identifies it with \(P\). The collapsed support-spanning portions have vanishing arclength, so fractional times converge as well. This reasoning is uniform over the finite recipe geometries and all their centers in the compact set: a failure of uniformity would give a contradicting sequence of shrinking supports.

In particular, the gate segment lies inside the same fixed interval \([t_1,t_2]\) of \(P'\) for all sufficiently fine windows, with compact clearance in \(U\). Apply the output upper comparison to the two remaining pieces of this interval and use the strict gain from Lemma 83. This proves the strict left inequality in (74).

For the other inequality, take a proposed output competitor in \(U\) with bounded cost. Cut it at its first and last encounters with a slightly larger ball than the observation buffer. Its exterior subcertificates have the original costs and can be retained in the old law. Replace the whole intervening portion by one old short connection, and correct its two moved endpoints by old short connections. Lemma 76 bounds all added costs by a deterministic function of the old compact reference modulus, the largest support radius, and the endpoint displacements, tending to zero. Repeated visits cost no additional error because only the first and last encounters are used. If the competitor never enters the ball, it transfers directly. Strict larger buffers ensure compact confinement of the retained exterior records even when an individual competing path has only its own positive clearance in \(U\).

An output competitor whose cost was below the old saturated value by a fixed positive amount would therefore contradict (65). This proves the required lower bound on its cost up to an error tending uniformly to zero on old reference-data cutoffs. The reference endpoint distances converge, and the fixed interval length has a positive lower bound. The resulting ratio error tends to zero in probability. Choose a deterministic sequence \(\epsilon_N\downarrow0\) by successive probability cutoffs; retain the source event on which all errors are at most \(\epsilon_N\). Its probability tends to one, proving the assertion. ◻

Proof. Let \(\mathcal X\) denote the measurable space of the complete joint observations, and let \(\mathcal I_N\) be the finite set of centers, scales, volume labels and recipes in the simultaneous annulus construction, and write \(w_i=(Mv_i)^p\). These weights are deterministic. Under the one common unmodified output law \(\mathbb P\), define \[R=\frac{F_U(P(t_1),P(t_2))}{D_h(P(t_1),P(t_2))}, \qquad E_N=\{0<C-R\leq\epsilon_N\}.\] Set \(R=+\infty\) unless \(P([t_1,t_2])\Subset U\) and the denominator is positive. Thus \(R\) is a single random variable, independent of the choice of ball or recipe.

For an index \(i\), let \(G_i\) be the following event of the output alone. The output anchor geodesic satisfies the fixed total-length cutoff and makes a complete transit of the recipe’s marker slab inside its enlarged shell, with subpath length at least \(a_3w_i\). Also require the alignment test, expressed using the output field minus that recipe’s fixed function \(f_i\). Choose \(a_3>0\) below the source marker-crossing cutoff times the minimum bounded Weyl factor. Every modified good hit has these properties. All marker faces, shells and functions are deterministic for index \(i\); the reference geodesic is measurable and almost surely unique. Hitting times, with the retained strict spatial margins, therefore make \(G_i\) measurable from the output field and geodesic. It refers to no discarded source path or auxiliary draw.

On the saturation event, intersected with the high-probability cover and error events, every point of the middle third of the interval has at least \(cN\) good covering holes. The time spent by a reference geodesic in any one ball is at most \(Aw_i\): its first and last outer-circle visits can be connected within the surrounding buffer at this cost. Integrating the covering multiplicity along an interval of length bounded below gives \[ \sum_{i\in\mathcal I_N}w_i\mu_i(\mathcal X)\geq c_1N \tag{75}\] for \(c_1>0\), where \(\mu_i\) is the source submeasure for the corresponding good hit and a selected valid recipe. Choose one valid recipe measurably from the fixed finite list for each hit. The source probability of the retained saturation event stays bounded below, since its original probability was positive.

Push each submeasure separately through its modification, obtaining \(\nu_i\). By Lemma 84, \(\nu_i\leq K\mathbb P\) for the same \(\mathbb P\) and \(K\). By Lemma 83 and Lemma 85, it is supported on the output event \(G_i\cap E_N\) defined above.

