Proof. This is a deterministic consequence of the distance-graph and flag-mesh assertions of Theorem 2. Work on a representation on which those assertions and the weak area convergence hold almost surely. Every endpoint occurrence has two side flags, so sending each flag to its primal vertex pushes \(m_n\) to \(\eta_n\), including at loops and bridges. The spherical displacement of this projection is at most the maximal flag mesh. Consequently \(\widetilde\eta_n=(\phi_n)_*\eta_n\) converges weakly to \(\mu_h^{\rm crit}\), and the embedded vertices become dense in \(S^2\).

Continuity of \(D_h\) and Hausdorff convergence of the full distance graphs give \[\sup_{u,v\in V(M_n)} \bigl|a_nd_n(u,v)-D_h(\phi_n(u),\phi_n(v))\bigr| \longrightarrow0.\] Indeed, a sequence violating this assertion has a convergent graph-point subsequence, whose limit belongs to the graph of \(D_h\). Choose \(r_n\downarrow0\) large enough to cover the maximal flag mesh and to couple \(\widetilde\eta_n\) with \(\mu_h^{\rm crit}\) so that all but \(r_n\) of the mass has spherical displacement at most \(r_n\). The relation \(\{(v,z):\rho(\phi_n(v),z)\le r_n\}\) covers both spaces. Its distortion for \(a_nd_n\) and \(D_h\) tends to zero by the displayed bound and uniform continuity of \(D_h\), and it carries all but \(r_n\) of the lifted coupling. The correspondence-and-coupling characterization of Gromov–Hausdorff–Prokhorov convergence proves the assertion. The same construction on each coupled subsequence proves the stated joint convergence in law. ◻

Proof. The infinite sphere measure underlying (Duplantier et al. 2021, Definition 4.21(ii)) has average process equal to \(2/\gamma\) times a log-Bessel excursion of dimension \(\delta=4-8/\gamma^2\in(0,2)\), on its unit quadratic-variation clock, independently decorated by \(G^\perp\). The excursion maximum in Bessel units has intensity proportional to \(s^{\delta-3}\,ds\). Conditional on that maximum, the two ascents, one read backwards, are independent \(\mathrm{BES}(4-\delta)\) paths from zero to the maximum; this is the Bessel excursion decomposition at the maximum (Pitman and Yor 1996, Theorem 1 and the Bessel specialization following it). For the scaled logarithmic maximum \(m\), the substitution \(s=e^{\gamma m/2}\) gives intensity proportional to \(e^{-2am}\,dm\).

Place the maximum at horizontal coordinate zero. On one ascent, the deficit below \(m\), starting at its first visit to a level \(K>0\) and ending at zero, is Brownian motion with drift \(-a\), started at \(K\) and killed at zero. This follows from Itô’s formula and the quadratic-variation clock. Its reversal from killing, as \(K\to\infty\), has the law of \(|W_t+at e_1|\). We include the transition-density calculation to fix this identification.

Let \(p_b(s,x,y)\) be the killed half-line transition density of Brownian motion with drift \(b\), let \(g_b\) be its time integral, and let \(q_b(s,x)\) be the first-hitting density of zero. At reversed times \(0<s_1<\cdots<s_k\), the subdensity for duration exceeding \(s_k\) is \[g_{-a}(K,x_k) \left(\prod_{i=2}^k p_{-a}(s_i-s_{i-1},x_i,x_{i-1})\right)q_{-a}(s_1,x_1).\] The reflected heat kernel with drift gives, for \(0<x\leq K\), \[g_{-a}(K,x)=\frac{1-e^{-2ax}}a.\] After the drift factors telescope, the limiting density is the zero-drift \(\mathrm{BES}(3)\) density—the same product with \(p_0,q_0\) and prefactor \(2x_k\)—times \[e^{-a^2s_k/2}\frac{\sinh(ax_k)}{ax_k}.\] This is exactly the radial Brownian change of measure for vector drift \(ae_1\). The duration tends to infinity in probability, and the limiting density integrates to one. This identifies the reversed deficit on every finite horizon. Both halves remain independent of each other and of \(m\); translation leaves the independent stationary lateral field unchanged.

For a fixed maximum-zero shape with total area \(A\), its area at maximum \(m\) is \(v=e^{\gamma m}A\). Therefore \[e^{-2am}\,dm =\frac1\gamma A^{2a/\gamma}v^{-1-2a/\gamma}\,dv.\] Disintegration at \(v=1\) tilts the shape by \(A^{2a/\gamma}\) and subtracts \(\gamma^{-1}\log A\) from the field. Replacing \(A\) by \(2(2-\gamma)L_\gamma\) changes the tilt only by a deterministic factor; adding \(c_\gamma\) leaves exactly (10). This is the continuous-by-constant-shift version of the area-one disintegration. Finiteness, positivity, and integrability of its density near two are proved in the next two lemmas. ◻

Proof. The normalized masses on slightly larger compact disks are tight by (8). Pack \(c k^2\) disjoint disks of radius \(c'/k\) inside the half-disk. Their Markov decompositions consist of independent rescaled zero-boundary fields and an independent collection of harmonic parts. With probability at least \(3/4\), all those harmonic parts are bounded below by \(-C\log k\) on the corresponding half-disks, uniformly for large \(k\). Indeed their variances on the three-quarter disks are at most \(\log k+O(1)\) by the Green function, or conformal-radius, formula. Harmonic interior estimates, the Gaussian exponential estimate and Jensen’s inequality bound each supremum tail; a union bound over \(O(k^2)\) disks proves the claim when \(C\) is large.

This event is independent of the inner Dirichlet fields. On it, scaling contributes \((c'/k)^{2+\gamma^2/2}\) and hence \[Z_\gamma\geq c_1 k^{-B}\max_{1\leq i\leq c k^2}Z_{\gamma,i}\] for a fixed \(B\), where the variables on the right are independent copies of \(Z_\gamma\). Tightness of the parent mass, together with independence of the harmonic event, yields \[\mathbb P\!\left(\max_i Z_{\gamma,i}>C_1k^B\right)\leq\tfrac12\] uniformly for all sufficiently large \(k\), after increasing \(C_1\). Thus \(\mathbb P(Z_\gamma>C_1k^B)=O(k^{-2})\) uniformly. Integrating this polynomial tail proves the first assertion for a sufficiently small positive \(s\).

On a compact cylinder patch, realize \(G^\perp\) as a zero-boundary field in a larger coordinate disk plus a continuous random field whose absolute supremum has every exponential moment. To do so, add an independent two-sided average Brownian motion to recover a whole-cylinder field, use its Markov decomposition modulo constants, and subtract that Brownian motion. The remaining harmonic part is smooth on the compact patch and all normalization constants are Gaussian. Hölder’s inequality, a finite cover, and the first assertion give a smaller positive moment for the lateral field. Stationarity makes this bound uniform over unit strips. ◻

Proof. On compact cylinder sets, the decomposition used in Lemma 7, the positive-measure limit in (8), and multiplication by the continuous correction field give convergence to \(m^\perp\). These limits agree on overlaps, are measurable from \(G^\perp\), and are diffuse with full support. They retain the strip moment bounds by Fatou’s lemma.

For each \(b>0\), uniformly in \(0\leq a\leq1\), \[ \mathbb E\exp\!\left(-b\inf_{[k,k+1]}R_a\right) \leq C_b k^{-3/2},\qquad k\geq1, \tag{11}\] and the same estimate holds on the negative half-cylinder. The density of \(W_k+ak e_1\) is bounded by \(Ck^{-3/2}\), independently of its mean. Integrating \(e^{-b|x|}\) against this bound and then allowing the independent one-unit increment supremum proves (11); those increments have uniform exponential moments when \(a\leq1\).

The lateral field and the radial processes are independent. The \(s\)-moment of the normalized area on strip \([k,k+1]\) is therefore at most \(Ck^{-3/2}\), using (11) with \(b=s\gamma\) and taking \(\gamma\) in a fixed interval near two. The same bound holds for \(e^{-2R_0}m^\perp\). Since \(s<1\), subadditivity shows that the area outside \(|t|\leq K\) has \(s\)-moment tending uniformly to zero as \(K\to\infty\). On compact sets \(R_a\to R_0\) uniformly almost surely. This proves the weak convergence, convergence of the totals, and the uniform positive moment. The limit total is finite by the tail bound and strictly positive by local full support; its ends have zero mass.

Finally \(2a/\gamma\to0\) and \(\log L_\gamma\to\log L\) in probability, so the displayed powers tend to one in probability. For large \(\gamma\) they are bounded by \(1+L_\gamma^{s/2}\), a uniformly integrable family because the \(s\)-moments are bounded. This proves \(L^1\) convergence. The proof of local convergence and the tail estimate also applies when the left-hand field is the fixed field \(X^0\). ◻

Proof. In the \(z=e^{t+i\theta}\) coordinates, the round-background scalar associated with (10) is \[G^\perp-R_a-\frac{\log L_\gamma}{\gamma} +Q_\gamma\log\cosh t.\] This follows from the flat-cylinder charge rule and \(g_{\rm round}(z)=4/(1+|z|^2)^2\). The varying radial terms converge in round \(L^2\) in probability. Indeed the common Brownian bound \(|W^\pm_{|t|}|+O(1+|t|)\) is square-integrable against the exponentially decaying round-area density at both ends; the random constant converges in probability by Lemma 8. The lateral field is already in \(H^{-1}\) and is the same in the coupling. Thus the fields converge in \(H^{-1}\) norm jointly with the finite measures. The normalized densities in (10) tend to one in \(L^1\), so the conclusion also holds under the tilted sphere laws.

Under the ordinary area-one DMS sphere law, the two end marks, conditional on the unmarked surface, are independent area samples (Duplantier et al. 2021, Proposition A.13). Appending a third independent area sample and sending the ends and that sample to \(0,\infty,1\), respectively, therefore gives exactly the three-marked law in the theorem. The area-sampling kernel is weakly continuous on probability measures on the compact sphere. The limit measure has no atoms, and the third mark avoids the ends, so the normalizing Möbius maps converge smoothly as sphere diffeomorphisms.

Composition of scalar distributions with a smoothly convergent family of sphere diffeomorphisms is continuous in \(H^{-1}\): the pullback operators have locally bounded norms by duality on \(H^1\), and the assertion follows first for smooth distributions and then by approximation. If \(f\) maps new coordinates to old coordinates, the scalar background correction is \[\frac{Q_\gamma}{2} \log\frac{(g_{\rm round}\circ f)|f'|^2}{g_{\rm round}},\] which is smooth on the sphere and converges smoothly with \(f\) and \(Q_\gamma\). The continuous mapping theorem now proves the stated joint limit and its sequence independence.

One may append any finite number of independent area samples before taking this limit. At each subcritical parameter, replacing the original three marks by a fresh triple gives the same law. The limiting marks remain distinct almost surely, so passing to the limit in that identity proves the resampling assertion with charge two. ◻

Proof. The translation changes neither the marked embedded surface nor the law of the independent stationary lateral field. On a fixed finite horizontal interval, consider the radial process before subtracting its area normalization. If its maximum is outside the interval, the restriction is absolutely continuous modulo constants with respect to Brownian motion, because a positive-start \(\mathrm{BES}(3)\) path has that property before approaching zero; time reversal handles the opposite side. If the maximum is inside, its location has a density and is therefore absolutely continuous with respect to the Brownian arcsine maximum-time law. Conditional on the location, its two drops are independent negative \(\mathrm{BES}(3)\) paths. The Brownian decomposition at the maximum (apply the minimum decomposition to \(-B\)) gives independent meanders for the corresponding drops, and Imhof’s relation makes each meander equivalent to \(\mathrm{BES}(3)\) on its prescribed finite time interval; see the sampling and meander identities in (Pitman 1999, Corollary 5 and Equations (5)–(6)). This proves the required one-dimensional absolute continuity modulo constants. Adding the independent lateral field proves the slab assertion, and hence the assertion for joint interior restrictions. The normalization \(-(\log L)/2\) is only a constant.

For the subcritical shapes, adding vector drift to the Brownian motions is absolutely continuous on every finite horizon. The subsequent absolutely-continuous shape tilt changes no null sets. The same argument therefore applies to (10). ◻

Proof. Consider a set having zero measure for that product reference law. Its conditional sampling probability is obtained by integrating its indicator against the product of the local critical area measures and dividing by a finite positive power of the total area. The assertion that its unnormalized product mass is zero is an event determined by the local field data modulo constants. By Lemma 10, it suffices to prove that event has probability one for a reference cylinder GFF.

For the reference field, condition modulo constants on the complement of disjoint slightly enlarged domains containing the \(U_i\), still separated from \(V\). The inside fields are independent Dirichlet GFFs. Their critical measures almost surely annihilate each deterministic Lebesgue-null set. Here is a way to see this without an integrability assumption on critical area. For a whole-plane GFF modulo constants, translation stationarity makes the probability of positive mass on a translate of a fixed null set independent of the translation. For each realization, local finiteness and deterministic Fubini show that the mass is zero for almost every translation. Thus it is zero for any fixed translation. Local equivalence transfers this to the interior Dirichlet measures. In particular their expectations after normalization to bounded total mass are absolutely continuous with respect to Lebesgue measure. Independence gives the same statement for their product and therefore for any product-Lebesgue-null set of locations.

The conditioned harmonic parts multiply the inside measures by positive continuous weights; they do not alter this zero-mass conclusion. The field on the observation patch is already fixed by the exterior conditioning, and the product-law null assumption says that the exceptional location section is Lebesgue-null for almost every such observation field. Integration proves the reference assertion, then local absolute continuity proves it for our field. Dividing by the total area does not change a zero event. The same reasoning, or the usual local first-moment formula, applies subcritically. ◻

Proof. We must transfer deterministic-chart statements to maps depending on area samples. Cover configurations of old and new marks and observation patches by countably many location boxes with the separations in Lemma 11 and compact margins for the coordinate maps. For each deterministic choice of locations in one box, the local Gaussian law and the chaos statements hold in the observation patch, after subtracting a test-function mean. Integrate these statements first against a bounded location density, then transfer them by the joint absolute continuity of Lemma 11. For convergence in probability the same argument first restricts the Radon–Nikodym density to be bounded and then exhausts. Finally restore the random field constant by its scalar exponential factor; the undifferentiated critical term still vanishes. This proves both local Gaussian absolute continuity after the random map and derivative convergence in the common nonsingular region.

For the chart using the two ends and one extra area sample, that region exhausts the sphere minus \(0,1,\infty\). The local derivative limit there is the pushforward of \(\nu_{\mathcal C}\), by the fixed-field construction and (9). The latter probability measure is diffuse and has full support, and charges none of the three marks. It is therefore the unique no-atom extension of the local derivative measures. Those measures are local measurable functions of \(h\), so the extension is measurable too. Proposition 9’s resampling assertion is consequently an assertion about samples of exactly this derivative-normalized measure. The same separated-patch argument proves its covariance under that re-marking operation. Finally a deterministic local field restriction has zero probability under a nontrivial Gaussian field law. Local absolute continuity modulo constants therefore precludes a deterministic limiting field and proves nondegeneracy. No convergence of subcritical metrics has been used. ◻

Proof. Take as reference \(\omega\) a normalized whole-plane GFF plus an independent Gaussian constant of positive variance. Its law is equivalent under deterministic constant shifts and charge-shifted similarities: the modulo-constant field is similarity invariant, and the remaining constant has a full-density conditional law. Locality supplies an internal metric functional on each patch. Normalize its input by a fixed smooth test-function mean, writing the normalized restriction as \(\eta\). Conditional on \(\eta\), the reference additive constant has an everywhere positive density. Weyl scaling for fixed constants, equivalence under shifts, and Fubini therefore show that \[c\longmapsto e^{-\xi c}D_{\eta+c}\] is essentially constant in \(c\), almost surely in \(\eta\). Indeed the evaluations agree for almost every pair of constants. Define the local functional of \(\eta\) to be this common essential value, then multiply by the exponential of the actual mean. Countable dense distance entries and continuity give a measurable version. It agrees with the reference metric and is exactly constant-homogeneous wherever defined. Thus it transfers to arbitrary laws absolutely continuous modulo constants, even when their actual constants have no density.

For each deterministic smooth addition, the reference Weyl identity is a local event invariant under a common additive constant. Smooth additions preserve the local reference measure class, so the same transfer gives Weyl scaling for the constructed patch metrics.

Restriction consistency and agreement on overlaps follow from reference locality and transfer on still smaller patches. Similarity covariance transfers in the same way: the transformed reference law uses the same internal functional by its measure-class equivalence. We use countable bases and overlaps within each fixed assertion.

Paste the compatible length data by subdividing paths into patches, or equivalently by taking infima of finite chain sums of internal distances. Intermediate chain points may be restricted to a countable dense set, which proves measurability. Connectedness gives finite chains. To exit a sufficiently small relatively compact neighborhood of a point costs a strictly positive local distance; hence the pasted distance separates points and induces the original topology. For sufficiently near endpoints, short minimizing sequences cannot exit a fixed compact neighborhood. Parametrizing them with bounded speed and applying compactness gives a local minimizing geodesic. This argument uses only interior distances and requires no extension of a patch metric to its boundary. ◻

Proof. We first specify the sufficient tests, in rescaled units. For (i), compare the two \(B\)-internal metrics on all pairs of old radii \(2\) and \(4\), evaluating the new entries at their \(\psi_{p,r}\)-images, with factor \(1+\delta_0\). For (ii), bound old \(B\)-internal entries on the \(2\)-circle by \(C\) times the infimum of old \(B\)-internal entries between circles \(2\) and \(4\). For (iii), require a short winding-one loop in the new closed ring \(2.25\leq|u|\leq2.75\), with length bounded by \(C\) times the corresponding infimum of old \(B\)-internal entries between circles \(1\) and \(2\). Containment and locality imply the conclusions in the definition: the internal-distance infima are lower bounds for the constrained crossing costs, and the new fixed ring lies in the old band \((2,3)\) for small \(r\).

These are measurable tests. In particular, bounded-length winding-loop existence in a compact ring can be tested using bounded-speed parametrizations. For continuous metrics inducing the ring topology, uniform convergence of metrics and compactness of such paths imply lower semicontinuity of the required length bound. Uniform convergence away from the center preserves winding number. Thus the loop test is a measurable condition on the internal metric.

Uniform comparison of the input laws. We claim that the two internal metrics in (13), evaluated at corresponding points, approach one another uniformly in probability on compact sets of entries in \(B\). First we compare their field laws. Choose domains with fixed margins, \[B\Subset W_0\Subset W=A(0.8,7)\Subset A^+\Subset A, \qquad W\Subset\psi_{p,r}(A).\] For either \(D=A\) or \(D=\psi_{p,r}(A)\), the Markov decomposition on \(W\) writes the relevant restricted field as \[X_W+H_D+u_D,\] where \(X_W\) is a zero-boundary GFF on \(W\), independent of the harmonic Gaussian field \(H_D\). The smooth term \(u_D\) includes the prescribed addition and, in the transformed coordinate, the logarithmic derivative. Choose and subtract a constant for which the prescribed addition has \(C^2\) norm at most \(M+1\). Both metrics acquire the same scalar factor, so all the annular tests are unchanged.

The domains \(D\) contain \(W\) with uniform margins and lie in one fixed larger domain. Differences of their Green functions therefore bound the variance of \(H_D\) uniformly on \(W_0\). Harmonic interior estimates give uniformly tight smooth norms there. Choose a smooth cutoff \(\chi\) compactly supported in \(W_0\) and equal to one near \(\overline B\). The assumed \(C^2\) bound for the additions, together with \(\psi_{p,r}\to\mathrm{id}\) smoothly, implies that the Dirichlet norms of \(\chi(H_D+u_D)\) are uniformly tight. Condition on \(H_D\). On the event that this norm is at most \(R\), apply the Cameron–Martin formula and Cauchy–Schwarz, then integrate over \(H_D\). For every measurable set \(E\) of restrictions to \(B\), this gives \[\mathbb P\{(X_W+H_D+u_D)|_B\in E\} \le \varepsilon+e^{R^2/2}\mathbb P\{X_W|_B\in E\}^{1/2},\] where \(R\) is chosen so that the norm bound fails with probability at most \(\varepsilon\), uniformly in all the parameters. Thus both input laws are uniformly absolutely continuous with respect to the same reference restriction law.

Continuity in probability of the metric comparison. Under their original coupling the two fields in (13) approach one another in a local negative Sobolev topology, uniformly over the stated centers and additions. Apply Lusin’s theorem to the measurable map from the restriction to its \(B\)-internal metric, viewed as a continuous distance function on each compact set of entries. On a compact set of arbitrarily large reference probability this map is uniformly continuous. The preceding uniform absolute continuity controls the probability that either input leaves that set, and closeness of the coupled inputs then controls the two metrics. Their tight continuity moduli also absorb the endpoint motions by \(\psi_{p,r}\). This proves the claim; it uses measurability of the metric functional and comparison of the input laws.

The three geometric tests. Internal-distance infima between two disjoint compact circles are strictly positive, and compact suprema are finite. Consequently the first two tests have probability arbitrarily near one when \(C\) is large and then \(r\) is small. The least length of an admissible winding loop is tight by the same uniform absolute continuity. For each continuous-topology length metric, some finite-length winding loop exists: approximate a middle Euclidean circle by joining sufficiently many successive nearby points using short local paths which remain close to the circle. This proves the third test, uniformly under the stated bounds on the additions. ◻

Proof. Fix \(\lambda>20\) and, for a deterministic center \(p\), use \(r_j=\lambda^{-j}\), \(m\leq j\leq2m\). The annuli \(p+r_jA\) are disjoint. Decompose the whole-plane field into their independent Dirichlet pieces and the independent projection harmonic on all of them, modulo constants. Let \(H_j\) denote the rescaled harmonic restriction, normalized by subtracting a smooth average on \(A''=A(0.6,7.8)\). For all sufficiently large \(N\), \[ \mathbb P\!\left(\sum_{j=m}^{2m}\|H_j\|_{L^2(A'')}^2>Nm\right) \leq e^{-cNm}. \tag{14}\] The constant is uniform in \(p,m\) on the compact patch. To verify the bound, the diagonal covariance kernels are uniformly bounded after the Dirichlet log singularity and the common average have been removed. For distinct annuli, the cross-covariance is the whole-plane logarithmic kernel double-centered by the averages; it is \(O(\lambda^{-|j-k|})\). The covariance on the direct-sum \(L^2\) space therefore has bounded operator norm and trace \(O(m)\). Diagonalization and a small fixed exponential moment of its squared Gaussian norm give (14).

Outside this exception, at least half the indices have bounded harmonic \(C^2\) norm on \(\overline{A^+}=\overline{A(0.75,7.25)}\), with a bound depending only on \(N\). Harmonic interior estimates give this bound from the \(L^2(A^{\prime\prime})\) bound, since \(\overline{A^+}\subset A^{\prime\prime}\). Thus the exact norm and fixed margins required by Lemma 15 hold. Conditional on the common harmonic projection, the corresponding tests use independent Dirichlet pieces. For those indices their failure probabilities are at most any prescribed \(q>0\), once \(C\) and then the minimum scale have been chosen. Hence the probability of no success at the center is at most \[e^{-cNm}+q^{m/2}.\] Choose \(N\) first, then \(q\), so that this is bounded by \(e^{-c_*m}\) with \(c_*>5\log\lambda\), increasing the lower cutoff on \(m\) if necessary.

At level \(m\), a lattice of spacing \(\lambda^{-2m}/10\) has \(O(\lambda^{4m})\) centers near the compact patch. The union bound and Borel–Cantelli show that, almost surely, every such center has a success for all sufficiently large \(m\). Every point is within \(\lambda^{-2m}/10<r_j/2\) of a lattice center and hence has a successful annulus enclosing it at a scale tending to zero. Only countably many deterministic centers and radii were used for similarity covariance. Finally exhaust compact patches and intersect over the prescribed countable sequence of errors. No quantitative rate in the Lusin argument was required: its failure probability was chosen before the minimum scale. ◻

Proof. First work with the reference field and a fixed \(\delta_0>0\) on the probability-one event in Lemma 16. We give two steps, since no Euclidean Hölder estimate is available or needed.

A finite Lipschitz bound along geodesics. Let \(P:[0,\ell]\to U\) be any nonconstant minimizing \(d\)-geodesic in the patch, parametrized by \(d\)-length. At each interior time choose a successful small loop surrounding \(P(t)\), with both endpoint images outside its enclosed region. Let the nearest hits on this loop before and after \(t\) delimit an open time interval. Throughout that interval \(P\) remains in a bounded complementary component with nonzero winding. The loop’s \(\widetilde d\) length is at most \(C\) times the interval’s duration: the geodesic crosses the corresponding band \([r,2r]\) there.

For \(0<\varepsilon<\ell/2\), finitely many such intervals cover \([\varepsilon,\ell-\varepsilon]\), with all loop diameters as small as desired. Prune to a minimal interval cover. Its intervals form an ordered chain without containments and have multiplicity at most two. Consecutive loops intersect. Indeed their hit intervals interlace, so a point of each loop lies in a bounded complementary component of nonzero winding of the other. If the connected loops were disjoint, each entire loop would lie in such a component of the other. This is impossible, for example by taking their extreme horizontal coordinates. The argument does not require simple loops. Concatenating along the intersecting chain costs at most \(2C\ell\). Let the loop mesh tend to zero and then \(\varepsilon\downarrow0\). Continuity gives \[\widetilde d(P(0),P(\ell))\leq2C\ell.\] The argument applies to every subsegment. Thus \(F(t)=\widetilde d(P(0),P(t))\) is Lipschitz on \([0,\ell]\).

The sharp bound at differentiability times. At an interior differentiability time \(t\) of \(F\), take successful annuli shrinking to \(P(t)\). Let \(u<t<v\) be the nearest visits on either side to old radius \(4r\), and let \(k,s\) be the first and last visits to radius \(2r\) between them. All four times tend to \(t\), by simplicity and compactness of a minimizing geodesic. Its minimality and test (ii) imply \[s-k=d(P(k),P(s))\leq C(k-u).\] Both \([u,k]\) and \([s,v]\) are traversals eligible for test (i), and \[v-u\leq(C+2)\max\{k-u,v-s\}.\] One eligible traversal therefore occupies a fixed positive fraction of the whole shrinking interval. Test (i) bounds the absolute change of \(F\) on it by \(1+\delta_0\) times its duration. Differentiability at \(t\) has error \(o(v-u)\), which remains negligible after division by that traversal duration. Hence \(|F'(t)|\leq1+\delta_0\). Integrating the Lipschitz function gives \[\widetilde d(P(0),P(\ell))\leq(1+\delta_0)\ell.\] Local geodesic accessibility proves this bound for all sufficiently near points. Pass through the countable sequence \(\delta_0\downarrow0\). Apply the same argument in the opposite coordinates, whose field is locally absolutely continuous modulo constants, to obtain the reverse inequality. The germs agree, and therefore all intrinsic lengths agree. Local absolute continuity transfers the result from the reference field to the stated class of fields. ◻

Proof. Use the third successful-annulus test with \(f=\operatorname{id}\). Fix a local finite minimizing segment ending at the point under consideration. The cost of crossing the shrinking displaced bands is bounded by the length of a tail of that segment and hence tends to zero. Lemma 16 holds simultaneously at every point, so this argument gives the same all-point quantifier for the loops. It transfers by local absolute continuity.

For random maps, use Lemma 11 on countably many configuration boxes and observation patches avoiding all relevant marks. Internal metric comparisons on smaller neighborhoods are measurable local tests, including as the Möbius parameters vary; one can use fixed countable coordinate bases. For each deterministic parameter the proposition holds almost surely. Integration against the product reference law and then joint absolute continuity transfer it to the sampled parameters. This is why fixed-map almost-sure covariance alone is not the argument for a mark-dependent coordinate change. ◻

Proof. Couple the original three-marked sphere with three fresh independent area samples, and express the same abstract sphere in the coordinate defined by the fresh triple. Diffuseness makes the two triples disjoint almost surely. By Proposition 9, the field in each coordinate has law \(\mathsf P_{\rm crit}\). Pull both local length structures back to the common abstract sphere. Each is defined away from its own exceptional triple, and their lengths agree away from the union of the two triples by Corollary 18. Its random-map assertion applies using cylinder observation patches separated from the ends and all sampled locations.

The two regular regions cover the abstract sphere. Their compatible local length structures therefore paste as in Lemma 13, giving a finite length metric inducing the sphere topology. This metric space is compact. Every point of either exceptional triple has arbitrarily short surrounding detours because it is regular for the other coordinate. It remains to show that removing one triple does not change intrinsic distances between the remaining points.

For a path whose endpoints avoid one triple, choose short separating loops in arbitrarily small disjoint neighborhoods of its three points, avoiding the path endpoints. Replace the path between its first and last encounter with each loop by an arc of that loop. The increase in length is at most the loop length, and no new visit to the exceptional triple is introduced. Sending these costs to zero shows that intrinsic distances avoiding the triple equal the pasted distances there.

The original punctured metric is thus exactly the restriction of this compact metric sphere. Its completion adds the three distinct missing points and no others. It is determined measurably by the original field: construct its punctured local metrics using countable entries and take their unique continuous extension. The auxiliary chart proves the extension property and contributes no metric randomness. Covariance on the common regular regions extends uniquely over the finite exceptional sets. The same applies to the already identified area, proving the asserted fixed-chart and re-marking rules. ◻

Proof. Use a centered flat-torus GFF as reference. It is translation invariant and has the same interior measure class modulo constants. Set \(T=\log\varepsilon^{-1}\). Circle-average variance is at most \(T+O(1)\), and the variance of an increment between centers at distance \(b\leq\varepsilon\) is \(O(b/\varepsilon)\). This follows from the logarithmic kernel plus its smooth remainder: one circle average already gives a Lipschitz bound \(O(\varepsilon^{-1})\) for the logarithmic part.

Gaussian union bounds on a grid of spacing \(\varepsilon T^{-2}\) give an upper bound \(2T+O(\log T)\) for the grid maximum with probability tending to one. Use dyadic grid refinements, with increment thresholds \(2^{-j/4}\) at refinement \(j\). Their variances are \(O(T^{-2}2^{-j})\); the Gaussian tails, including the number of grid edges, are summable in \(j\). Continuity then gives \[\max h_\varepsilon\leq2T+O(\log T) \quad\hbox{with probability tending to one}.\] Seneta–Heyde convergence gives undifferentiated total critical mass \(O_{\mathbb P}(T^{-1/2})\). The negative derivative mass is at most this mass times \((\max h_\varepsilon-2T)_+\), hence is \(O_{\mathbb P}(\log T/\sqrt T)=o_{\mathbb P}(1)\). Convergence of the signed total mass now implies tightness of the total variation.

Divide the variation measure by one plus its total mass. The resulting bounded random measure is stationary, so its expectation is a constant multiple of Lebesgue measure with uniformly bounded density. Together with variation tightness this gives uniformly small mass in probability on deterministic sets of small Lebesgue measure. The limiting measure has the same property after the analogous normalization. Approximate a bounded Borel test function in Lebesgue measure by bounded continuous functions; these small-set bounds and tightness control the error uniformly, and local weak convergence handles the continuous approximations. This proves the Borel assertion. Applying it to a compact-set indicator times a continuous test function, then using tight variation, proves weak convergence on each deterministic compact restriction.

Normalize by a local test-function mean to transfer through absolute continuity, then restore the actual constant using the identity in the proof of Lemma 5. The undifferentiated term still vanishes. A finite cover of a compact patch completes the transfer. ◻

Proof of Theorem 1. Proposition 9 constructs the unique limit of the prescribed subcritical probability laws in the required norm field and weak measure topologies. Proposition 12 supplies its nondegeneracy, derivative normalization, full support, diffuseness, unit total mass, measurability, and area-mark resampling rule. Lemma 21 gives the stronger compact-restriction interpretation if used. Proposition 19 constructs the required field-measurable finite critical metric and identifies its completion and coordinate rules.

Finally take the product of the entire marked \((H,\nu)\) law with the fixed whole-plane nested \(\mathrm{CLE}_4\) probability. Its Möbius invariance is (Kemppainen and Werner 2016, Theorem 1), whose parameter range includes four. This defines the specified independent decorated continuum law. No discrete subsequence or assumption about convergence of subcritical metrics enters any part of this construction. ◻

Proof. Slitting an oriented occupied primal root edge to make a root face of degree two, with the same orientation on the copy with slit on its right, does not alter the number of triangles or interfaces. Even a self-edge gives a degree-two face by inserting two copies in the map’s rotations; the boundary can pinch at the vertex. Conversely a disk of degree two with internal triangles always has two distinct boundary edges (if both boundary incidences are the same bridge no boundary length is left for either attachment, hence no triangles). With boundary blue, close the degree-two face to an occupied edge to undo the slit. Since primal-dual exchange preserving weight gives exactly half the weight to occupied roots, and the bridge-only disk has weight one, \[ F_2(x)=1+\frac12\sum_{n\ge1}Z_n(4)x^{2n}. \tag{16}\]

The outer gasket, obtained by retaining edges accessible without crossing a loop, has the ordinary boundary-map face weights \[g_j(x)=2\sum_{l\ge0}\binom{j+l-1}{l}x^{j+l}F_l(x)\qquad(j\ge1).\] Indeed each internal face of the gasket is replaced with a triangle ring carrying an outermost loop and then an independent disk of opposite boundary color. With a specified first outer base, the ring is determined by its interleaved cyclic list of \(j\) outer and \(l\) inner bases. Radial links between successive triangles encode the boundary correspondence, with a rooted inner disk read starting at a prescribed inner base (or a single point if no such base); corners of faces and triangle ring boundaries need not have distinct labels in the assembled map. Deletion/retention of all the edges not belonging to the ring gives the inverse, including in pinched cases. In particular peeling the root edge of the gasket satisfies \[ F_p=\sum_{i=0}^{p-2} F_i F_{p-2-i}+\sum_{j\ge1}g_j F_{p+j-2}. \tag{17}\] The first step either deletes a root bridge incident to the root face on both sides or deletes a root edge merging an internal face with the root face. Each exposed disk (and each loop interior) has fresh Boltzmann law given its length, by the factorization of the weights with distinguished boundary edge occurrences. ◻

Proof. Indeed divide \(\Phi(z)=\int r(s)/(z-s)\,ds\) by \(D(z)=\sqrt{(z-A)(z-B)}\sim z\); the jump from lower to upper value (upper minus lower) of that quotient is \((\sqrt t-c)/(i d(t))\), so \[\Phi(z)/D(z)=\tfrac12\,\mathbb E[(U-c)/(z-U^2)].\] Here one can apply Cauchy’s formula deforming onto the cut (Sokhotski–Plemelj in the regular interior; equivalently distributional boundary values). Endpoint circle integrals vanish by boundedness of \(r\): the integral of \(|\Phi|\) along an endpoint circle of radius \(\epsilon\) is \(O(\epsilon\log(1/\epsilon))\) by integrating the absolute kernel first. Decay at infinity gives \(m_1=c\), and taking the jump of \(\Phi\), using the vanishing interior Hilbert transform of the arcsine law, yields (18). For independent \(U,V\) with this same law put \(h(U)=-U^3+(A+B)U-AB/U\). Normalization in (18) now reads \(1=\mathbb E[h(U)/(U+V)]/4\); symmetrizing and using \(m_2=(A+B)/2\) gives \[c^2-AB m_{-1}^2=8.\] It follows as \(x\uparrow X\) that \(\alpha\to0\), \(\beta\to\pi c_*/2\) (by \(m_1=c\)). In particular the left endpoint is strictly away from \(-c_*\), including for the limiting cut of the moment formula at \(X\), obtained by continuity or dominated convergence in (18). ◻

Proof. We mean here a holomorphic continuation to small \(\delta\ne0,\ |\arg\delta|<\pi-\epsilon_0\) for each fixed \(\epsilon_0>0\), with uniform equivalents. Indeed \(J_+\to2/\pi,\ J_+'=O(\log(1/|\sigma|)),\ J_-\sim\log(1/\sigma)/\pi,\ J_-'\sim-1/(\pi\sigma)\) uniformly in such sectors. For example after \(v=\tan\theta\) the integral for \(J_-\) uses kernel \((\sigma+v^2)^{-1/2}(1+v^2)^{-1/2}\); after differentiation one uses \(\int_0^\infty(\sigma+v^2)^{-3/2}dv=1/\sigma\) and the sector bound by powers of \((|\sigma|+v^2)\). Rouché on small fixed relative-radius circles about \(4\delta/\log^2(1/\delta)\) gives a unique solution there and the implicit derivative gives the second estimate. This branch agrees by (19) with the real solution near criticality.