The output events satisfy \[ \sum_{i\in\mathcal I_N}w_i\mathbf 1_{G_i} \leq c_2N \tag{76}\] on the fixed output total-length cutoff. At a fixed spatial radius, choose the first qualifying marker transit specified by each \(G_i\). Its length is at least \(a_3w_i\) and its whole parameter interval stays in that enlarged shell. The mesh balls have bounded overlap, so integration along the output geodesic bounds the sum of these subpath lengths by a fixed multiple of its total length. There are \(O(N)\) spatial radii. The alignment coarse value is measured outside the modification support; alternatively, its change is bounded by the fixed amplitude. Consequently only boundedly many volume labels can be aligned with one spatial ball. The finite number of recipes contributes another fixed factor. This proves (76).

Combining the two weight bounds and the common-law domination gives \[c_1N \leq\sum_iw_i\nu_i(\mathcal X) \leq K\int\mathbf 1_{E_N}\sum_iw_i\mathbf 1_{G_i}\,\mathrm d\mathbb P \leq Kc_2N\mathbb P(E_N).\] For a fixed random variable \(R\), the probability \(\mathbb P(0<C-R\leq\epsilon_N)\) tends to zero. This is a contradiction. All modifications were applied separately; neither independence of selected hits nor a simultaneous modification of overlapping balls was used. ◻

Proof. If \(c<C\), Lemma 81 constructs a saturated tangent with fixed-gap shortcuts. Lemma 86 excludes it, so \(c=C\). Positivity and finiteness follow from Lemma 79.

The upper inequality for the displayed internal distance follows by following internal reference paths. For the reverse inequality, start with a bounded-cost certificate compactly confined in \(U\) and its actual discrete approximating paths. Before taking limits, add ordered marks whenever their accumulated normalized cost first crosses a multiple of \(2^{-k}\), retaining the previously specified marks. The single-edge cost tends to zero, since the initial-growth argument gives \(A_j\to\infty\). For each fixed \(k\) the number of marks is bounded by the cost budget. Lemma 31 lets us retain all these finite refinements consistently along a diagonal.

Apply the lower comparison to the subrecord between any two retained marks. Their \(D\)-distance is at most \(1/c\) times their intervening cost. This is the modulus that was unavailable before \(c>0\). The marked points therefore extend to a continuous path parametrized by accumulated cost; flat cost intervals have identical spatial endpoints. Its image stays in the retained compact confinement. For every finite subdivision, the sum of its reference endpoint distances is at most the certificate cost divided by \(c\). Taking the supremum over subdivisions bounds its \(D\)-length by that value. Thus the certificate costs at least \(cD(x,y;U)\), and infimizing gives the reverse inequality. This construction uses consistent finite refinements and the positive lower bound; no continuous path is inferred from a spatial range alone. Exhaust compact confinements and retain the arbitrary positive open losses to obtain the assertion for every open \(U\). ◻

Proof. Fix \(t\in\mathbb Z\) and put \(v=4^t\). In the joint extraction retain the raw observations at effective time unit \(v4^j=4^{j+t}\) and cost denominator \(A_jv^p\); denote their local kernel array by \(K_t\). Retain the shifted-base-coordinate description and the prescribed circle-average cone description jointly, as spatial relabellings of the same time-labelled raw records in the common aggregate reference extraction, before changing any field law. This is the effective-time array supplied by Theorem 32(1),(4). Evaluating \(K_t\) with the fixed embedded cone input and band convention used in (63) therefore gives the limiting buffered observations of \(X_{j+t}/(A_jv^p)\). The time-labelled raw records are those of that same word model at time \(4^{j+t}\), and their spatial relabelling uses this fixed input. No identification of \(K_t\) with the kernel at time \(4^j\) is asserted.

The indexed unit identity of Proposition 39 instead identifies \(K_t\) at the shifted input \(h_v=h-\gamma^{-1}\log v\): its passage costs and reference metric are both \(v^{-p}\) times those at the original input \(h\). The comparison constant in this shifted-input experiment is thus \(\lambda\). On the fixed dependency buffer, the shifted cone restriction and the unshifted circle-average cone restriction are mutually absolutely continuous. The same kernel \(K_t\) is used under these two input laws. Hence their probability-one local comparison with constant \(\lambda\) transfers to the fixed unshifted cone input. This step uses only an almost-sure transfer for each fixed \(t\); it does not require a density bound uniform in \(t\). Internal metrics and all circuit and traversal tests here are confined to the retained buffer. Countably many fixed offsets may be retained together.