For real \(c>c_*\) near \(c_*\) differentiate at fixed \(t\) in the cut: \[\partial_c r(t)=\frac{t-\eta}{2\pi d(t)},\qquad \eta=\frac{m_1}{m_{-1}}=\frac{c^2}{J_+ J_-}.\] Indeed endpoints depend differentiably by (19), and (18) lets us differentiate under the integrals (integrable endpoint inverse square-root behavior, smooth on compact interior). Thus \(\partial_c\Phi\) has sum of its two boundary values \(-1\) there. Multiplication of \(\partial_c\Phi+1/2\) by \(D\) removes this cut, leaving a polynomial by the endpoint bounds and the expansion at infinity, necessarily of the form \((z'-\eta)/2\) in the Stieltjes-transform coordinate \(z'\); normalization differentiated fixes \(\eta\). In particular with primes in the next line meaning \(c\) derivatives of the moments \(F_p(x)\), \[F_1'=1+(\eta-m_2)/4,\qquad F_2'=2F_1+(m_3-2c m_2+c\eta)/4.\] These identities continue \(F_2(X\sqrt z)\) holomorphically by integration of the moment expressions using (19). In the slit sectors the \(z\) derivatives of \(m_2,m_3\) are bounded (arcsine integral and \(d\sigma/dz\) above), but \(d\eta/dz\sim-\pi^2 c_*^2/(2\delta\log^2(1/\delta))\). Consequently \[ \frac{d^2}{dz^2}F_2(X\sqrt z)\sim \frac{4\pi^2}{(1-z)\log^2(1/(1-z))}. \tag{20}\]

There are no other dominant circle singularities: in (16) iteratively prune pendant tree edges in the original primal map, except the root edge, which is protected. The unique remaining rooted core (with no removable leaves off that edge) has \(m\ge1\) edges; each corner carries an arbitrary attached plane tree (including the empty choice). Uniqueness also follows since the pruning keeps the cycles, the root and the paths needed to connect them, removing the attached unprotected trees. Appending an open leaf keeps \(\ell\) unchanged, a closed one multiplies the weight by 2, by the displayed FK formula. Rooted cores have no residual rotation symmetry. Thus \(\sum_{n\ge1} Z_n(4)w^n=C(wT(w)^2)\), where \(T=1+3wT^2\) counts the attachments and \(C\) is a nonnegative power series counting the cores. Since \(X^2=1/32<1/12\), \(T\) is regular, \(C\) is finite at \(X^2T(X^2)^2\) by (16), and all other points of \(|w|=X^2\) have strictly smaller \(|w T(w)^2|\), hence regular extensions. This gives the indented-disk analyticity required in the Flajolet–Odlyzko singularity transfer theorem (Flajolet and Odlyzko 1990), together with (19). Applying the logarithmic transfer (power exponent \(-1\)) to (20) and then (16) proves the first equivalent of (15). ◻

Proof. Indeed for \(x=X e^{-\lambda v_p}\), fixed \(\lambda>0\), \(p\alpha/c_*\to z_0=\pi\sqrt{\lambda/2}\). In the moment integral in \(u'=c-y\in[\alpha,\beta]\), use \(v=p u'/c_*\); \((y/c_*)^p\to e^{-v}\) for bounded \(v\). For \(v>z_0\), \[\frac{p}{c_* \log p}\,r((v c_*/p)^2)\ \longrightarrow\ \frac{\sqrt{v^2-z_0^2}}{\pi^2}.\] In fact the arcsine expectation of \(1/(U+v c_*/p)\) in (18) is asymptotic to \(2\log p/(\pi\beta)\), by its sine-integral representation; insertion of \(v c_*/p\) in the denominator changes \(m_{-1}\) only by \(O(1)\), since \(\alpha\) has order \(1/p\). The rescaled densities here are bounded by \(C v\) from (18) and \(m_{-1}\), so exponential domination where \(y\ge0\) suffices; the region \(y<0\) contributes exponentially little. Dividing by \(F_p(X)\), the limiting Laplace transform is \(\int_{z_0}^\infty e^{-v}\sqrt{v^2-z_0^2}\,dv\), equal to \(\int_0^\infty\exp(-u-z_0^2/(4u))du\) (change variable \(v=u+z_0^2/(4u)\), both branches, and integrate by parts). This gives (21). ◻

Proof. Parity follows by counting triangle incidences. Here are details to control Fourier inversion. With \(\widetilde g_i=c_*^i g_i(X)\) one has

\[|f_{i+1}-f_i|\le C f_i/i,\qquad \widetilde g_i=2f_i+O(f_i/\sqrt i).\] The first bound follows using the moment formula at \(X\), the negative endpoint gap, \(t^i(1-t)\le (C/i)t^{i/2}\) on \([0,1]\), and the two-sided moment equivalent used above. The second follows by representing \(\widetilde g_i/2\) as the expectation of \(f_j\) at \(j\) the sum of \(i\) iid geometric variables (mean one, distribution \(2^{-l-1}\) at \(l\ge0\)); use exponential deviations, first absolute fluctuation \(O(\sqrt i)\) and the first bound.

In (17) follow always the larger-perimeter disk at a negative (bridge) step, with deterministic tie breaking, and the external remainder at a face step. At current perimeter \(P=m\) large the next perimeter \(P'\) is \(m+d\); \(d=-i-2\) for \(0\le i\le(m-2)/2\), with probability \(c_*^{-2}w_d^- f_{m+d}/f_m\) where \(w_d^-=2 f_i\) (just \(f_i\) for the tie); or \(d=k-2\ge-1\) with probability \(c_*^{-2}w_d^+ f_{m+d}/f_m,\ w_d^+=\widetilde g_k\). For fixed \(a>0\), \[\mathbb E[(P')^{-a}\mid P=m]\le m^{-a}(1+C_a\log m/m).\] Indeed in the difference from \(m^{-a}\) one can drop \(d>m\) (negative terms). In the remaining range \(1+d/m\) is bounded away from zero, \(f_{m+d}/f_m=1+O(|d|/m)\), and Taylor expansion gives linear contribution bounded relatively by \(C_a\log m/m\) and remainder by \(C_a m^{-2} O(m\log m)\). Here \(\sum d(w_d^-+w_d^+)=O(\log m)\) on the indicated ranges (sum separately over each sign’s stated indices), by \(\sum i|\widetilde g_i-2 f_i|<\infty\) and the log-over-square tail, which likewise gives \(\sum d^2(w_d^-+w_d^+)=O(m\log m)\).

Let \(\phi_p(\theta)=\mathbb E e^{i\theta N_p}\). For sufficiently large \(K\), sufficiently small fixed \(\delta>0\), and \(K v_p\le\theta\le\delta\), take \(q>e\) solving \(\theta=\log^2 q/q^2\). Put \(s=\sqrt{pq}\) and run at most \(\lfloor s/\log s\rfloor\) steps, stopping on crossing below \(s\); \(p\ge s\ge10q\) by choice of \(K\). The probability of crossing is \(O_a((s/p)^a)\), by the stopped inverse-moment bound iterated (overshoot stays positive). While above threshold, with probability at least \(c_1\log q/q\) a bridge step splits off an independent disk of perimeter \(i\in[q,2q]\). By (21), \(\sup_{q\le i\le2q}|\phi_i(\theta)|\le1-c_2\) for all large \(q\), since the limiting characteristic function has modulus strictly below one on the relevant compact positive frequency interval. Conditional on the peeling step types/perimeters, the swallowed disks are independent, contributing additively to the volume. Hence by successive conditioning (drop the threshold survival indicator at each last step for the nonnegative product bound) \[|\phi_p(\theta)|\le C_a(q/p)^{a/2} +(1-c_1 c_2\log q/q)^{\lfloor s/\log s\rfloor} \le C'_a(v_p/\theta)^{a/4}.\] We used that \(\sqrt{p/q}\log q/\log s\) grows faster than \(\log(p/q)\), and \(q/p\le\sqrt{v_p/\theta}\). For \(\delta\le\theta\le\pi/2\) the same argument with fixed \(q\) and just perimeter \(i=2\) swallowed gives the last bound, since \(N_2\) can be 0 or 2. This gives integrable domination on the rescaled frequency axis (take \(a>4\)); Fourier inversion on the parity lattice and (21) now give (22) by \(L^1\) convergence on that axis. ◻

Proof of the intensity assertion in Proposition 22. To mark a loop, cut out its triangle ring, leaving two disks with boundary degrees \(k,l\), colors blue and red. This is the same ring decomposition as for gasket faces but now with two fillings. Reroot at one of the \(k+l\) successive links distinguished in the ring (orientation specified by colors), replacing the directed FK root (with \(2n\) choices); counts have the corresponding ratio \((k+l)/(2n)\) since both markings kill sphere-map automorphisms. Traversing from the link with blue side in a given orientation specifies the list interleaving both bases and hence disk boundary roots (if positive lengths). Thus for \(k+l>0\), \[\mathbb E\#\{\text{loops of type }(k,l)\}= \frac{4n}{k+l}\frac{\binom{k+l}{k}2^{-k-l} f_k f_l}{X^{2n}Z_n(4)} \ \mathbb P(N_k+N_l=2n-k-l),\] where the fillings are independent before total conditioning. Also \(u-N_k/(2n)\) is between zero and \((k+l)/(2n)\), since each Tutte triangle has exactly two flags. For \(k/B_n\) in compact subsets of \((0,\infty)\), summation over \(l\) is exponentially concentrated at \(l/k=1+o(1)\) by the binomial kernel of total mass 2 (tails outside e.g. \(k\pm k^{3/4}\) dominate no polynomial factor; \(f_j\) are bounded). For \(k\sim sB_n\), the perimeter and area units satisfy \[v_k\sim\frac{\pi^2}{16s^2n},\qquad \frac{N_k}{2n}\Longrightarrow\frac{s^2}{E},\qquad f_k,f_l\sim\frac1{s^2n\log n}.\] The span-two lattice in Lemma 27 therefore gives \[\mathbb P(N_k=j)=\frac1n\bigl(d_s(j/(2n))+o(1)\bigr), \qquad j\equiv k\pmod2,\] uniformly in \(j\) and for \(s\) in the stated compact range. Thus the area mesh for one filling is \(1/n\). This lattice factor is distinct from the loop fugacity \(2\), the rerooting factor \(2n/(k+l)\), and the total mass \(2\) of the ring kernel summed over \(l\). The total constraint leaves the first fraction on a grid of spacing \(1/n\) with the complementary parity automatically satisfied. Its constrained probability measure multiplied by \(n\) thus tends weakly to \(d_s(v)d_s(1-v)\,dv\) along every convergent \(s\), with the same limit using \(u\). Inserting and then summing with \(s\)-spacing \(1/B_n\) proves the second formula of (15). ◻

Proof. It is enough to prove this for mixed half-disks on a compact subinterval of the flat edge with nonempty interior; continuous Gaussian-tail corrections and Hölder handle the local applications. For positive moments pack \(c k\) disjoint translated scaled copies of the domain along the subinterval with scale \(c'/k\) using the bulk packing argument of Section 2. The Markov complementary harmonic field is independent of the small fields; its restrictions to the copies, estimated on the observed subintervals, have variances \(O(\log k)\) there and on fixed relative enlargements. The probability of a minimum below \(-B\log k\) is \(O(k^{1-cB^2})\) by reflected harmonic interior estimates. (We only claim independence of the whole complementary field from the small GFFs, not independence between its restrictions.) Tightness of the mass on the original interval follows by the local limit. Thus exactly as before the maximum of \(c k\) iid masses gives a polynomial tail estimate for the single mass. Conversely absence of one of these masses \(\ge\delta>0\) has exponentially small probability by uniform positivity. On the good harmonic event this bounds the original mass below by \(c'' k^{-C(1+B)}\); taking \(B\) large proves every desired inverse moment. Constants and laws on all rescaled copies use the original standard flat-coordinate normalization. We will use such inverse estimates also independently of a semicircle coarse tilt: at scale \(e^{-t}\) about a boundary point \(x\) of the Neumann half-plane, write \(H_t\) for the semicircle average at that radius (fix the additive constant by \(H_0=0\) at some given center, or subtract the local starting average). On tilting by \(\exp(H_t-\operatorname{Var}(H_t)/2)\), the normalized length of an interval at distance and with size comparable to \(e^{-t}\) from \(x\), divided by \(\exp((\gamma/2)(H_t-Q_\gamma t))\), still has inverse moments of all orders uniformly. This follows by rescaling: the tilt adds a covariance which, after subtracting its semicircle average shift, is bounded on the fixed rescaled interval, by the averaged logarithmic kernel; the residual random field there is a standard rescaled free field with tight Gaussian choice of its residual constant. The statement is unchanged with tilt by a bounded multiple of \(H_t\) with its variance compensation. ◻

Proof. In fixed interior nested disks pack \(\asymp k^2\) disks on a mesh of order \(1/k\), of disjoint larger closure, and test copies of a fixed scaled compact inner-disk mass. The harmonic outside fields \(Y_i\) at centers have variances \(v_i=\log k+O(1)\), with pairwise covariances bounded above by \(\log |x_i-x_j|^{-1}+O(1)\). Oscillations on the inner disks about the centers have uniform Gaussian tails, also when individually tilted by \(\exp(\alpha Y_i-\alpha^2 v_i/2)\) for fixed \(\alpha\); this uses the Green kernels of the disk and subdisks and harmonic interior bounds. Choose small \(\alpha>0,\ \alpha^2<2\). The sum of these weights divided by \(k^2\) has bounded second moment (mesh integral of the covariance exponential). Keeping only terms with \(|Y_i-\alpha\log k|\le\epsilon\log k\) and oscillation at most \(\epsilon\log k\) its first moment stays bounded below. Thus with positive probability independently of \(k\) there are at least \(c k^{2-\alpha^2/2-\alpha\epsilon}\) good indices, determined independently of the small GFFs. On them the scale factor for the small normalized mass to the original mass is at least \(c' k^{-4+2\alpha-C\epsilon}\) uniformly for \(\gamma\) sufficiently close to 2. Tightness of the original mass and the iid copies now bound the single tail at \(C' k^{4-2\alpha+C\epsilon}\) by \(O(k^{-2+\alpha^2/2+\alpha\epsilon})\), as the above good-count probability stays bounded below. For small \(\epsilon\) this proves the moment assertion. ◻

Proof. Consider one unit length of straight boundary in the half-plane, extending tangentially as necessary, with Neumann GFF modulo constants. Use the free-field representative with reference semicircle average zero of a large radius \(R\) at the midpoint, strictly enclosing the test neighborhoods. Then \(H_n(x)\), the semicircle averages at radii \(e^{-n}\), are Gaussian with independent Brownian increments of variance rate 2 at each fixed center and uniformly bounded starting variances. This follows from even reflection of the log kernel; in the test neighborhoods the reference-circle logarithmic potential is constant. We will use two elementary uniform barrier estimates. Tilt at \(n\) by \(\exp(H_n(x)-\operatorname{Var}(H_n(x))/2)\); then \(X_i=2i-H_i(x)\) for \(1\le i\le n\) are a centered-diffusion variance-2 Gaussian walk with bounded initial shift law. Uniformly, \[{\mathbb P}_*(X_n\in[-y+j,-y+j+1],\ X_i\ge-y,\ 1\le i<n) \le C(1+y+|j|+\log(2+n))^C n^{-3/2},\quad y\ge0.\] This is just the killed Gaussian bridge estimate: between integer times, a Brownian bridge with these endpoints above \(-y\) (except possibly at \(n\)) stays above \(-y-|j|-2-\sqrt{C'\log(2+n)}\) with probability bounded below; then reflection on the whole interval and integration of the initial Gaussian give the bound. The oscillation \(O_n\) of \(H_n\) over center displacements \(\le C'e^{-n}\) from \(x\) has uniform Gaussian tails, including under the tilt. Indeed rescaled log-circle-average fields have increment variances bounded by \(C|\Delta x|^\eta\) for some \(\eta>0\) (average the log potential \(-\log(1\vee|z-u|)\)); the tilt shift in increments is bounded.