Now take the reference cutoffs under that one fixed cone input law. A circuit in the middle band contains a reference advance across a strict smaller band. Its positive lower cutoff can have probability sufficient to contradict the upper quantile if the ratio in (77) were too small. The all-certificate lower comparison gives the left bound. Conversely a rectifiable reference winding path in the narrower band has a finite-length cutoff with probability greater than \(\tau\), with the needed probability slack. Following and stitching it gives a middle-band circuit and hence the right bound. Both cutoffs depend only on this fixed input law and the test geometries, so the constants \(k_-,k_+\) are independent of \(t\) and of the extracted law. Strict buffered tests and the two quantile inequalities after (63) justify these transfers, including possible quantile atoms. Setting \(t=0\) gives the stated compact interval for \(\lambda\). The same bounds exclude zero or infinite forward ratios before renormalizing at that offset. ◻

Proof. The adjacent bounds give positive finite limits for every fixed offset after a diagonal extraction. Fix \(\theta<1/4\). Choose a large \(L_0\), uniformly in the starting offset \(i\). If, for arbitrarily large \(N\), more than \(\theta N\) values of \(k\in[N,2N]\) satisfy \(r_{i+k}>L_0r_i\), the preceding lower tests at these deterministic labels give barriers of size at least a fixed constant times \(L_04^{-pk}\) in the start-\(i\) units. The simultaneous annulus theorem and Lemma 74 force a lower comparison proportional to \(L_0D\) at the starting scale. For sufficiently large \(L_0\) this contradicts its upper circuit quantile, because a fixed reference band crossing has positive distance with the required probability. All constants are uniform in \(i\): at the starting scale use its own \(A_{j-i}\) and evaluate its effective-time kernels with the same fixed cone input and band convention as in (63). The ordinary local laws used by the simultaneous annulus theorem are likewise fixed at every label. Thus the reference probability cutoffs in this contradiction are chosen once; no uniform absolute-continuity bound for an unbounded family of additive shifts is being used.

Likewise, if more than \(\theta N\) labels have \(r_{i+k}<r_i/L_0\), their upper circuit quantiles and upper chaining give a following constant proportional to \(L_0^{-1}\). A reference traversal with a fixed finite-length cutoff then contradicts the starting normalized lower traversal test. Increase \(L_0\) once to handle both contradictions. Consequently, for every fixed \(i\) and all sufficiently large \(N\), more than \((1-2\theta)N\) indices \(\ell\in i+[N,2N]\) satisfy \[ r_i/L_0\leq r_\ell\leq L_0r_i. \tag{78}\]

For two fixed starts \(i,i'\), the shifted intervals differ in at most \(2|i-i'|\) indices. Since \(1-2\theta>1/2\), the two sets of indices in (78) intersect for sufficiently large \(N\). At an intersection index, \(L_0^{-2}\leq r_i/r_{i'}\leq L_0^2\). Take \(i'=0\), for which \(r_0=1\). The constant is independent of \(i\), as required. ◻

Proof. We propagate a finite list of tests. Choose a broad adjacent-ratio interval \((L^{-1},L)\) which is strictly passed by the ratios of Lemma 88. Put \(s=p/2\) and choose an integer \(m\) so large that those rigid ratios strictly pass \[\frac{A_{j-m}}{A_j}<4^{-sm}.\] Finally choose a sufficiently small fixed normalized lower traversal threshold so that every rigid law with \(\lambda\in[k_+^{-1},k_-^{-1}]\) strictly passes its high-probability lower test. Use the continuous buffered-test convention above, with enough probability slack that limits of passing tests still suffice for the simultaneous annulus theorem. The upper circuit quantile is already supplied at every index by (63). The remaining finite test family consists of the adjacent-ratio bounds, the displayed \(m\)-step bound, and the buffered lower traversal test. We use two different implications: limits of passing tests retain the closed inequalities needed for Lemma 89, whereas every rigid law passes the tests with strict margins and therefore forces nearby approximants to pass. The first-failure argument below uses these implications in that order.

There are arbitrarily long success runs at arbitrarily large indices. Indeed, by Lemma 78, for every \(\varepsilon>0\) the sequence \(4^{-(p-\varepsilon)j}A_j\) has arbitrarily late record maxima. Take record indices with \(\varepsilon\downarrow0\). For every fixed preceding offset \(i\) their ratios satisfy \[\frac{A_{j-i}}{A_j}\leq4^{-(p-\varepsilon)i}.\] Every joint limit therefore satisfies (64). Theorem 87 and Lemma 88 imply that all the chosen tests strictly pass at every fixed backward and forward offset. If some bounded window of tests failed along these record indices, another joint extraction would contradict that strict passage. A diagonal choice yields success runs whose lengths tend to infinity.

Suppose there were infinitely many later failures. Starting from the success runs just obtained, extract at the level immediately before the first later failure. Every fixed preceding offset passes the tests. Lemma 89 supplies the uniform exact-power bound (64). The extracted law is rigid, and Lemma 88 includes the first forward offset. Thus all the tests strictly pass at the alleged failure as well. The continuity and strict-loss convention for the tests gives a contradiction. Hence all sufficiently large indices pass.

In particular, for those indices, \(A_{j-m}/A_j\leq4^{-sm}\) and the adjacent ratios are bounded. Write \(j-j'=km+r\), with \(0\leq r<m\). Iterating gives \[\frac{A_{j'}}{A_j}\leq L^r4^{-smk} \leq L^m4^{sm}4^{-s(j-j')}.\] Changing the constant handles the finite initial indices. Passing from \(j,j'\) to \(\lfloor\log_4n\rfloor, \lfloor\log_4n'\rfloor\) changes this bound by at most another fixed factor, proving (79). Taking \(n'=1\) proves \(B(n)\to\infty\).

For an arbitrary sequence of powers of four, all fixed preceding tests now pass. Extract their ratios and apply Lemma 89 again. Theorem 87 identifies the local port law, and Lemma 88 puts its deterministic factor in the fixed positive compact interval. For a general time unit \(n\), extract also \(n/4^{\lfloor\log_4n\rfloor}\in[1,4]\). The bounded time-unit re-expression of the contour, field and certificate arrays absorbs this converging factor into \(\lambda\); its possible \(p\)th powers remain in a fixed compact interval. No convergence rate is needed for this extraction.

The simultaneous circuit and finite-system assertions follow from Lemma 76 and the robust topological intersection statements. Finally, the finite-volume part of Theorem 32 uses the same kernels and the same deterministic cost units. Lemma 41 dominates the actual normalized sphere field, with its actual additive height, by the ordinary field law on each such patch. Together with Proposition 39, it transfers the probability-one comparisons there. A countable chart exhaustion makes them simultaneous off the exploration root.

For the sphere with \(n\) edges the duration-one letter clock has unit \(2n\), so this first gives the comparison in \(B(2n)\) units. The propagated adjacent-ratio bounds imply that \(B(2n)/B(n)\) lies in a fixed positive compact interval: the two base-four integer parts differ by at most one. Retain a deterministic subsequential limit \(r\) of this ratio and absorb it into the comparison constant. Thus the \(B(n)\) convention used in Section 11 has factor \(r\lambda\), which we again denote by \(\lambda\), with all passage and circuit conclusions scaling together. Neither an exact ratio limit nor an additional time-scaling factor is required. Uniform costs to prescribed lattice endpoints are supplied separately in Section 11. ◻

Proof. A correspondence of distortion tending to zero in a Gromov–Hausdorff realization gives (82): choose one related point of \(Y\) for each vertex. The image is a net with error tending to zero. The space of nonempty compact subsets of the compact space \(Y\times\widehat{\mathbb C}\) is compact, so a subsequence of \(R_n\) converges. Net density in the two factors proves the two surjectivity assertions.

For the upper inclusion in (83), take any converging sequence of triples from \(K_n(a_n)\) and extract the two \(Y\)-coordinates of its vertices. They give two pairs in \(R\), and (82) identifies the third coordinate. For the reverse inclusion, approximate any two pairs of \(R\) by pairs of \(R_n\) at the same indices. Their distances again converge by (82). This proves both Hausdorff inclusions. ◻

Proof. Choose a continuous path \(P\) from \(y\) to \(y'\) in \(Y\setminus\{y_0\}\). If the endpoints coincide, take the constant path. Its compact image has distance \(b>0\) from \(y_0\). By uniform continuity, choose a finite partition of \(P\) into pieces whose endpoints are at \(d_Y\)-distance less than \(b/10\). Approximate the partition points by \(j_n(V_n)\), using \(u_n,v_n\) at the ends. For large \(n\), consecutive chosen vertices have \(\delta_n\)-distance less than \(b/5\), and each chosen vertex has \(Y\)-distance at least \(4b/5\) from \(y_0\).