Except for probability \(C(1+y)^C e^{-y}\), \(X_n(x)\ge-y\) simultaneously on our boundary interval for every \(n\ge1\). At a global first failing level take a nearest grid center of spacing \(\le e^{-n}\) on the interval; it has survived on all earlier levels and in the endpoint band \(j\) failure needs \(O_n\ge j\). Undo the tilt and sum over \(O(e^n)\) centers at cost \(C e^{-y+j}\) times the tilted probabilities. For \(j>0\) or when needed below, Hölder keeps a power \(b>2/3\) of the ballot bound, while retaining a Gaussian oscillation tail; now sum over \(j\in\mathbb Z,n\ge1\). Compact horizontal extensions introduce no change. ◻

Proof. Test the area in interior Whitney cells near this line at layer \(n\) (distance \(\asymp e^{-n}\)) with \(O(e^n)\) cells, of sufficiently small fixed relative diameter. Choose associated semicircle center on the line and average at a comparable outer scale (e.g. \(n\) replaced by \(n-C_0\) for all cells, an integer). The cell is in a larger interior disk disjoint from the semicircles at these and all earlier levels. Up to bounded factors its normalized area is bounded by \[\exp(\gamma(H_{n-C_0}-Q_\gamma n))\,Y\,Z ,\] where \(Z\) has the independent rescaled compact Dirichlet mass law and \(Y\) has all moments uniformly under the tilt at this last level (supremum of the complementary field minus \(H_{n-C_0}\), by rescaled Green estimates); all barrier and endpoint tests at this cell center are independent of the Dirichlet field. Constants can depend on \(C_0\). Choose \(p\in(1/2,1)\) from the moment just proved, and keep \(\gamma p>1\) uniformly. On the global good-barrier event, bound the tilted expectation of the \(p\)-power of the cell mass bandwise using independence for \(Z\), Hölder with all moments for \(Y\), and power \(b\in(2/3,1)\) of the ballot bound at \(j\ge0\). After summing centers this gives at each large \(n\) an upper bound \[C e^{(\gamma p-1)y}(1+y+\log(2+n))^C n^{-3b/2}.\] Here \(Q_\gamma\ge2\); no strictly positive charge gap is used. Thus at fixed \(y\), tails over Whitney layers vanish in probability on the good event. Taking thresholds of order \(e^{\gamma y}\) also gives uniform small positive moments of total mass by subadditivity, including the finitely many deeper-interior coverings. This proves the desired estimates for the lateral strip field as well by local Neumann decomposition/comparison (adding a continuous Gaussian-tail term, Hölder); it proves convergence of masses to the interior limit without boundary atoms.

Over the infinite strip, independence and \(\mathbb E\exp(-b\inf_{[k,k+1]}U_a)=O_b(k^{-3/2})\), as in Section 2, now finish the joint convergence of finite measures, fields locally, and boundary labels up to the ends. The tilt \(V_\gamma^{2a/\gamma}\) converges to 1 in \(L^1\). Disk area at fixed initial-units perimeter therefore has a strictly positive finite limiting distribution. For clarity, the convergence of the boundary conversion scalar follows already from the area distributions. If \(b_\gamma\) is that scalar, a disk of perimeter one in the initial units has area law \(b_\gamma^2/E_r\), where \(E_r\) has the Gamma\((r,1)\) law. The area variables just proved to converge have a finite strictly positive limiting law, while \(E_r\) converges to an exponential variable as \(r\downarrow1\). These facts first bound \(b_\gamma\) away from zero and infinity. Every subsequential limit \(b\) then gives the law \(b^2/E\); its strictly monotone distribution function, or any one fixed quantile, determines \(b\) uniquely. Thus \(b_\gamma\) converges to a finite positive constant. In the limiting normalization, given perimeter \(s\), unbiased disk area has density \(d_s=d_s^1\) as in (15). The end-mark symmetry passes to criticality; alternatively re-embed using three sampled boundary points by continuity and apply the resampling test as for the spheres in Section 2. These disk laws and estimates make no assertion of subcritical LQG metric convergence. ◻

Proof. Approximation in standard charts.

In disk or strip charts as above the boundary chaos is intrinsic and is determined by the field as an abstract disk (with or without the marks), and on genuinely shared open boundary segments with the same side and conformal collar it agrees between charts. We need this also for the disk laws occurring as marginals in cuttings, with coordinate identifications not necessarily independent of the fields. Work first in straight local half-plane charts off the distinguished ends, and at a strictly subcritical test exponent \(\eta/2\). Positive length estimates/measures here refer to \(\epsilon^{\eta^2/4}\exp(\eta h_\epsilon/2)dx\) (deterministic normalizing factors suppressed), and may equivalently be obtained (up to a fixed kernel constant) by smooth approximate averages centered a distance \(\asymp\epsilon\) inside the chart, with support in an interior Whitney ball of radius comparable to \(\epsilon\). Indeed the mixed GFF has the reflected log kernel and the change from Wick normalization at these boundary-scale averages is asymptotic to the same regular-part correction up to a kernel-dependent constant; use standard subcritical GMC uniqueness by mollification (or the reflected Dirichlet field construction and the same subcritical approximation theorem). Subcritical approximation is in \(L^1\) for the mixed GFF; continuous and normalization corrections give convergence in probability for the fields here. Lebesgue log-scale averages of these approximants over \(\log(1/\epsilon)\in[N,2N]\) converge in probability too (use \(L^1\) before corrections, transferring by localization/truncation), and integrals divided by \(N\) over bounded-width bands about the endpoints vanish in probability. All are local assertions within regular open intervals on the boundary.

Transport between collars.

If charts on a common collar differ by a possibly random analytic map of straight boundary segments, with nonzero derivative, localize it on countably many compact subarcs with analytic extension and deterministic smooth bounds as needed there. Transport a smoothed test kernel to the other chart by change of variables. It agrees, up to field evaluation error \(o(1)\), uniformly on these localized subarcs, with the standard kernel at radius \(\epsilon\) times the local magnification (using positively oriented flattenings with inward normal up). To see this, in rescaled coordinates the test density after pullback has the usual kernel as leading term, an \(O(\epsilon)\) linear combination of a bounded family of finitely many smooth derivative-polynomial kernels at the magnified size (Taylor expansion), and a compactly supported smooth remainder \(O(\epsilon^2)\) with fixed arbitrarily many derivatives. For local mixed Neumann GFFs evaluations of the first-order kernels at centers in a bounded interval and bounded magnification range are \(O(1+\log(1/\epsilon))\) with probability tending to one simultaneously for all small \(\epsilon\): log variance at most \(C(1+\log(1/\epsilon))\), Gaussian tails and unit rescaled parameter boxes with smooth increment bounds, then a union bound on dyadic levels. The remainder is \(o(1)\) even by local \(H^{-s}\) of the reflected field for small \(s\in(0,1)\), since a rescaled unit-mass smooth density costs \(O(\epsilon^{-1-s})\). Continuous corrections cause no problem. The estimates likewise cover random compact subarcs by exhaustion. Bounded log magnification then disappears in the log-scale averages by positivity and the endpoint-band observation, even when it varies along the line. The smooth charge shift remains: if the fields are related by \(h\mapsto h\circ f+Q\log|f'|\), the residual factor against the pulled-back measure is \(|f'|^{\eta Q/2-1-\eta^2/4}\); extra continuous fields just give the exponential weight. This proves agreement for \(Q=Q_\gamma,\eta=\gamma<2\).

Critical passage and coordinate-independent versions.

At criticality use \(\eta\uparrow2,Q=2\) and then the normalized parameter limit in each chart. The same argument, easier without the straight boundary condition (radially symmetric smooth interior kernels), applies to bulk masses on common conformal neighborhoods. These comparisons require both fields/masses in the individual charts to satisfy the stated approximation facts; they don’t require a theorem of simultaneous covariance for arbitrary laws on both sides of a random coordinate change. Missing finitely many marked points are handled by diffuseness. To view a measure as a field function on abstract standard disks, one can integrate versions over Hausdorff charts parameterized by disk automorphisms with a full Haar-class density, keeping the a.e. common pulled-back value; fixed automorphism covariance and Fubini in the standard disk laws give agreement. Here only the measure class in averaging charts matters. Thus this function may be made exactly coordinate independent on a saturated set of full disk law. Aperture coordinates on non-Jordan domains, when used, always refer to prime ends. ◻

Proof. Strip likelihood.

Randomize horizontal translation in the strip with Lebesgue measure. Consider first \(\sigma_W\), radial part \(m\) with two-sided variance-2 Brownian increments with drift \(+a\), \(m(0)\) Lebesgue independently; and \(\sigma_D\), changing this to initial intensity \(e^{-a m(0)}dm(0)\), independent variance-2 Brownian increments away from 0 on both sides with drift \(-a\). Both get the independent lateral field. These are translation invariant, and on the strip up through real coordinate \(v\) their likelihood ratio is \(\exp(-a m(v))\). Modulo translation they are, up to constants, the DMS quantum half-plane measure (a multiple of probability) and the twice-marked disk measure. For example divide by translations using a unit-location cutoff on first reach of average height \(k\). For \(\sigma_W\) this has finite positive mass (integrate the initial Lebesgue height against Brownian first crossing, or use the bound by the expected one-unit range and transience on the left), independent of \(k\). The path after first reach of \(k\) has the unconditioned forward increment law by the strong Markov property (localize to finite windows); for the past center first at first reach of \(-K\), use translation invariance of the quotient, and let \(K\to\infty\) in the ascending Brownian segment, reversing at \(k\) as in (10) to obtain its deficit \(U_a\) before first reach. This is the usual first-exit embedding of the weight-2 wedge (\(\gamma\)-insertion parameter). For \(\sigma_D\) restricted to reaching \(k\), transfer the path to this stopping time by the likelihood \(\exp(-ak)\) (on bounded time windows by Brownian optional stopping), then use the forward drift \(-a\). This is exactly the description of the scaled log Bessel excursion up through a fixed level (upward part \({\rm BES}(4-\delta)\) ascent, followed by unconditioned \({\rm BES}(\delta)\) descent to zero in Bessel units, \(\delta=3-4/\gamma^2\)). Vary \(k\); independent lateral projections remain stationary. Thus after dividing by translations the \(\sigma_D\) perimeter-pair intensity is a constant times \(S^{-1-r}dx\,dy\) by the maximum and marking rules above (Duplantier et al. 2021).

On \(\{T>t\}\) where \(T\) is duration of the branch, map the strip to the remaining region, mapping ends to tip and target by \(F\) with \(F(u)-u\to0\) at the right end (which has an untouched collar). This difference and all its derivatives in the tail are \(O(e^{-\Re u})\) by reflection in exponential coordinates. Denote the charge-transformed field by \(\tilde h\), its average by \(\tilde m\). Under \(\sigma_W\), MSW gives the product of the half-plane cut Poisson law up to \(t\) (including the signed length changes) and \(\sigma_W\) for \(\tilde h\), with exact constant one: at the quotient level this is the MSW normalized probability assertion, and the gauge of \(\tilde h\) is equivariant under parent strip translation, hence has the same translation factor (Miller et al. 2022, Proposition 4.1, Corollary 4.2, and Proposition 4.4).

Domination.

We claim domination for \(\sigma_D\) on \(T>t\) of these same outputs, by the cut Poisson law times \(\sigma_D\) for \(\tilde h\). Test a nonnegative functional of all cut marked disks and signed changes up to \(t\) and \(\tilde h\) up to horizontal level \(K\). Localize temporarily to the event that the hull up to \(t\) and these observed field restrictions come from before a large horizontal level \(v\). This can be tested there, with independent full bSLE: run up to leaving a slightly smaller tail cutoff, using counts of discovered loops and their quantum lengths to read the clock, requiring that trunk time already exceeds \(t\). Any partial cutoff excursion can be discarded here. Indeed the count convention recovers the same duration under both measures on all completed portions by the public laws above, simultaneously by testing rational chronological bounds. Loop lengths on portions compactly before the cutoff are read on their insides using the field restrictions and canonical univalent charts; under the wedge they are disk lengths, and the same locality holds under the disk by finite-cutoff absolute continuity (or the MSW unmarked disk rule). The signed change paths themselves can be read from the jumps by compensated stable summation on positive-survival compact intervals, using the same subcritical path laws. Thus there are such field-before-\(v\) tests exhausting the survival event as \(v\to\infty\). In this use of a measure likelihood all event/test definitions need only agree within their local measure classes.

With \(\chi_v\) a smooth probability kernel on \([v-2/v,v-1/v]\) require also that \(F\) maps this slab before \(v\) and \[|m(v)-\int\tilde m(y)\chi_v(y)dy|\le\epsilon .\] Along integer \(v\) this extra localization is eventually immaterial under \(\sigma_D\) on survival. Indeed local fixed-negative-order distribution norms of the lateral field on unit slabs, including the Neumann edges by reflection, are eventually polynomial, as are radial suprema; \(F\) is exponentially close to identity in every smooth norm, so changing the averaged test density on the slab (of only polynomial smooth norms) and adding the charge correction causes a vanishing error. Brownian oscillation on the \(2/v\)-window vanishes along these integers. By Fatou and the parent likelihood the \(\sigma_D\)-integral is bounded by \(\liminf\) of \(\sigma_W\)-integrals with the likelihood bounded on the localization by \(e^{a\epsilon}\int e^{-a\tilde m(y)}\chi_v(y)dy\). Drop localization on the latter side; Fubini and the precise wedge product give \(e^{a\epsilon}\) times the claimed product integral by the same Brownian likelihood in the remaining strip (where field tests use only up to \(K\)). Send \(\epsilon\) to zero. Cylinder domination extends to the full output by exhaustion/monotone class (use finite product-mass cylinder cutoffs, e.g. restricting the remaining average and any needed translation gauge).

Disintegration.

Divide by translations using a unit cutoff on the remaining disk gauge. In the dominating product the putative \(Z_t\) is the remaining split length; in the survival law that equality and the signed change law hold by the definition and the cited disk theorem. Keep only paths with valid positive backtracked pair \(Z_s\), \(0\le s\le t\). On projecting just to the absolute lengths/path, this restricted dominating mass is exactly saturated by Kammerer’s law: indeed its density with respect to stable changes and \(dx\,dy\) at time zero is the disk constant times \(S_t^{-1-r}\) by translating the two lengths. The disk constant is the same one as for the initial disk. Thus the domination is equality. Disintegration proves the full marked-factorization rule for almost all initial split lengths: given the path up to \(t\) on survival, the offspring are independent once-marked disks at their perimeters, and independently the remaining surface with tip and target is a twice-marked disk conditioned on the two arc lengths. This assertion concerns abstract marked surfaces, not independence of embedded maps. All such explorations may carry along the unobserved CLEs with their component Markov law, by first conditioning on the branch/CPI geometry and field (Kammerer 2025, Proposition 3.6, version 5).

Clock and stopping-time conventions.

A few details about times and measurability in this transfer. All laws of surfaces/cut disks in the MSW input are modulo conformal re-embedding, retaining the indicated pinch points as boundary marks; a possible additional mark/record of original boundary on a negative disk is not needed as another independent sample. Clock constants in wedge and disk can separately be calibrated exactly by the number of completed positive excursions whose inside boundary has length in shrinking bins (in the same deterministic length units). In particular their calibration agrees on any common portion under the local likelihood comparison, without referring to how much absolute length there may be far to the right. Indeed the stable-process law of large numbers for those bins (e.g. almost surely along a fast subsequence and at rational trunk times, then by monotonicity) applies also up to each fixed compact surviving positive path horizon by the disk path law. This gives simultaneous elapsed trunk time on all observed portions, with the clock flat during any loop tracing in expanded time. Absolute jump marks can likewise first be tested on countably many lists at positive thresholds, and the stable change process recovered by compensation (including the strictly stable centering). Boundary measures on already completed disk interiors are local field functions in those disk charts. Thus in using the parent likelihood one may require common observed clock strictly beyond \(t\) before the cutoff and rational slightly larger hull bounds, discard incomplete excursions there, and exhaust; definitions of the outputs up to \(t\) agree on these tests under the two parent laws. For any surviving bounded initial piece all needed earlier trace and completed interiors fit before a sufficiently far cutoff. Only the full independent exploration geometry, not field data later than the cutoff, is allowed for free in this likelihood. For the quotient interpretation of \(\sigma_W\), one could equally change to the gauge of the last reach of 0, by translation and reversal with sign (symmetries of its radial model); then the free and barrier-conditioned halves occur in the opposite order. These are the two exit descriptions of the same drifted wedge, and only the translation quotient matters (Miller et al. 2022).

Continuation of the full exploration at the zero-driver-gap trunk times considered here (including just after a disconnection or completion of a selected loop excursion) uses the target-component bSLE Markov property. One can regard all parent field/end data as known initially in applying geometric strong Markov in the independent exploration; tip times defined by the length process on observed paths are adapted using the count clock or its right-continuous inverses as above. For example first jump times past fixed thresholds can be tested by disconnections and completed excursions with their component marks/lengths, and deterministic trunk horizons can be located by when completed-path trunk time first exceeds the prescribed value, in right-continuous filtration. We never ask for a fresh full bSLE at a stopping tip in the middle of tracing a loop. Future unrevealed CLE geometry in the components has the corresponding CLE-exploration conditional law. These uses are the ordinary domain Markov/branching and zero-gap restart properties of the MSW/Sheffield bSLE construction, combined with the independent field, in disk or half-plane conformal type as above (Miller et al. 2017, 2022). ◻

Proof. Stopping just after jumps. One may stop just after jumps, selected as stopping times of the length path (including crossing specified absolute jump thresholds). Approximate by dyadic times from above on survival. In the canonical prime disk of the true remaining region at that jump, the subsequent hulls shrink to the post tip. Indeed clocks parametrize the continuous trunk, attached loop excursions are locally finite at positive diameter, and continuation in this disk is continuous at its starting prime end by bSLE Markov at the jump (this is a stopping time for the exploration given the independent parent field, or use the completed loop-return time for a positive jump). Normalized maps to the slightly later domains in this disk (normalizing at 0 or another fixed interior point) converge to identity uniformly up to the boundary: reflected convergence on boundary arcs off the tip, and on the remaining sector confinement by small crosscuts about that tip, or the maximum principle. Pullback charts of the later remaining marked disks, using the post tips there and added independent boundary marks for standard twice-typical coordinates, differ from these maps by automorphisms which cannot escape compacta in probability on compact ranges of the two lengths. Indeed an open arc about the original target is untouched, with the same length by the arc comparison in later marked charts. Choose a third comparison point in that arc a sufficiently small fixed length away from the target (exhaust by countably many such lengths). Its locations, the targets and tips stay separated in the true remaining canonical disk, while their preimages in the standard twice-typical charts are separated in probability by diffuseness/tightness of the boundary measures and by their positive length spacings. A degenerating automorphism would collapse two. Here at the limiting jump the positive split values are supplied by the signed càdlàg paths; their identification with actual post lengths is not presupposed. Interior distribution pullbacks converge through the normalized maps since all derive from the same parent field. Thus by subsequences and the dyadic laws the true post field and marks in prime coordinates have the same asserted law at the limiting split, including true post lengths; actual post tip and target are the prime limits through these shrinking continued hulls. One obtains joint independence with all cuts up to that jump by the same test limit at dyadic times (disk laws with prescribed splits vary continuously by common twice-typical coordinates and length scaling/labels; keep first finitely many already completed jumps and exhaust). Stopping tests can be localized to compact positive survival; boundary or axis hits not under this assertion.

Prescribed starting pairs. For the single-branch subcritical law we only use almost everywhere initial perimeter/split prescriptions (or AC randomizations of them). In deriving the law later for any critical prescribed positive length pair, one simply approximates the prescribed start by such good subcritical pairs, or by compactly supported diffuse laws on the pairs tending to the prescription. The parent is a once-typical-mark disk at this perimeter with the target at the relative oriented length label given by the pair; for bounds or taking limits embed using an extra fresh independent typical boundary point with the source, not using the target itself as the extra point. Thus given the start lengths this chart has the perimeter-scaled twice-typical field and genuinely has the convergence and estimates above/below uniformly for converging positive pairs, including convergence of the target by diffuse full-support chaos. Conditional on that data the unclocked bSLE/CLE is just the independent conformal-domain law with the indicated endpoints. Almost-sure properties requiring, for example, other fields/charts simultaneously can first be proved by averaging at each subcritical parameter before choosing the approximate starts. No uniform estimate in the widths of small split intervals by dividing general error probabilities by those widths is needed. ◻

Proof. Protected prefixes and negative losses.

Work just after the specified start in its own canonical disk. The target has an untouched neighborhood on every pre-terminal compact path segment. Before any first intrusion, untouched open arcs use exactly the length inherited from this disk by collar comparison. A first hit into the interior of such an arc, coming from outside, separates the rest toward the tip on that side, making a disconnection event, i.e. a negative jump. To formalize for a protected prefix, test with an intermediate endpoint \(h'\) slightly farther than \(h\), still strictly below the graph; such test points can be chosen untouched. Indeed in starting prime coordinates the continued exploration has the ordinary bSLE law, for which specified other boundary points are not hit (only the trunk could hit, as complete future loops are simple CLE loops). Here one can use MSW’s trunk law at skew zero, chordal \({\rm SLE}_{16/\kappa}(8/\kappa-3;8/\kappa-3)\), and the standard SLE with force points boundary-intersection theorem (in particular the boundary dimension bounds of Miller–Wu and related works): away from start and target, for an initial same-side weight strictly above \(\kappa'/2-4\), with \(\kappa'=16/\kappa\in(4,6)\) and the two tip weights \(>-2\) here, it is not boundary filling and has zero boundary Lebesgue measure there (Miller and Wu 2017, Theorem 1.6). Indeed the boundary dimension at \(\rho=\kappa^\prime/2-3\) is \(1/2+1/\kappa^\prime<1\). The one-side local bounds suffice (near a boundary swallow on that side localize away from opposite force points); Möbius covariance then gives avoidance of a given other point. Even empty interior of the hit set suffices to choose untouched intervening labels, since boundary chaos in prime coordinates has full support without atoms. These continuations need only be used from the countable post-state times under consideration. A first hit on the arc from target to \(h'\) then cuts off a nonempty boundary interval including \(h'\) on that side. The post tip in the target component is at this hit with the access on the target side; that side length is exactly the coordinate of the hit. For the last assertion, along the arc from the target the post domain has untouched access all the way up to the hit, converging there to the tip prime end, and in the cut domain the just lost part has access from the other direction also tending to its pinch prime end. There is no extra split into different access labels at the hit in either component: in these starting-disk coordinates the prior trace avoids the hit until this first contact, is connected to the earlier start, and any two inequivalent accesses there joined by a simple interior curve from and to the point would bound a pocket in the disk with no prior trace inside (the trace is disjoint from that joining crosscut and the point until its terminal visit, and starts outside). Hence the pocket lies in the component; equivalently the boundary interval between those accesses on the pocket side in Riemann coordinates would have constant image, impossible for a univalent map. This also follows by trimming away from the approaching final trace and the untouched side arc. Full bSLE excursions before contact are included in the connected trace argument. Positive loops never intrude onto the starting boundary when they are in a fresh starting disk. If a drop crosses \(h\) while a slightly longer prefix was preserved up to just before it, it must hit into that longer arc at the drop: otherwise the prefix would still give the lower bound (a disconnection swallowing the arc also requires such a hit between it and target). The disconnection point’s post length gives the claimed difference coordinate in the negative disk by the same shared arc rule. This argument uses disconnection-post and cut markings at the jumps, covered by countable size-threshold lists (Miller et al. 2017).

The inserted loop and its two boundary measures.

For a positive discovery, the loop is attached at the trunk and in the post boundary, walking back from the tip on the insertion side traverses just the inserted loop first (two copies of the pinch); the interior is the corresponding \(C_t\). These are the excursion and side conventions of bSLE, the loop other than its attachment being away from the previous compact trace. More explicitly complete positive-excursion loops discovered strictly later than an ordinary surviving trunk time are still simple CLE loops in the unexplored target domain at that time, by the bSLE exploration property in those coordinates. Use also countably many such times increasing toward the discovery; earlier contacts even at an eventual pinch would be excluded on these earlier pieces. The only accumulation from the immediate approach is at the attachment by path continuity. There is no simultaneous other detached open component at that time by the cut/jump correspondence and the Poisson/stable laws (two such positive-perimeter events do not coincide). Thus away from the pinch along the simple loop the exterior access belongs to the post frontier, and the two disk sides there have local collars for comparing this actual seam (Miller et al. 2017, 2022).

Its length agrees on both sides not merely as a total but on open subarcs: use the DMS/Sheffield subcritical chordal quantum zipper (Duplantier et al. 2021)(Sheffield 2016a, Theorem 1.8 and Remark 5.7) (ordinary independent chordal \({\rm SLE}_\kappa\), \(\gamma=\sqrt\kappa\), weight-4 wedge into weight-2 wedges), which identifies lengths on the two sides. This transfers locally to the present loop away from its start/end. Indeed, viewed conditionally from rational capacity times during its tracing (also before any prescribed strictly interior subarc), bSLE until the excursion closes and away from the return point is ordinary simple chordal SLE with a boundary force/drift, locally absolutely continuous up to localized stops with separated driver points. Its fresh open segment is interior in the then domain. The parent field is independently, on interior compact subsets, a GFF plus a continuous correction up to change of law, and can there be compared by Markov coupling to the weight-4 zipper input (difference continuous on the neighborhood; local wedge chart and both fields couple via the same Dirichlet projection). Condition first on the geometry up to the initial time for this comparison, then localize to subarcs in those neighborhoods and stopping bounds for ordinary chordal absolute continuity. Zipper equality there concerns only that open part of the seam: completed ordinary-SLE collars from each side carry mixed-Neumann plus continuous regular arc charts, and other completions with the same local germ, including the actual loop cut/post charts, use exactly the same measures up to the exponential weight of the continuous field correction on the seam, by arc comparison. One can phrase the localized transfer just as an assertion about smooth-kernel approximations on common side collars and thus needs no absolute continuity for the rest of the loop. To detail that local comparison, condition at a geometry-only initial capacity time in an excursion, use an interior domain library for compact segments lying ahead away from the force coincidence, and couple the parent and zipper fields on the chosen neighborhoods by the same interior zero-Dirichlet projection there, first in reference field laws restoring independent average processes. Those projections can be taken identical; the discarded harmonic fields, average substitutions and conformal terms are continuous in each interior neighborhood, and the disk bias can be included by absolute continuity. Thus the zipper copy still satisfies its length equality there a.s. One can first draw a common ordinary SLE segment independently, transfer up to localized separated-driver stops by chordal SLE drift absolute continuity, and then complete the actual exploration (the zipper conclusion was local to the compared seam with its own completion). Simple-arc subcollars share the same side as these completions and use only those field restrictions, with the common continuous-surface correction on the seam. Actual post and cut arc charts qualify for the earlier arc comparison by their disk marginals just proved; zipper charts away from their end marks similarly by wedge laws. Countably many such compact-segment localizations cover the open loop arcs. This applies first in typical-target coordinates, where the whole branch is independent of the parent disk field before clocking, to every excursion indexed using also field-independent rational times; almost-everywhere statements suffice for the above choice of approximations. Endpoints have no atoms. Together with the full signed jump rule this identifies the inserted interval as claimed.

Position and order of the inserted interval.

The open loop excluding the attachment is inside the pre-loop target component with closure based at a fresh tip, and is adjacent externally to the post component as above. The attachment, fresh even through the immediate approach, has only one access in the pre-loop target component. Indeed two accesses joined by an interior Jordan arc from and to that point would enclose a pocket missing the earlier connected trace (which starts on the outer boundary and avoids the point until then), again impossible by nonconstancy on prime intervals. Thus in pre-loop prime coordinates the circle can only add a boundary detour from and back to that tip, with the post tip at one of its ends. Its open arc carries equal interior/exterior length by the preceding comparison, so the detour at the top of that post side has exactly the jump magnitude. The rest of each post side, before this detour as applicable, has length at least the corresponding pre-value: indeed work in a sufficiently close preceding post-state (ordinary times increasing to the event suffice), where prefixes strictly short of those pre-values stay protected up to insertion and persist still outside the detour since the new loop is interior relative to that state. Total length increase now gives equality and the side/order convention as claimed. Horizontal comparisons from a positive aperture just start in its post-state with inherited labels. ◻

Proof. On interior compacta this is ordinary metric positivity and tight continuous corrections; in bounded horizontal slabs up to the strip edge it suffices to rule out arbitrarily cheap paths close to the boundary crossing a fixed interval tangentially. For each fixed horizontal coordinate \(x\) on that interval, consider interior Whitney boxes above \(x\) at small depths \(e^{-n}\), with finitely many per level, arranged so any path hitting the normal above \(x\) at such depths and exiting its Whitney neighborhood (size at most \(C\) times depth) must cross one interior annulus of size \(\asymp e^{-n}\). In the straight local chart their proxy charge scaling is \(e^{\xi_0(H_n-Q_0 n)}\) up to tight coarse continuous factors, times annular costs with uniform inverse moments. The latter fact follows by rescaling the Neumann field minus semicircle average (bounded harmonic Gaussian corrections on the rescaled interior boxes) and the stated interior metric estimates. The deterministic parent chart derivative correction on the slab is bounded. Since \(H_n/n\to0\) in probability uniformly over all large \(n\) at fixed \(x\), with summable Gaussian error tails, and \(\xi_0Q_0<1\), with uniformly high probability at fixed \(x\) all these small crossings cost at least \(e^{-n}\) (the DMS tilt and random coarse corrections can first be truncated). Fubini gives a good set covering most of the fixed interval in Lebesgue length. Hits by a path confined to the thin collar and traversing tangentially thus force total length bounded below: select disjoint horizontal Whitney neighborhoods of hits on good normals with sum of widths bounded below by interval covering (use a strictly interior subinterval to ensure exits). Paths that do not remain in a thin collar already have an interior macroscopic traversal. Any macroscopic movement on the compact disk boundary contains tests away from the two ends, covered by finitely many such slab tests. This proves the estimate. ◻

Proof. We prove (23). On a unit slab use first the mixed/free GFF boundary model above, writing \(H_n(x),X_n=2n-H_n(x)\) with the previous reference. Cover a boundary interval by mesh points at each scale \(e^{-n}\). Chains to larger scales above any point of the interval can be made by short crossing and surrounding paths in interior Whitney boxes: for instance place annuli about an interior point at height \(c e^{-n}\) above each mesh center, with radii small multiples of that height, and connect by chains of overlapping annuli in the comparable-depth neighborhood to the possible next coarser centers, using forced crossings and staying at comparable height. Windings at successive levels and these bridges join; finitely many configurations at each center suffice. Centers of boxes need not lie directly to the right of the desired point. Charge the estimate at scale \(e^{-n}\) against the mass of a short boundary interval at displacement \(+C e^{-n}\) on the same side from its grid center, so all charged intervals are within a bounded distance to the right of the limiting \(z\). Under the endpoint tilt by \(H_n\), the mass divided by \(e^{(\gamma/2)(H_n-Q_\gamma n)}\) has inverse moments of all orders as proved above, and the crossing costs (summing the finitely many tests at this center) with the derivative upper weight, divided by \(e^{\xi_0(H_n-Q_\gamma n)}\), have positive moments of all orders. Indeed without the derivative weight use the standard rescaled interior metric estimate with \(Q_0\); interior harmonic-field corrections after subtracting the coarse semicircle, including the tilt shift, have uniform exponential moments. Thus the cost bound at the center divided by the indicated short mass to power \(\theta 2\xi_0/\gamma\) is at most \(Y_n e^{-dX_n}\), with \(d=(1-\theta)\xi_0>0\) and \(Y_n\) having uniform moments of all orders under the tilt. Indeed, the exact remaining exponential factor is \(e^{-d[X_n+(Q_\gamma-2)n]}\), which is at most \(e^{-dX_n}\) because \(Q_\gamma\geq2\).

Uniform control of all chains and visit counts.

Here are details for summing these charges on all boundary chains. Use the simultaneous barrier with parameter \(y\ge1\) already proved. Except for probability \(C_d(1+y)^C e^{-y}\), at all grid points in the band \(X_n\in[-y+j,-y+j+1]\) with barrier survival, \(j\ge0\), we have \(Y_n\le C_d e^{d j/4}\) and oscillation \(O_n\le C+j/3\) up to fixed mesh multiples. Indeed sum the undo-tilt bounds with cost \(C e^{-y+j}\), using the high-moment/Gaussian tails and saving ballot power \(b>2/3\). Thus along chains using nearest mesh centers at each scale, ratios are bounded by \(C_d e^{dy}e^{-d(X_n(z)+y)/3}\). We next bound the number of levels at which a chain comes close to the barrier. Work first on the simultaneous barrier and oscillation event just constructed. Suppose that, for some boundary point \(z\), the quantity \(X_i(z)+y\) visits \([0,J]\) at more than \(A(J+1)^4\) levels, and let \(n\) be a level at which this bound is first exceeded. Choose a mesh center \(x_n\) within \(e^{-n}\) of \(z\). At every earlier level \(i\leq n\), both points lie within a fixed mesh multiple of the same level-\(i\) center. The oscillation bound, solved for that center’s height, shows that every counted visit gives \[0\leq X_i(x_n)+y\leq C'(J+1).\] The lower inequality at all other levels follows from the simultaneous barrier. In particular the endpoint at level \(n\) belongs to this same bounded band. This is a pathwise implication on the good event. Having obtained it, we drop that event and estimate the resulting event at the fixed mesh center \(x_n\).

Under the endpoint tilt by \(H_n(x_n)\), put \(W_i=X_i(x_n)+y\). Its increments form the centered variance-\(2\) Gaussian walk, and its starting height is \(y\) plus the bounded Gaussian correction used in Lemma 30. Let \(V_J\) count its visits to \([0,C'(J+1)]\) before killing below zero. This count has an exponential upper tail on the \((J+1)^2\) scale, uniformly in the starting height. Indeed, start the first block at the first visit to the band, give it \(L_J=\lceil C''(J+1)^2\rceil\) steps, and start each subsequent block at the first later band visit. From every point of the band there is a probability at least \(c>0\), independent of \(J\), of being killed during that block. Each block contains at most \(L_J\) counted visits. The strong Markov property therefore gives \[\mathbb P_*(V_J\geq k)\leq C\exp\{-ck/(J+1)^2\}.\] At \(k=A(J+1)^4\), combine this estimate with the endpoint ballot bound by Hölder, retaining a ballot power \(b>2/3\). Undoing the tilt and summing over the \(O(e^n)\) possible centers gives the same factor \(e^{-y}\) as before. The endpoint bands cost at most \(e^{C'(J+1)}\), whereas the visit estimate contributes \(e^{-cA(J+1)^2}\) after adjusting \(c\). The sums over \(J\) and over \(n\), the latter bounded by \[\sum_{n\geq1}n^{-3b/2}(1+y+\log(2+n))^C,\] are finite. Thus, outside an event of probability \(C_d(1+y)^Ce^{-y}\), every boundary chain has at most \(A(J+1)^4\) visits to \([0,J]\), simultaneously for all integers \(J\geq1\). Let \(A_*\) be the supremum of the summed cost ratios over all boundary chains in the slab. The preceding bounds give \[A_*\leq C_d e^{dy}\sum_{J\geq0}(J+2)^4e^{-dJ/3}<\infty.\] The bound \(\mathbb P(A_*>C_de^{dy})\leq C_d(1+y)^Ce^{-y}\) gives \(\mathbb EA_*^q<\infty\) for every \(q<1/d\). Since \(d=(1-\theta)\xi_0\), choosing \(\theta\) sufficiently close to one supplies any prescribed moment, with room for the Hölder inequalities used to restore the radial part.