Join consecutive vertices by graph shortest paths. Every vertex of such a path is at \(\delta_n\)-distance less than \(b/5\) from its initial vertex, so (82) keeps its \(Y\)-image at distance at least \(b/2\) from \(y_0\). It therefore avoids \(S_n\). Concatenation proves the assertion. The same proof shows the following useful version: for this fixed path \(P\), it is enough that all of \(j_n(S_n)\) eventually lies within distance \(b/4\) of \(y_0\). ◻

Proof. Work first with \(D^{\mathrm o}\) in the original oriented sphere. A fixed orientation-preserving inversion puts \(p_*^{\mathrm o}=\infty\) at zero. By Proposition 10, near this mark the field is represented by a GFF with a logarithmic insertion of strength \(\gamma\), a locally continuous term, and the finite random additive constant that normalizes its area; the change to the unit-area law has a positive integrable density. We may first work with \[h^\gamma=h^0-\gamma\log|\cdot|,\] where \(h^0\) is a whole-plane GFF normalized on the unit circle. The local continuous term changes lengths by factors bounded above and below on a fixed smaller disk, by Weyl scaling.

For \(0<r<1/4\), consider the annulus \(r<|z|<2r\). The around-annulus estimate for \(h^\gamma\) gives a disconnecting closed path of length at most \[ r^{\xi(Q-\gamma)}e^{\xi h^0_r(0)}T_r, \tag{84}\] where the nonnegative variables \(T_r\) form a tight family. This is the around-distance estimate, not just a point-to-circle estimate; see Definition 3.7 and the proof of Proposition 3.14 in (Dubédat et al. 2020). It also follows by scaling a unit annulus, using the spatial coordinate rule and the tight unit-annulus internal distances. The normalized field \(h^0(r\cdot)-h^0_r(0)\) has the same annular law at every \(r\).

Now \[Q-\gamma=\frac{4-\gamma^2}{2\gamma}>0, \qquad \frac{h^0_r(0)}{\log(1/r)}\longrightarrow0 \quad\hbox{in probability}.\] Indeed the numerator is centered Gaussian of variance \(\log(1/r)\). Thus (84) tends to zero in probability; no independence between its factors is needed. Bounded local continuous perturbations, the finite area-normalizing factor, and an absolutely continuous probability weighting preserve this convergence. Choose deterministic radii \(r_k\downarrow0\) so that the probabilities of a length exceeding \(2^{-k}\) are summable. Borel–Cantelli gives such paths almost surely in the original oriented sphere. Their diameters tend to zero because they stay in the corresponding shrinking annuli. Transport them by \(f\). Equation (80) preserves their metric lengths, and continuity of \(f\) gives vanishing canonical spatial diameter. This obtains the asserted \(L_k\) without changing any field law.

Let \(z,w\ne p_*\). Take a \(D\)-geodesic between them. If it passes through \(p_*\), replace the portion between its first and last intersections with \(L_k\) by a portion of \(L_k\); both endpoints lie outside the annulus for large \(k\). The replacement avoids \(p_*\) and adds at most the length of \(L_k\). A closed path with nonzero winding can, for this purpose, be replaced by a separating simple subcircuit of its trace; equivalently one can use the first and last crossing of its separating trace and either connecting portion of the closed path. The internal distance is therefore at most \(D(z,w)\) after letting \(k\to\infty\). The reverse inequality is immediate. Finally the punctured sphere is \(D\)-dense by the topology of the LQG metric, and \((\widehat{\mathbb C},D)\) is complete. This identifies its completion. ◻

Proof. Choose a fixed strictly buffered annular test in an ordinary patch of the original oriented coordinate \(x_n\), with vertices on both sides. The positive lower port comparison in \(B(n)\)-units implies that, for some \(b,\theta>0\) and all sufficiently large \(n\), every traversal of that annulus has cost at least \(bB(n)\) on an event of probability at least \(\theta\). To obtain fixed \(b,\theta\), use the positive reference distance between its compact boundary sets and the fixed positive lower bound for \(\lambda\); then truncate to a positive-probability event. If the initial choice of patch depends on the location of the exploration mark, choose one member of the countable ordinary atlas with positive probability and restrict to its strict containment event.

Every path between the two sides contains an annular traversal. Hence the ambient graph diameter is at least \(bB(n)\) on this event. If \(a_nB(n)\) were unbounded, a deterministic subsequence tending to infinity would contradict the tightness of \(\mathop{\mathrm{diam}}(V_n,\delta_n)\) in Theorem 2. This proves boundedness and (85). Multiplying the upper and lower comparisons by \(a_nB(n)\) gives the stated factors.