Finitely many larger-scale slab-interior crossings to link these chains and move right one slab are handled the same way without level summation, charged against a fixed interval still to the right. They can use common surrounding paths to connect the choices. Enlarge \(A_*\) to include these finitely many links; the same moment bounds continue to hold. All these estimates hold for \(P^\perp\) by the compact slab continuous Gaussian corrections.

Restoring the radial part and the perimeter.

Restore the radial and normalization factors: per slab needed to go right from \(z\) (including the starting one) the cost is at most the right side of (23) with \(K\) replaced by \[C\,A_k V_\gamma^{-(1-\theta)2\xi_0/\gamma} \exp\{-d\,U_a(k)+C\,{\rm osc}_{[k-C',k+C']}U_a\}.\] Indeed each compared mass interval is inside the indicated tail; \(A_k\) depends only on the lateral field with the above moment control and bounded normalization-unit constants. Sum the bounds over all \(k\): the sum of the displayed exponential occupation factors alone has all positive moments uniformly. For integer moments order tuple times, group overlapping bounded windows (groups still bounded for fixed order), and use at each next separated group the Gaussian density bound \(C(1+m)^{-3/2}\) for the 3-dimensional drifted motion across gap \(m\), integrating the exponential radial decay with bounded-window supremum-increment exponential moments. The two halves cause no change. Conditional Minkowski over the lateral field therefore preserves the desired large moments of the \(A_k\)-weighted sum, with slack for Hölder with inverse \(V_\gamma\) (all orders by one slab and radial bounds), and the small positive DMS tilt. This proves (23). The same proof works at limiting charge, with limiting normalization constants. ◻

Proof. The candidate length law is the Doob transform of two independent symmetric Cauchy processes (each density \(du/(2u^2)\)), killed outside the positive quadrant, by \(S^{-2}\), run to the corner. Here are lifetime details. At \(S=1\), twice the killed generator of \(S^{-2}\) is \[\sum_{\ell=x,y}\left\{{\rm PV}\int_{-\ell}^\infty \frac{(1+u)^{-2}-1}{u^2}\,du-\ell^{-1}\right\}=0\] (each term is \(2\log(\ell/(1-\ell))+1/(1-\ell)-1/\ell\)). Replacing \(-2\) by \(-1\) gives \(\sum(\log(\ell/(1-\ell))-1/\ell)\le-4\). Scale homogeneity therefore gives transformed drift of \(S\) bounded above by a negative constant. These calculations apply up to quadrant compact exits by Lévy optional stopping, including overshoots with the actual function value (zero outside; exits from pre-sides \(\ge\epsilon\) leave at least one side \(\ge\epsilon\), so the harmonic density stays bounded). Thus there is a consistent probability law until leaving all compacta, with mean lifetime bounded by \(C S_0\) and \(S\) a supermartingale there. Exits cannot run to infinity, nor leave through an individual axis while \(S\) stays positive: localized with \(S\) away from zero the density at inner side exits with \(S\) still bounded below is bounded, and on finite time windows a raw one-dimensional Cauchy process almost surely never hits a specified point by value or left limit (polarity, including jump pre/post values by the Lévy measure). Jump exits outside are given zero weight. Finally from arbitrarily small \(S\) returns to fixed large size before lifetime have probability bounded by the supermartingale estimate. Thus almost surely \(S\to0\) at finite lifetime \(T\), and the localized laws exhaust unit mass. ◻

Proof. Localized convergence and stopping selections.

As \(r\downarrow1\), the subcritical strictly stable pairs converge locally in \(J_1\) before change of measure (the positive imbalance is \(O((r-1)^2)\), so even the compensation from large jumps converges without residual drift). Hence the Doob laws on any pre-terminal compact path cylinder with null threshold boundaries converge with their weights, and the same is true jointly for the abstract jump and post disk laws on these pieces, by the preceding disk limits. Here the (good subcritical) initial splits can tend to any desired critical pair in the open quadrant; positivity tests use strict interior compact ranges, or one conditions/exhausts by these. There is no loss of probability by considering just such pieces up to post-jump times where \(S\) is small, because the candidate above has total mass one, reaches such times in countable threshold tests with probability tending to one (jumps are dense), and the entire remainder at the corresponding subcritical jumps has diameter tending to zero in probability as the threshold for \(S\) tends to zero by (23). This last assertion uses post disk law and a selectable stopping test bounded in time while the pre-stop path stays in a fixed positive compact set for each fixed cutoff test; one may also exhaust the selection by a bounded list and use disintegration at each first suitable jump. Estimate (23) conditional on the split/path supplies the upper costs with probability tending to one at rate a power of the small perimeter, independent of the number of steps for a stopping choice (the parent lower proxy test is made separately). The remainder contains the terminal target in its closure.

For clarity these localized convergence arguments can be implemented by coupling along further subsequences of any input sequence: keep countably many finite time windows, side-range cylinder cutoffs and enumerated jumps bounded away from zero therein (choose thresholds off atoms), with marginal convergence on these compact-survival events. One can place states at rational time horizons in a stopped path space with a default symbol if the path has already left the corresponding cylinder. Cylinder masses and joint laws for all coarser initial pieces inside such cylinders converge by the bounded-density path transfer, including the stopped first-exit prefixes on smaller cylinders when needed. Equivalently, for each accuracy first couple on finitely many disjoint compact-survival tests whose limit probabilities already sum to \(1-o(1)\) and which together continue through a selected tiny-\(S\) jump; then diagonalize accuracy. Such disjoint tests are formed by rational bounds on duration and side extrema (including left limits), and testing the tiny-\(S\) entry using paths up to that bound (or first passing a jump threshold); priority for testing earlier entries can be imposed using only prefixes. Failure to extend farther in the approximants never removes mass from a common prefix in a strictly positive cylinder. This is a liminf/saturation use of compact convergence; it does not posit convergence of terminal durations at the singular corner. Jointly bounded-coordinate field objects, paths and embeddings on the pieces need tightness, as follows.

In these localizations bounded-weight transfer works up to a first selected jump before a finite bound while pre-path and post-value stay in positive compact side ranges. One way to see it just using deterministic horizon laws is to round that first jump time from above by dyadic times, requiring survival until then in a slightly expanded cylinder. Right-continuity gives exhaustion at the actual stop for every counted compact-surviving path, and on the raw side the terminal density converges boundedly to the value using the jump-post position. Equivalently stop the localized Doob law there. For convergence across parameters, threshold/entrance and range tests up to the stop (including pre/post side sizes, the absence of earlier qualifying jumps etc.) can be chosen off all discontinuity boundaries in the raw limiting path; a fixed jump threshold has only finitely many candidates there before the time bound. Thus this convergence of restricted finite measures applies also to their paths, and the disintegrated marked laws at those stops use the disk estimates uniformly in the selected lengths. In particular one can use a single stopping selection for each desired tiny bound \(\delta\) on \(S\), e.g. first jump above a lower threshold in absolute magnitude with small enough positive post-\(S\), before a sufficiently large fixed time and under pre- and post-compact survival bounds. Choose these subsidiary thresholds (with strict margins, then allowing the compacts to increase for each fixed desired accuracy) so that existence has probability tending to one under the limiting transformed law, by jump density and \(S\to0\) only at terminal. Conditional upper-chain bounds at this selected stop need not pay once per candidate. The selected times in the candidate law eventually cover each initial compact piece as the tiny bound vanishes. For post-map tests at other individual jumps on a fixed compact window, extra future success conditions need not be imposed when bounding bad-map probabilities: sum bounds at the jump stops themselves restricted to positive compact prefixes, controlling candidate counts there by the bounded-density and jump intensity estimates. On projecting to a fixed matched post or cuts and length data up to their horizon, the marginal marked rule in any final subsequential limit follows by testing on these fixed compact-survival events in the approximants, whether or not maps from several different times were jointly kept. Omitting outcomes failing the final tiny-\(S\) tests costs vanishing probability. This use of exhaustion/couplings needs only convergence on initial segments; in expanding by excursions and passing to full curves the terminal remainder is controlled spatially, not by bounding its duration.

Tight closure maps and noncollapse.

Embed the original disk in \(\Delta\) in standard perimeter-normalized twice-typical coordinates (ignore the prescribed target for this choice if needed). All individual jump offspring and post states on the tests above, mapped into \(\bar\Delta\) in their own corresponding standard coordinates, are tight in the uniform topology by (23); they are noncollapsed in subsequential limits. Indeed their areas on inner compacta are tight with nonzero limiting mass, these are actual parts of the original area by interior covariance, and the parent measures converge to diffuse measures. Interior field pullbacks therefore have the genuine charge-2 relation in the limit by univalent compact convergence, and lengths in each chart are the stated critical disk lengths. Different jumped-away disk interiors remain disjoint and are avoided there by all past/future remaining traces (interiors persist by uniform compact convergence). Taken collectively, small jump-disk maps on compact path intervals shrink in maximum diameter: use (23) for the whole boundary with expected power sums of small jump sizes on the positive compact path tests bounded by stable-process sums of powers \(p>r\), tending to zero for a vanishing jump threshold. The same argument can transfer high-probability short-arc tests on countable jump-post lists; moment orders in (23) can be chosen arbitrarily high. Shrinking here and below takes place jointly using Lemma 36 once in the ambient chart, then short-cost upper tests.

Oscillation of the actual trunk.

We verify tightness of the actual trunk in trunk time, not just of the domains. On compact-survival intervals partition at successive jumps of size \(>\delta^R\), with fixed \(R>1\); include the initial state and, if stopping at a selected jump, end the last interval there. For each fixed compact cutoff under consideration, with probability tending to one as \(\delta\to0\) there are \(O_{\mathbb P}(\delta^{-Rr})\) such intervals and both side lengths oscillate by \(<\delta\) between the initial post-state \(a'\) of each and just before its last jump \(b'\) (different constants in front of \(\delta\) would work too). For example under the base laws the longest gap before a bounded horizon is at most \(O(\delta^{Rr}\log^2(1/\delta))\) with high probability. Compensated smaller increments have variance rate \(O(\delta^{R(2-r)})\) and jump bound \(\delta^R\), hence exponential maximal bounds by Lévy exponential tilting on blocks of that length; the remaining skew drift has size at most \(O(\delta^{R(1-r)})\) uniformly. Transfer on the cylinder. By (23), all length-\(O(\delta)\) arcs on the post and terminal jump frontiers under test have simultaneously vanishing image diameter in probability. For jump disks of sizes \(s\) possibly much smaller than the side lower bounds, use the whole-boundary estimate if \(s\lesssim\delta\), otherwise (23) with relative size \(C\delta/s\), bounded perimeter factor and the grid union. The polynomial number of tests costs nothing with sufficiently high moments.

In the canonical starting disk at \(a'\), near-full prefixes (leave \(C\delta\) gap to the tip) on both sides are then untouched until \(b'-\). The non-inherited arcs of the boundary of \(D_{b'}\) are short, with the following jump qualifications.

• At a positive terminal jump one inserted aperture may be large; join back \(C_{b'}\) along that aperture, whose open seam outside the pinch has a collar on both sides and was not encountered earlier.

• At a large negative jump the hit is on an untouched prefix; the long segment of \(\partial C_{b'}\) from that hit back toward the old tip (ending on the prefix) is inherited, and all the rest of that detached boundary has length \(O(\delta)\). At a small negative jump the whole detached boundary is small.

In all cases the omitted \(O(\delta)\) arcs at \(a'\), all the changed short arcs at \(b'\), and both tips are inside a parent ball of small radius tending to zero under our tests: they link by common inherited prefix endpoints and common pinches. The descriptions follow from the strict stack rules of Lemma 35, choosing margins off the fluctuating side lengths before the last jump; in the borderline case of a negative loss within that margin use only shorter prefixes and the small jump bound.

No intermediate macroscopic finger of the trunk or removed material (except the cell at the last jump itself) can then lie outside this ball, enlarged a little. Indeed in the open canonical disk of \(D_{a'}\), consider each component outside its pullback under the parent embedding. The boundaries of the post and terminal jump domains there have no interior portions except, in the positive case, the shared open seam just described. Thus a component is wholly retained in their union (including this seam), or would be wholly discarded. Any discarded component must have an access adjacent to an untouched/inherited old boundary point mapping outside the ball: otherwise its whole boundary in the closed disk maps inside the closed ball, impossible by the maximum principle. At such a boundary point the post or terminal jump domains have genuine inherited collar access on this starting-disk side, contradiction. Boundary contacts outside the ball at earlier times are likewise excluded by prefix protection. The interiors of post and last-jump domains are never touched on the trunk interval and the large positive seam is first reached at the last tip only. This proves the claimed tip oscillation up to and including the terminal pinpoint on each interval. For fixed mesh parameter successive large jumps are separated in time in the localized limit. Thus these oscillation controls prove tightness with continuous limits under the \(J_1\) path couplings of lengths. At the end, after tiny-\(S\) localized stops as above, all remaining tips and jumps are confined near target.

Pass now to joint limits on these pieces, using time homeomorphisms as needed for \(J_1\). We have a continuous limiting trunk \(P\) up to and including \(T\), continuous closure maps with inherited disk laws at jumps, and a full continuous path obtained by interspersing the positive loops, traversed in boundary-length order of the corresponding orientation. This last path also has uniform convergence modulo time change: give positive auxiliary durations to the positive jumps summably in their countable matched order, or match just finitely many large ones at each accuracy, using small diameters and trunk continuity for the rest. The trunk avoids jump-disk interiors by approximation. Every positive loop here is a distinct simple outermost loop of a genuine \({\rm CLE}_4\) in \(\Delta\). Indeed consider also the parent non-nested CLEs through their domains at countably many interior tests. These domains converge in kernel law to those of \({\rm CLE}_4\): by the Sheffield–Werner Brownian loop-soup construction (Sheffield and Werner 2012) they may separately for this assertion be coupled increasing at each interior point as soup intensity increases, and the Schramm–Sheffield–Wilson conformal-radius law (Schramm et al. 2009) is continuous up to \(\kappa=4\). Thus the increasing kernels exhaust the final interiors (strict radius monotonicity); prescribed interior points lie in loop interiors almost surely. This is tight joint data before any localization. Each positive jump domain contains a rational interior point stably, so equals in its interior limit the corresponding kernel, by its uniformly convergent univalent map, and distinct ones have disjoint interiors. Its boundary map thus parametrizes the simple CLE boundary homeomorphically. Consequently \(P\) has no constant intervals, since distinct disjoint simple loops attach on any such interval by positive-jump density.

Consistency of the limiting data.

The compact extractions can retain all the path data consistently. Localized extractions as above need only tightness of joint embedded data; for the post/cut laws they use the marked rule tested up to each piece’s own horizon or stopping jump (not a claim that post surfaces at different times are conditionally independent). For finite unions of compact path tests with varying horizons, intersections are again tests on a longest compact-surviving path and inclusion-exclusion gives path probabilities. Thus one can keep the path variables consistently in these joint extractions. ◻

Proof. The auxiliary prime-side disk.

The following quotient argument identifies all contacts of these maps; it is important not to infer an inverse welding rule just from interior kernel convergence. Make an auxiliary clean closed disk with a simple cross-diameter parametrized by \([0,T]\) from source to target. Put at every jump a small closed round disk tangent at its time on the diameter, on the jumping side, otherwise away from original boundary and diameter, with diameters tending to zero and disjoint from all the others (choose them successively). In the remaining open strip region of each side use its closed prime disk. Its prime boundary goes from the target along the old boundary back to source (label by height increasing from 0 to \(Z^i_0\)) and then along the diameter to the target detouring around each tangent circle on that side, labelled with height \(Z^i\) using linear interpolation from pre- to post-height over the detour. This is a continuous excursion once the detours are given circle parameters. The prime-to-physical map on this side folds only the two copies at each pinch on the diameter.

For detail, the region is simply connected (the excluded tangent disks join the exterior) with locally connected boundary by their vanishing sizes. In its boundary tour all open outer arcs and circle arcs away from pinches have a unique access. A diameter point which is not a tangency has a basis of separating crosscuts tending to it enclosing all near accesses: for any small neighborhood first isolate it between diameter points away from the finitely many large disks, connect across slightly above the diameter avoiding those disks, and approach at the two chosen points (not tangencies) with short paths staying on this side by detouring around smaller circles. Equivalently paths in the unobstructed half-neighborhood may be adjusted around intervening small tangent disks of diameter tending to zero; one can use their outer semicircular boundaries with arbitrarily small clearance, as each positive-height compact piece stays away from all but finitely many, and approach the line by successive such depths. At a tangency the same argument on the two separated sides of its excluded disk gives one access each.

More explicitly for a diameter point off all tangencies, at small horizontal scale one can truncate at a fixed nearby horizontal height and delete finitely many disks reaching that height. In smaller rectangles centered at the point all interfering disks are as small as desired. To build barriers, choose two tangency-free base points on either side with upward access obtained by connecting a sequence of free interior points converging to each one, sliding horizontal/vertical segments into the exteriors of interfering disks (all sliding distances vanish as one approaches the point), then simplifying to a crosscut arc. Since such disks are disjoint, each detour at positive height along one stays locally a positive distance from all the others. This also works separately on each flank of a fixed tangent disk at its pinch. These crosscuts may tend to the point with the bounded side containing all sufficiently near accesses in the stated sector. This gives the ordered prime circle with the indicated folds, also by cyclic order of the crosscuts and the open detours. In the prime-side geometry description, another precise way to check uniqueness of near access used there is the following. At a diameter point not a tangency, any two interior points tending to it can be joined by paths of vanishing diameter: start with interior segments near it at positive height, replacing portions hitting the finitely many tangent disks reaching those portions by detours along their outer arcs avoiding the line. These arcs at positive height have clearance from all other disks and can be slightly pushed off; sizes of interfering disks go to zero. At a tangency the same works separately in each flank (make the initial paths in the flank outside the one fixed disk before adjusting for all the others). Two genuinely inequivalent prime labels over the point can be separated in the round prime disk by a crosscut with endpoints mapping away from the point (no boundary interval is constant), thus cannot have this local joinability. This verifies the access counts including in the case of densely spaced jumps.