Away from the original exploration root, the simultaneous circuit conclusion is Lemma 76, multiplied by the bounded factors \(a_nB(n)\). Transporting the circuits by \(f_n\to f\) gives the asserted conclusion away from \(p_*\). Its simultaneous quantifier is relevant here: a point chosen from a later metric relation is allowed. The countable strict port library and local internal Hölder bounds in that lemma provide the assertion on every compact set off \(p_*\). At \(p_*\) use Lemma 94, cover each of its finitely long reference paths by finitely many ordinary patches, and apply port following and Corollaries 19 and 20. The resulting primal circuit has a strict central hole and lies in a shrinking neighborhood of \(p_*\). Its limiting cost is at most \(\alpha\lambda\) times the reference length, with arbitrarily small open losses. These lengths tend to zero.

For the last assertion, choose finitely many strict central holes covering the sphere, each bounded by one of the preceding reference circuits and with filled side of spatial diameter less than \(\varepsilon\). Join the circuits by finitely many reference paths off \(p_*\). Such paths have finite \(D\)-length by Lemma 94. Use transverse ports at all required intersections. Lemma 21 realizes this finite construction as a connected primal skeleton. Each complementary component lies on the filled side of one of the covering circuits, so has the stated diameter, allowing the fixed strict margins to be chosen in advance. The total cost in \(B(n)\)-units is bounded along the extraction for this finite construction. Multiplication by \(a_nB(n)\to0\) makes its \(\delta_n\)-diameter tend to zero. ◻

Proof. Write \(c=\alpha\lambda>0\). We first prove that every two related pairs satisfy \[ d_Y(y,y')\ge cD(z,z') \qquad\bigl((y,z),(y',z')\in R\bigr). \tag{88}\] Choose representing vertices and graph shortest paths between them. Their \(\delta_n\)-lengths are bounded. Parameterize them by this length and keep them constant after their endpoints, on a common bounded time interval. Project the edges by the surface homeomorphisms and the canonical coordinate.

For each \(r>0\), any projected path of spherical diameter at least \(r\) makes a fixed positive advance in one member of a finite ordinary annular cover. Such a cover can avoid a ball of radius much smaller than \(r\) about \(p_*\): a path with that diameter has a subpath making a comparable advance outside the smaller ball. The compact boundary sets in these finitely many tests have positive \(D\)-separation. The lower certificate comparison therefore gives, for all sufficiently large \(n\), a positive lower \(\delta_n\)-cost for every such advance. If this uniform lower bound failed, a sequence of failing traversals would yield a lower-certificate violation in one of the finitely many tests. The maximum single-edge spatial diameter tends to zero and \(a_n\to0\), so the length-parameterized projections are equicontinuous. A subsequence converges uniformly to a continuous path \(P\) from \(z\) to \(z'\).

Every subarc of \(P\) compactly off \(p_*\) pays at least \(c\) times the \(D\)-distance of its ends. Indeed a finite ordinary cover and subdivision reduce it to the local lower comparisons, with costs added over disjoint subpaths. For \(t>0\), remove the portion of \(P\) between its first entry into and last exit from the \(D\)-ball of radius \(t\) about \(p_*\). The retained pieces are compactly off the root and their endpoint distances sum to at least \(D(z,z')-2t\). The same statement, with one retained piece, applies if an endpoint is \(p_*\). If the ball is not visited there is no removal. Letting \(t\downarrow0\) proves (88).

Taking \(y=y'\) in (88) shows that \(R\) assigns each \(y\) a unique spatial point. Its graph is compact, so the resulting surjection \(\pi:Y\to\widehat{\mathbb C}\) is continuous. In fact (88) gives its continuity directly in the \(D\)-topology.

Suppose that a fiber over \(z\) contains more than one point. Take the shrinking primal circuits about \(z\) from Lemma 95. For each fixed circuit test, extract its \(j_n\)-image as a compact subset of \(Y\). Choose the tests so that the diameters of these limiting subsets tend to zero. After a further subsequence of tests they converge to one point \(y_0\). Their spatial diameters tend to zero at \(z\), so \(\pi(y_0)=z\). Choose \(y\in\pi^{-1}(z)\setminus\{y_0\}\) and choose \(x\in Y\) with \(\pi(x)\ne z\). Inequality (88) keeps \(x\) away from \(y_0\).

There is a path in \(Y\setminus\{y_0\}\) from \(y\) to \(x\) with positive metric clearance from \(y_0\). Fix a sufficiently fine circuit test that its limiting \(Y\)-image lies within one fourth of this clearance from \(y_0\). Then take \(n\) large. Representatives of \(y\) lie in its strict central hole, and representatives of \(x\) lie on its exterior side. The last version of Lemma 93 joins those representatives while avoiding the entire circuit, contradicting its separation. Thus the fibers are singletons. A continuous bijection between compact metric spaces is a homeomorphism.

To obtain (87), a contrary sequence would have spatially approaching pairs at distances bounded away from zero. Extract both \(Y\)-coordinates. The relation then assigns the same spatial point to two distinct \(Y\)-points, contradicting the singleton-fiber conclusion. This argument is uniform over all vertices.

It remains to prove the upper bound in (86). For \(z,z'\ne p_*\) and any \(e>0\), Lemma 94 supplies a rectifiable path compactly in the punctured sphere, from \(z\) to \(z'\), of length at most \(D(z,z')+e\). A finite cover by ordinary patches and the port upper comparison give discrete following paths with arbitrarily small spatial errors at their ends and limiting cost at most \(c(D(z,z')+e)\). Equation (87) now connects their ends to any prescribed representing vertices at vanishing additional cost: first make the fixed port errors small, then take \(n\) large, and finally let those errors tend to zero. Let \(e\downarrow0\). Endpoints equal to \(p_*\) follow by the continuity of \(D\) and of \(\pi^{-1}\). This proves the reverse of (88), and hence (86). ◻

Proof. For each \(k\ge1\), take a connected skeleton from Lemma 95 with complementary spatial mesh at most \(2^{-k}\). Its \(\delta_n\)-diameter tends to zero. Diagonally extract all these countably many skeletons, so that their \(j_n\)-images converge respectively to points \(y_k\in Y\).

Fix \(k\). For any \((y,z),(y',z')\in R\) with \(y,y'\ne y_k\), Lemma 93 joins representing vertices in the complement of the skeleton. They therefore lie in the same spatial complementary component. Passing to the limit gives \[ \rho(z,z')\le2^{-k}. \tag{91}\] This reasoning also applies to two representations of the same point \(y\ne y_k\).

Pass to a subsequence of \(k\) with \(y_k\to y_0\). Any two fixed points of \(Y\setminus\{y_0\}\) differ from \(y_k\) for all sufficiently large \(k\). Equation (91) shows that all their spatial representations agree with a single point \(z_0\). There is at least one such point of \(Y\) because \(Y\) is nontrivial. The surjectivity of \(R\) onto \(\widehat{\mathbb C}\) implies \((y_0,z)\in R\) for every \(z\ne z_0\); closure gives it also for \(z=z_0\). Surjectivity onto \(Y\) and closure likewise give \((y,z_0)\in R\) for every \(y\in Y\). This proves (89). Its exceptional row and column are unique because both factors contain more than one point.

For \(z\ne z_0\), the only \(Y\)-point related to \(z\) is \(y_0\), so the only third coordinate above \((z,z)\) in (83) is zero. At \(z_0\) the entire space \(Y\) is related, and a pair realizing its positive diameter supplies (90). The criterion is measurable: for a closed spatial set \(C\), its exceptional point lies in \(C\) if and only if, for some integer \(m\ge1\), the compact graph hits the closed set \(\{(z,z,r):z\in C,\ r\ge1/m\}\). These are Borel conditions in the compact-set topology. Thus \(z_0\) can be retained without reference to an exploration root or a choice of GH realization. ◻

Proof. Keep the original canonical coordinate for this argument. The finite pair \((\mu_n,K_n(a_n))\) is determined by the unrooted decorated map and the three normalization samples. Conditional on those data before forgetting incidences, the initial vertex of a uniform oriented root edge has law \[\nu_n^{\mathrm{root}} =\sum_{v\in V_n}\frac{\deg(v)}{2n}\,\delta_{z_n(v)}.\] Lemma 4 couples this distribution to \(\mu_n\) at spherical error at most the maximum flag diameter. Consequently, for bounded continuous \(g\) on \(\widehat{\mathbb C}\) and bounded continuous \(F\) of the root-free pair, \[\begin{align*} &\mathbb E\bigl[F(\mu_n,K_n(a_n))g(z_n(v_*))\bigr]\ &\hspace{12mm}- \mathbb E\left[F(\mu_n,K_n(a_n))\int g\,\mathrm d\mu_n\right] \longrightarrow0. \end{align*}\] The difference is bounded by \(\|F\|_\infty\) times the modulus of continuity of \(g\) at the maximum flag diameter, which tends to zero in probability and is bounded.