We compactify outside the auxiliary closed disk to a sphere.

On each prime side circle impose the direct linkage relation of Definition 39. Collapse the closed convex hull of each such class in the round prime disk. Here are properties holding almost surely for our Cauchy paths:

• No class on a side contains two distinct real diameter times (count pre/post at a jump as the same time). Apart from the trivial target/source conventions, at any real time at most one nontrivial such linkage to other labels occurs among the different side/time copies (a single nontrivial side class, which may itself have several labels).

• Thus in the two-sided physical auxiliary disk the collapsed sets, combining across folds and the common diameter, are nonseparating compact connected sets making an upper semicontinuous decomposition, with no identifications on any single outer or tangent boundary circle between distinct physical points of that circle. Equalities between different contour occurrences away from the fold/identical-time conventions are just the indicated direct stack linkages to the circles, arc labels or trunk time (when at a folded/time tip, use the incident copy making the link).

Record-range polarity.

For the first assertion use Cauchy fluctuation theory on intervals strictly before terminal via localized absolute continuity. At a rational split between two linked diameter times their common level would be in both closed ranges of the descending record heights from that split in the two directions, relative to the split level (vertical jump endpoints, not the filled-in jumps, for real diameter times). These independent ranges for raw Cauchy paths are polarized apart off their origin: the descending ladder heights, like the descending ladder times, have the range of a stable subordinator of index \(1/2\). We use the standard Cauchy ladder-process identification from Lévy fluctuation theory (Bertoin 1996, VI and VIII). For the range estimate, Taylor–Wendel’s exact Hausdorff-gauge theorem and its stable-subordinator interpretation imply zero \(\mathcal H^{1/2}\) measure for each bounded closed range piece; adding the countably many left limits does not change this conclusion (Taylor and Wendel 1966, Theorem 1 and Section 3). Such a range avoids any independent translated or reflected range away from a deterministic starting value: hitting an interval of width \(\epsilon\) on a compact positive-axis subrange has probability \(O(\sqrt\epsilon)\) by the stable overshoot law (Taylor and Wendel 1966, Lemma 1), hence sets of zero \({\cal H}^{1/2}\) there are polar. At the split itself the path immediately goes below, so the common split height does not qualify. This proves no distinct-diameter chord. Similarly there is no time simultaneously sustaining a real-time record interval (in either direction) for the two sides, using the descending ladder time ranges viewed toward such a time from rational points independently for each side. At a jump, on the same side the pre-value cannot sustain a linkage backward, nor the post-value forward (immediate oscillations by jump Palm/independent increments); only one of the reverse choices can extend over the jump depending on sign. The opposite side has no record there. At source the path immediately dips on each side, and target has zero height, excluded everywhere else. This proves the assertion.

The closed decomposition and its quotient.

A side hull is compact and projects injectively in the auxiliary disk (the two folded endpoints of a jumping side have different heights); side hulls are disjoint before folds by noninterlacing and closed as a relation by continuity. After folds any merging is only at a single diameter point for a cluster of hulls (each hull has at most one such physical point); thus chain length is uniformly bounded, preserving closedness. Each set is nonseparating on the sphere, being a planar embedded contractible compact convex set or finitely many such attached at one point (e.g. planar Alexander duality). Injectivity on a tangent circle now follows by monotonicity of its aperture heights and absence of two different nontrivial hulls merged through another time. An interior aperture label also cannot tie back to its own pinch via folding, by the same observation and the unequal heights. Injectivity on the original outer circle follows likewise (no cross-side nontrivial linking at one time, including the endpoint conventions). All points of the side prime disk already belong to the union of hulls: for any two nonequivalent boundary labels a level chord strictly separates them, taking endpoints of a connected component of the path above an intermediate height. If a point in the round disk were not covered by the closed union, for each class hull keep the closed circular segment outside it on this point’s side (containing the point). The intersection has interior, with extreme points only on the circle (bounding chords do not cross; any active chord in the closed limit stays uncut in the disk, so its relative interior is not extreme). All these extreme points are pairwise not strictly separable by any level chord, hence equivalent, contradiction. These facts give the claimed decomposition; interior points of tangent disks and outside are left untouched. By Moore’s decomposition theorem (Moore 1925, Theorem 22, p. 425), the quotient is a sphere. To apply its plane formulation, take an untouched exterior singleton as infinity. Every other class is a bounded plane continuum; its complement remains connected after removing that singleton. The restricted decomposition is upper semicontinuous. Moore gives a plane quotient, and restoring the singleton in the compact Hausdorff quotient gives its one-point compactification, a sphere.

Transfer of all stack linkages.

Every stack linkage just collapsed really has common image in the cutting limit above. Strict ties from an open outer/positive aperture label into an open loss aperture have this property by persistence and the strict subcritical comparison, also with non-strict intervening equality by perturbing both heights a little downwards within the apertures. For a tie to or from a real time use the same perturbation into a strict prefix at a large-jump partition time, within \(O(\delta)\) of its top height, observing that (by taking a sequence of meshes and the preceding simultaneous arc estimates) all such short arcs have vanishing image size in the limit on compact pieces. More explicitly at a continuous time in that coordinate use the preceding post-state \(a'\) nearby, whose height differs by \(O(\delta)\); transmission from an earlier aperture extends to \(a'\), or forward transmission from \(a'\) to a later aperture holds, on lowering by a sufficiently large multiple of \(\delta\). The interval from \(a'\) to the continuous time also has oscillation \(O(\delta)\). Partitions can be assumed to have these properties and arbitrarily small time mesh eventually along further subsequences, pathwise. \(J_1\) comparison with the approximating path then transfers the strict labels, by first fixing mesh margins/endpoint perturbations then passing to that subsequence limit. If at a jumping time, use its own post-state: either the linked height is the post-value and just below is near its top, or there is a forward tie from the pre-value over a positive insertion, in which case just below is adjacent to the pre-height end of the inserted arc, also mapping to the tip. A pre-jump backward record does not occur here. A nontrunk earlier occurrence tying to a later label/time has to be on an outer increasing arc or open positive jump; a later nontrunk occurrence on the detoured path in a linkage has to be on an open loss interval. Thus the perturbations suffice for all cases by aperture endpoints mapping to the tips and positive compact exhaustion. Source and target have only their automatic endpoints. This proves the image claim simultaneously (the pathwise strict-margin transfers allow arbitrary labels on good samples).

Excluding additional contacts.

Map the quotient sphere to the actual compactified disk using \(P\), the inherited outer boundary, and the jump maps with their boundary labels (using any homeomorphic abstract identification of each inserted auxiliary disk with the prime disk of its jump matching these labels); extend over the exterior homeomorphically. The map is continuous: it descends from the closed exterior, diameter and all tangent disks including their interiors taken together (a closed subset hitting all classes); continuity there follows from shrinkage at accumulating jumps. It is a local homeomorphism with singleton fibers at the images of all jump interiors and the exterior, by noncollapse/disjointness and avoidance by trunk and other boundaries. These regular points are dense on the source side by jump density and the hull description. Here is why no extra contacts form. Degree on the sphere is nonzero by any of those singleton regular fibers, so the map is onto. Moreover any disconnected fiber could be separated into two compact portions with open neighborhoods whose boundaries avoid the fiber. Local degree on each neighborhood would be constant on a common ball about the value. By regular point density and their singleton fibers, each degree must be nonzero, which now contradicts existence of nearby singleton fibers. Thus the map is monotone. But each remaining fiber is zero-dimensional: it lies on the countable union of outer circle, diameter, and tangent circles embedded injectively as just proved in the quotient; its closed intersection with each contains no interval, by absence of trunk plateau and by nonconstancy on intervals of continuous univalent boundary maps (reflection or boundary uniqueness). Each intersection is a zero-dimensional closed subspace of the fiber. The countable closed sum theorem for separable metric spaces (Engelking 1978, Theorem 1.3.1) therefore makes the entire fiber zero-dimensional. Hence every fiber is a singleton. This proves homeomorphism and the exact no-extra-contact stack rule, in particular simple trunk, Jordan offspring including the cut-offs, and no original boundary visit by an inserted positive circle. ◻

Proof. Loewner hulls, excursions, and the intrinsic clock.

Now use the upper half-plane Loewner chart (source and target at 0 and infinity), normalized also by any nondegenerating parent-coordinate convention. Away from the terminal point the capacity of full-trace initial segments converges under the uniform path couplings above. Indeed the limiting trace avoids infinity up to every pre-terminal time by the stack rule and the interior location of the positive loops. Bounded hulls then have kernel convergence as viewed from infinity by uniform trace convergence (paths in the unbounded component persist on compacta, and interior boundary points on the trace prevent a larger kernel), which gives hydrodynamic convergence and convergence of capacity. Capacity is continuous and strictly increasing in the expanded parametrization (include positive-loop intervals, no inserted holds). For any inserted positive loop at time \(s\), its post disk map has the whole loop as a boundary subarc, and an open true collar toward target \(y\). Indeed actual untouched prefixes on both original arcs of fixed short lengths near \(y\) persist strictly through \(s\), with exact length labels; the limiting post map on that open arc is onto an original boundary interval, hence reflects there. Its interior is unvisited so is accessible from \(y\). No part of the positive loop was visited before its attachment: a strict insertion label cannot tie backward over its jumping time, and the tip there cannot tie backward by any side. During the loop traversal, forthcoming distinct interior-of-parent loop points are therefore still reachable (in the closure of the post domain and not in the earlier trace). This gives strict hull growth on loop intervals, hence everywhere by positive-jump density. These checks can also be used for continuity of the unbounded kernels at any pre-terminal path time.

Jointly keep the driver pair of the independent bSLE, tight and converging to the standard symmetric critical driver with its excursion intervals. In detail \(X=(W-O)/\sqrt\kappa\), driver minus force-point location divided by \(\sqrt\kappa\), is the symmetric signed Bessel of dimension \(\delta=3-8/\kappa\), and \(O=-(2/\sqrt\kappa){\rm PV}\int dt/X\) in chordal Loewner time, with symmetric cutoff (Sheffield’s side-swapping convention). The nonzero excursions trace the discovered loops. As \(\delta\uparrow1\), use common Brownian motion in scale coordinate \({\rm sgn}(X)|X|^{2-\delta}\), with elapsed bSLE time per Brownian time at height \(u\) of the motion equal to \((2-\delta)^{-2}|u|^{-2(1-\delta)/(2-\delta)}\) (speed-scale construction without holding). Clocks converge compact-uniformly by Brownian local time. Principal-value integrals likewise converge using the difference of local times at \(u,-u\), uniformly Hölder at zero of any fixed order less than \(1/2\) on compacts; the remaining kernel has power \(-(3-2\delta)/(2-\delta)\) there. This proves the asserted marginal convergence jointly with zero sets/excursions, and independence of the limiting critical driver from the parent field given the endpoint prescription. Here Loewner time uses \(\partial_t g=2/(g-W)\), i.e. half-capacity.

The limiting hull domains throughout are driven by that pair’s \(W\), by driver and time convergence (standard chordal Loewner kernel convergence, or solve the inverse flow on compacta). Total Loewner time is infinite, since otherwise the continuous driver on a bounded interval could not generate the arbitrarily far interior loop points as the trace tends to target. A continuous capacity-parametrized path giving these strictly growing hull domains is unique: if two such paths separated at a tip time, the domains at slightly later times are strictly smaller; their boundary in the preceding open domain is nonempty and must be on both short incremental traces, impossible in disjoint neighborhoods. Thus by the bSLE continuity results the limit is exactly symmetric \({\rm bSLE}^{0,0}_4\) (i.e. the symmetric chordal \({\rm SLE}_4(-2)\) construction without added local-time drift). Excursion intervals and their signs match, by strict capacity increase and speed-scale excursion convergence (nonvanishing loops have nonvanishing limiting time interval there, and vanishing loops away from infinity only vanishing growth). In particular we can use the MSW critical GFF/CLE-percolation coupling: in a Dirichlet disk with constant boundary level and independent GFF \(J\) with that constant added, this bSLE has the realization tracing signed CLE\(_4\) loops on first encounter with the central level line (boundary-height level, law \({\rm SLE}_4(-1;-1)\)) as trunk. The exploration/branching Markov coupling gives fresh CLEs in unexplored cut domains, and in this realization Dirichlet GFFs there at the unchanged constant in negative regions and at increments \(\pm2\lambda\) in positive loop interiors (\(\lambda\) the Schramm–Sheffield half height-gap), with future branches using the corresponding level lines. This uses the Dirichlet disk coupling (pull back by conformal maps), not an assumption about geometry of LQG cuts. See also the level-line/local-set construction in the MSW proof. (Miller et al. 2017, Proposition 5.3)

Jump disks really are exactly the indicated complementary domains, including the negative cuts in this unclocked description. All negative boundary labels link to earlier labels or tips by time of the cut (trace the height from earlier on that side), so their Jordan boundary lies on the trace and/or original boundary by then. At any matched jump time \(s\) the post-domain map’s interior is in the accessible target component as seen above (the collar argument works for either jump), and contains interiors of all strictly later jump disks: by uniform post-map convergence it contains them at least in closure, and its boundary lies on the earlier trace/original boundary by approximation, hence outside their interiors. Thus later disks were not removed prematurely. Conversely there are no additional detached open components by the homeomorphism: the jump interiors are dense, and earlier ones already have their full boundary obstacle by their event times. More explicitly the terminal complement in the original open disk of the trunk and positive-loop circles is precisely the union of all jump interiors by the quotient representation. This identifies disconnections at negative jumps and their pinch tips, and the target post-domain interior equals the actual remaining component (also by the kernel theorem from infinity since the convergent map has interior there). Orientations and inherited length labels are joint physical ones from the limit. At a negative jump the side can also be read at the tip of the approaching trunk in the directed quotient disk. The truncating time change on this bSLE is now intrinsic at least in the almost-sure sense needed here: lengths of its encountered positive loci from the inside, as field disks, are measurable from the parent and the trace by the disk length identification. Counting the large ones recovers trunk time by Cauchy absolute continuity, simultaneously along the branch. Similarly signed changes in the limit here are read from their assigned jumps (symmetric compensated Cauchy summation), or measured on the post frontiers. The jumping side of a detached interior is intrinsic: the directed limiting trunk is a simple diameter-type arc with the two parent sides specified (one may close it externally by an arc from target back to source to define its sides even with contacts to the original boundary). By the quotient homeomorphism this agrees in actual disk coordinates with the side assignment. Cut and continuing fields in achieved domain interiors are the actual restriction pullbacks, since on all interior compacta univalent map convergence transports the tested field distributions (localize images to compacta in the parent). Boundary lengths needed here agree with their field-as-surface-function versions by the disk marginals. Thus to realize the one-branch law with the prescribed decoration one uses the limiting pair driver/excursion data, independent of the parent field conditional on endpoints, and the MSW joint trunk/encountered-loop coupling with exactly that symmetric bSLE excursion law. Curve and trunk/discovery order are fixed by these driver/hull/excursion data as above. One may condition that ordinary geometric coupling on them to add its unobserved decoration, without retaining any tentative coupling with unhit CLE kernel tests from the subcritical extraction. The marginal of \(J\) used to generate this branch is the ordinary independent constant-boundary Dirichlet one conditional on the parent. There is no residual freedom in the matched lengths/coordinates other than harmless chart conventions. (Miller et al. 2017, Proposition 5.3)

Remaining disks at deterministic times.

At any ordinary deterministic compact-surviving trunk time the same remaining-disk rule holds. For instance take nearby later jump stopping times (first jumps over vanishing thresholds after that time), exhausting with positive compact bounds. Their actual critical post maps just identified are tight uniformly up to closure even in this varying list: for each threshold the laws arise by the matched subcritical limits and (23) and the single parent lower test with uniform constants, conditional split bounds as needed. Normalized maps are noncollapsed by area; as times decrease, kernel convergence as viewed from target and convergence of traces identify the interior at the deterministic time, using an untouched target collar as before to exclude a trapped normalization. Tips and targets converge to the appropriate prime ends: target remains untouched, and at deterministic time almost surely no record linkage extends to other labels, hence the point is new and there is only one access there in the target component. Indeed an inequivalent access would again give a joining interior crosscut from and to this point enclosing a pocket disjoint from the prior trajectory from the outer start up to its terminal visit, forcing a constant boundary interval. Interior fields therefore identify the abstract post law and its marks by the converging split disk laws; joint testing with completed marked cuts and lengths gives the claimed conditional independence statement by localization and right-continuity. One can approximate general length stopping pieces analogously when required, or work with jump and dyadic times. The lengths/clock and contact law just proved hold for an independent MSW branch in an actual critical disk, not only an unspecified geometric realization of the Cauchy data, by the above driver identification and recovery. In particular one can resample its geometry by the \(J\) coupling instead; countably many Markov branches can be routed together, exploring onward in detached domains with relative targets prescribed from transmitted boundary labels or freshly length-sampled, under the same one-branch rules. Conditioning for lengths and offspring here retains only the abstract previous length data/pinch prescriptions as tested in the factorization; embedded data need not factor.