Pass to the joint limit and use bounded continuous tests followed by the monotone-class theorem. Conditional on \((\mu,\mathcal K(R))\), the point \(p_*\) has distribution \(\mu\). The point \(z_0\) is measurable from \(\mathcal K(R)\) and \(\mu\) has no atoms by Corollary 56. Hence \[\mathbb P[p_*=z_0] =\mathbb E\bigl[\mu(\{z_0\})\bigr]=0.\] ◻

Proof. Suppose that the deterministic subsequence in (85) has \(\alpha=0\). The cross relation (89) then holds almost surely for the finite-sphere extraction. We will use an orientation-preserving reembedding by fresh area marks and produce two separated positive escape readouts in that reembedded sphere law. The cross relation is preserved by the reembedding, so it must hold in the same experiment.

Proof. For any bounded measurable \(F:E\to\mathbb R\), the unordered pairs \(\{X,\mathfrak rX\}\) and \(\{Y,\mathfrak rY\}\) coincide. Hence \[F(X)+F(\mathfrak rX)=F(Y)+F(\mathfrak rY)\] almost surely. Taking expectations and using both invariances gives \(2\mathbb EF(X)=2\mathbb EF(Y)\). ◻

Proof. If \(c>1\), equality of laws implies, for every finite \(t>0\), \[\mathbb P[X\le t]=\mathbb P[X\le t/c^k]\longrightarrow\mathbb P[X=0]=0.\] This contradicts the almost sure finiteness of \(X\). If \(c<1\), apply the same argument to \(1/c\). ◻

Proof of Theorem 91. Start with an arbitrary deterministic subsequence of positive integers. The preceding sections and Theorem 2 give a joint further extraction of the reference data, local readouts and intrinsic spaces. Their marginals are tight; adjoining the compactified local arrays preserves tightness. The full distance graphs are tight as well: their first two coordinates lie in a fixed compact sphere, and their third coordinates are bounded by the tight intrinsic diameters. Take a Skorohod coupling for these retained data, and then apply Lemma 92 pathwise.

Lemma 95 allows the deterministic subsequence to be chosen so that \(a_nB(n)\to\alpha\). Proposition 99 gives \(\alpha>0\), and Proposition 96 identifies the location relation as a homeomorphism satisfying \[(Y,d_Y)\cong(\widehat{\mathbb C},\alpha\lambda D).\] Both \(\alpha\) and \(\lambda\) are deterministic: the first is a limit of numerical normalizations, and the second is the law-wide rigid comparison constant in Theorem 90. By (80), \(f\) is an isometry from \((\widehat{\mathbb C},D^{\mathrm o})\) to \((\widehat{\mathbb C},D)\) on each sample, regardless of its orientation or dependence on the data. Therefore the unmarked space \((\widehat{\mathbb C},D)\) has the ordinary unit-area marginal of \((\widehat{\mathbb C},D^{\mathrm o})\). This is the same marginal as \((Y,d_Y)\) by Theorem 2. Their diameters are finite and strictly positive. Lemma 101 gives \(\alpha\lambda=1\); no assertion about the law of an adaptively reembedded field is used here.

Equation (83) is now exactly (81). In particular, it identifies every vertex pair in the Hausdorff limit, rather than only a countable or area-typical set of pairs. The area convergence is already joint in Corollary 56. The resulting canonical pair is \[X=(\mu,\mathcal G_D) \in\{X^+,\mathfrak r X^+\},\qquad X^+=(\mu^+,\mathcal G_{D^+}),\] where reflection acts simultaneously on the measure and both spatial coordinates of the distance graph. We now remove this possible adaptive sign at the level of the pair’s law.

The law of each discrete pair \((\mu_n,K_n(a_n))\) is reflection invariant: reflection is a weight-preserving bijection of the oriented finite map ensemble, preserves graph distances and flag areas, and conjugates the normalized coordinate because \(0,1,\infty\) are fixed by conjugation. Reflection is continuous in the weak-measure and compact-graph topologies, so \(\mathop{\mathrm{Law}}(X)\) is invariant as well. The law of \(X^+\) is reflection invariant by the reflection symmetry of the ordinary quantum sphere, its area and metric, and its three independently sampled marks. Lemma 100, applied to this pair, gives \(\mathop{\mathrm{Law}}(X)=\mathop{\mathrm{Law}}(X^+)\) even though the choice between \(f=T\) and \(f=\mathfrak r\circ T\) may depend on every retained observation.

Every deterministic subsequence therefore has a further subsequence with the prescribed pair law. The subsequence criterion proves the asserted full-sequence convergence. The metric identification itself was made in the joint extraction, so it remains (81) with any additional retained observations; the common orientation sign is kept with those observations. The normalization \(a_n\to0\) is the one from the companion intrinsic theorem. ◻