Product cut-disk laws given the full parent length path follow already by compact survival exhaustion: for finitely many selected cuts observed on a surviving bounded prefix, test the deterministic-horizon formulas also against later bounded surviving path events. Finite combinations (including later nonsurvival by subtraction) and monotone class then give the claimed cut marginal even conditional on the whole length path. This does not claim such independence for post disk fields given their future path. ◻

Proof of Proposition 40. Approximate the prescribed starting pair by good subcritical pairs as in Lemma 34. Lemma 41 produces the limiting actual maps, traces, and marked disk laws. Lemma 42 identifies every contact, and Lemma 43 identifies the exploration as the specified independent critical branch. The shrinking terminal remainder extends the compact-survival comparison to the complete ordered loop tours. These conclusions give all the assertions of the proposition. ◻

Proof. We detail the consistency here with multiple targets and \(J\). We use the signed CPI version of MSW in a constant-boundary Dirichlet disk: jointly with the signed non-nested CLE the central level line plus tracing of its encountered loops is the symmetric bSLE exploration. Unencountered non-nested signed loops in negative cut components (and in a continuing target component at a branching restart) have the fresh signed CLE Markov law; when branching to two still-transmitted boundary targets the ordinary bSLE exploration coupling follows a common past up to separating them, then restarts toward the targets in their respective components. Here only full explorations including the loop excursions are considered for the cut components, and separation on the original boundary is at a negative disconnection with a common tip, not during a partial positive-loop excursion. These uses are in ordinary disk conformal type, with countably many ordinary endpoint choices independent of the current Dirichlet GFF before exploration (one can condition initially on independent Liouville data). The statements use the CLE-percolation/branching domain Markov rule, not a claim that a slit disk obtained by uncovering the trunk alone has constant data. (Miller et al. 2017, Proposition 5.3)

Here is also a reconstruction explanation for using this coupling with Dirichlet fillings and level continuations. In the constant-data CLE\(_4\)/GFF coupling the signed carpet is the two-valued thin local set with offsets \(\pm2\lambda\); conditional on it, fields in loop interiors are independently Dirichlet at their labels. A central level line trunk to an ordinary target is a deterministic local set of \(J\) staying outside all these interiors. Thus it does not bias their fillings even on conditioning on it as well as on the signed loops; indeed it is determined by those signed loops. To see the locality implication, for a fixed finite union of test domains \(O\) compactly in the starting disk, on avoiding its closure the level line is measurable from the field off its zero-Dirichlet projection in \(O\) (deterministic-local-set avoidance property; its simple trace fixes order). Conditional on the signed loops, use increasing such test sets with closures inside their interiors and union exhausting all the interiors, chosen from countable libraries. Conditional avoidance there has probability one. In that conditional Gaussian Dirichlet-filling law the observed outside projection uses only the labels and the complement of the same interior projection in the filling GFFs: projections onto test functions supported inside \(O\) are just the corresponding Dirichlet filling projections since boundary shifts there are constant. These decreasing complementary Gaussian sigma-fields have trivial intersection conditional on the signed loops, by exhaustion/density of the interior Dirichlet spaces in the sum of the loop Dirichlet spaces. This proves the assertion (and the full bSLE path/traversal is then fixed by its trunk and the oriented loops). Consequently in negative complementary domains with the fresh signed-loop law the field is the corresponding constant-data Dirichlet GFF, and positive loop interiors also retain their prescribed independent fillings. Indeed restrictions to unencountered loop interiors agree with the independent fillings shifted by their signed labels in an ordinary signed CLE/GFF of each new domain, by the signed Markov rule and the preceding conditional independence. No discrepancy on the remaining carpet inside the open domains is possible: its neighborhood volume decays as a positive power on compacta a.s. (exponential tail of the single CLE conformal-radius drop at interior points and Koebe, integration and dyadic Borel–Cantelli), while both the ordinary disk GFF and the restriction of the original GFF there have arbitrarily small negative Hölder order. Cutoffs on fine squares meeting the carpet with smooth rescaled derivative bounds thus have vanishing pairings with the difference. This works jointly across the cut domains. Alternatively this is the constant-data complementary-region rule of the MSW full level-set exploration. (Miller et al. 2017, Proposition 5.3)

In particular continuations can be taken in the same \(J\). At a target fork one can use the ordinary target-consistent signed bSLE coupling; its branch with either target and the parent signed loops has the one-target CPI joint law, so must agree with the corresponding deterministic level-line exploration in this field. In a detached negative disk branching into it from the common tip uses the ordinary signed bSLE in that disk, hence by the same joint-law determinism agrees with the corresponding level exploration there. Equivalently, after a full branch redirect to swallowed targets by these rules in its fresh components. Only targets on open surviving/transmitted intervals, not ambiguous corner targets, need be redirected. The signed geometry and recursive continuations can be conditioned also on the initially independent Liouville input throughout these applications. For length laws one nevertheless conditions only on the tested abstract path/mark data to get fresh field disks, not on embedded geometry. Conditional on explored geometry and all independent Liouville input the appropriate fresh geometric experiments in cut disks then use the CPI laws just described; in particular their conditional law on each given cut surface only needs the abstract surface/target, permitting iteration under the path disintegrations. (Miller et al. 2017, Proposition 5.3)

Proxy covariance for prescribed targets.

For use of (23) directly and recursively in these critical laws, let us clarify the random-map metric comparison when an actual target is at a prescribed relative length from the typical source. (The laws/contact rules above at such starts used only good approximating subcritical splits, not this critical comparison.) Couple countably many extra typical targets by the ordinary target-consistent bSLE/GFF construction given the independent parent disk. At every compact initial part strictly before terminal in the prescribed branch there remains a neighborhood of its original target free of the trace (in particular by the contact rule); all targets in a sufficiently small open arc remain unseparated there. Thus sufficiently close members of the typical list, dense a.s., follow the very same full exploration there, with common clock by positive jump counts and the same cut and post domains. The immediate parent-to-jump/post comparisons for those countable branches hold almost surely by the earlier fixed-geometry and Devlin argument, since they are ordinary independently targeted in twice-typical parent charts and now have the critical cut/post disk field marginals. On a common portion one can thus use the same comparisons for the prescription. If additionally changing standard parent or child charts, these differ by conformal maps on the fixed standard open domains; all charts used for the chain estimates qualify there in their own marginal laws (fresh typical mark with the source/tip conditional on the specified perimeter data). Devlin with the proxy charge and the continuous Weyl log-derivative correction, or the local LFPP comparison above for that correction between qualifying marginal fields, therefore identifies the length elements also across this change. This proves the needed comparison within the prescribed-start marginal, hence also when this step is used at countably many nodes conditional on abstract previous perimeter data giving that marginal. Multiplication of the successive local derivative weights uses only actual open interiors. In their product the upper bound in (23) comes from the total composition map into a bounded original chart, without per-generation distortion constants. (Miller et al. 2017; Devlin 2026) ◻

Proof. Here is one useful full tiling by such branches. In fresh jump disks always use the pinch as first mark and a fresh independent length target, sampling initially typical marks. Recurse in all jump interiors. For negative-only steps there is strict moment contraction in perimeter. At \(S=1\), compute twice the killed generator on \(S^{p-2}\) with the compensated added cost of negative offspring sizes to power \(p\) under the transform. It is \[\sum_{\ell=x,y} \left\{{\rm PV}\int_{-\ell}^\infty\frac{(1+u)^{p-2}-1}{u^2}\,du-\ell^{-1} +\int_0^\ell u^{p-2}(1-u)^{-2}\,du\right\}.\] Independent of the split by differentiating (symmetric endpoint terms), this equals \(-2\) at \(p=2\), hence stays negative just below 2. After division by \(S^{-2}\) the drift/cost rate is \(-c_p S^{p-1}\). Stopped compensation on compacts and exhaustion give \(\mathbb E\sum_{\Delta S<0}|\Delta S|^p\le1-c_p\mathbb E\int_0^T S_t^{p-1}dt\); hence averaging over the uniform split gives a factor strictly less than 1. Positive offspring power sum also has finite mean by its compensator proportional to \(\int S^{p-1}\). Full-path offspring independence as used in iteration follows from the marked rule and geometric domain Markov (future in the post disk independent of past offspring given the post surface and length data); exhaust by compact times increasing to terminal.

Using (23) with \(\lambda'\xi_0=p\) at criticality gives summable upper-chain probability bounds for macroscopic diameters in the negative-only generations by this contraction (apply it in the critical marginal laws and compose comparisons to the same ancestor parent chart). Thus those generations shrink almost surely uniformly in Euclidean diameter, by Lemma 36 and its almost-sure positive proxy threshold for fixed macroscopic size. Every infinite chain with infinitely many positive steps likewise shrinks by nested CLE local finiteness in the closed disk; here one uses the actual independent CLE critical coupling just described. Countably many negative-only subtree starts and the shrinking of siblings at every fixed node by (23) and power sums now imply uniform mesh shrinkage down the full tiling (otherwise follow an infinite large-diameter chain). All true macroscopic loops occur as positive disks, since a not-yet-hit CLE loop has to survive intact in remaining complementary domains. Area adds exactly over jump interiors at each cutting: off the outer boundary the complementary skeleton has zero area, by Lebesgue nullity of the independent critical branch trace and interior diffuse LQG that charges no independent Lebesgue-null set (as in Section 2). Use typical targets or their common pre-fork explorations. Disk boundaries themselves carry no area. Thus the joint domain tiling and its volume labels refer throughout to an actual critical field disk with independent signed CLE, without needing determinism of that quantum surface from any encoding processes. ◻

Proof. The strictly subcritical identity.

We use the Ang–Holden–Sun SLE loop welding theorem together with the SLE loop measure/whole-plane CLE intensity identification (see Ang–Cai–Sun–Wu): in the simple CLE range \(8/3<\kappa=\gamma^2<4\) the unmarked quantum sphere measure times the independent canonical SLE loop measure (up to constant the expected counting intensity of Kemppainen–Werner whole-plane nested CLE) corresponds on cutting to the uniform-boundary-rotation weld with law proportional to \(\int s\,ds\,{\rm QD}(s)\otimes{\rm QD}(s)\), where \({\rm QD}(s)\,ds\) disintegrates unmarked ordinary \(\gamma\)-quantum disk measure. We only need the cut and actual boundary identification direction. The DMS twice-marked disk intensity was used above, so the unmarked density has mass a constant times \(s^{-2-r}\); likewise unmarking the ordinary twice-marked sphere reweights by inverse area squared, irrelevant under area disintegration. These public inputs are used strictly before the endpoint. Constant shifts and deterministic measure units preserve this identity up to a scalar, so at unit area of the ordinary sphere in (10) it yields the one-loop split intensity density \(C_r s^{-2r-3}d_s^r(v)d_s^r(1-v)\,ds\,dv\), where \(v\) is blue volume. Conditional surfaces are the two disks with these volumes and perimeter and independent uniform phase alignment (equivalently match independent typical boundary marks), as abstract matched surfaces, not independently positioned in the sphere. Conditioning here and below at unit total area can be done by continuous area scaling via field constants in the infinite measure identity. Conditional on the marked loop in the nested intensity the remaining geometric CLEs are independent ordinary nested CLEs on both sides by the whole-plane nested CLE interior Markov and inversion/Möbius invariance (these also give independence from the field before conformal pullback). (Ang et al. 2023, Theorem 1.1)(Ang et al. 2024, Theorem 7.1 and Proposition 7.2)

Next restrict to loops separating the two cylinder ends. This adds the factor \(2v(1-v)\) on the intensity side, with the ends then area-marked in the respective disks. Write \[m_r=C_r\,2v(1-v)s^{-2r-3}d_s^r(v)d_s^r(1-v)\,ds\,dv .\]

Stationary chain intensity and loop spans.

On the cylinder side of (10) we have an independent whole-plane CLE, and now just count the chain enclosing the origin of the plane chart \(z=e^u\). Write \(t\) for the log conformal radius at 0 of such an interior. The Palm density in \(t\) is \(I_\kappa=1/\mathbb E\tau_\kappa\), with \(\tau_\kappa\) the disk outermost log conformal-radius drop at a fixed point. Indeed the intervals to the next smaller loop partition the real line; their Palm drop law is the disk CLE law by nesting. Translation invariance and this mass transport give the formula (finiteness follows as well by charging an initial bounded interval of each gap). Here we use the whole-plane nested Markov construction of Kemppainen–Werner (Kemppainen and Werner 2016), in particular nesting toward 0 after a chain loop is independent given that loop and the exterior, also for the translation Palm intensity (equivalently stop after descending across deterministic radius bounds and count). By the Schramm–Sheffield–Wilson conformal-radius theorem (Schramm et al. 2009, Theorem 1) these drops have uniform exponential tails near \(\kappa=4\), and \(I_\kappa\to I_4=1/\pi^2\) (the limiting transform at argument \(\lambda\) near zero is \(1/\cos(\pi\sqrt{2\lambda})\)).

In cylinder horizontal coordinate the whole boundary of the typical chain loop lies in \([t-w,t+w]\) with \(w\) of stationary law having some uniform positive exponential moment. One flank follows by the quarter theorem. For the other, the normalized steps from containing parent interiors to the child containing 0 (interior charts with derivative positive at 0) are iid ordinary disk CLE steps backwards for any finite number of generations under translation Palm. Indeed shift the count from a loop to its descendant by that number of generations and integrate \(t\) against Lebesgue, using the forward nesting property. Each step has probability bounded below to lie inside a fixed compact subdisk of its parent chart: use simple loop-soup monotonicity up to 4 at the center. Go back to the first such good step; its image size in the plane is bounded by a constant times that parent’s conformal radius by distortion. The geometric generation bound and the exponential moment of each log-radius increment (use a small exponential tilt/Hölder) prove the assertion. Also the law of the normalized interior map at fixed \(t\) converges on interior compacta to the corresponding Palm map at \(\kappa=4\). Indeed each normalized step has kernel convergence in law by the increasing soup and the conformal-radius theorem. Compose finitely many; on compacta the normalized map from a parent many generations back contributes an asymptotically negligible nonlinear part by the distortion theorem and the product of the radii ratios tending to zero uniformly in probability. This characterizes the limiting map and loop interior, independently of the fields.

Determining the normalization constant.

First let \(r>1\) fixed close to 1 and compare expected counts with both volumes \(\ge\epsilon\). Integrating \(s\) explicitly gives leading order \[2 C_r\Gamma(r-1)\Gamma(r)^{-2}\log(1/\epsilon).\] In the cylinder, normalized area exponentially far in either end has almost logarithmic rate \(-\gamma a|t|\) there: more precisely the logarithm of the small-side area of the chain loop at \(t\), divided by \(|t|\), tends in probability to \(-\gamma a\) as \(|t|\to\infty\) under the tilted Palm law. Indeed sandwich by radial tails at \(t\pm w\), use \(R_a(t)/|t|\to a\), and the uniform finite small moment on unit slabs of the lateral mass from Section 2 (and its stationary positive law for lower bounding by one slab). \(w\) is tight. Moreover on rescaling \(t=z_0\log(1/\epsilon)\), the count probabilities are integrably dominated for large \(|z_0|\): except for exponentially small probability of a huge span \(w\), use the area in the tail beyond \(|t|/2\); its unnormalized small moment decays exponentially by the nonzero radial drift and the slab estimates. Normalization and the \(L_\gamma^{2a/\gamma}\) tilt preserve an exponential bound by truncation or Hölder; \(L_\gamma\) has the requisite small positive moments with slack near criticality as shown earlier, and its inverse has uniform positive moments (any fixed order by the same packing/inverse argument on one interior slab). This gives the other leading coefficient \(2 I_\kappa/(\gamma a)\). Hence \[C_r=r\Gamma(r)I_\kappa/2.\]

No loss of perimeter at zero.

We check carefully that the boundary length \(s\) is not lost at zero for loops at finite \(t\) as \(\kappa\uparrow4\). For a translation Palm loop about \(t\) before the sphere field tilt, let \(B_\gamma\) denote its length in the cylinder lateral field alone, in the same units as \(s\). This length is well defined Palm-a.s. at each parameter under consideration by the disk cut rule, undoing the continuous radial and normalization factors. It is a function of the lateral field and the loop alone, since one can read the seam from the disk interior field pullback via the boundary chaos construction, continuously corrected by these factors. Its law jointly with \(w\) is stationary in \(t\) and independent of \(R_a\) before tilting. We have bounds \[B_\gamma L_\gamma^{-1/2} e^{-(\gamma/2)\sup_{[t-w,t+w]}R_a}\ \le\ s\ \le\ B_\gamma L_\gamma^{-1/2} e^{-(\gamma/2)\inf_{[t-w,t+w]}R_a}.\] For fixed \(p>0\), \(\int s^p m_r\to\int s^p m_1<\infty\) by the explicit formulas. Integrating the lower estimate on \(|t|\le1\) and applying Hölder thus gives \(\sup\mathbb E B_\gamma^{p_0}<\infty\) for some small fixed \(p_0>0\). In detail use \(B_\gamma\le s L_\gamma^{1/2}\exp(\gamma\sup_{[-1-w,1+w]}R_a/2)\), allocate \(s^{2p_0}L_\gamma^{2a/\gamma}\) to the cut-law integral, and bound the remaining small powers of \(L_\gamma^{\pm1}\) and exponential supremum by the cited moments and the span exponential tail independent of the Gaussian radial input.

Along a subsequence with \(B_\gamma\Rightarrow B\), use the exact identity \[p^2\int s^p\,dm_r/I_\kappa =\int_{\mathbb R}\mathbb E[L_\gamma^{2a/\gamma}s^p]/\mathbb E L_\gamma^{2a/\gamma}\,dT_0,\qquad t=T_0/p^2\] (where \(dm_r\) denotes the above density). The integrands as \(\gamma\uparrow2\) are dominated by \(C(1+|T_0|)^{-3/(2q)}\), for some \(1<q<3/2\), uniformly for small enough \(p>0\). Indeed allocate Hölder power \(q\) to \(\exp(-p\gamma\inf R_a/2)\), with the other powers accommodated by the above bounds. Its expectation to power \(q\) is \(O((1+p^2|t|)^{-3/2})\) using independence of \(w\) and the three-dimensional Gaussian density bound at \(|t|-w\) when \(2w\le |t|\), integrating the radial exponential and allowing oscillation over \(2w\) (small-\(p\) exponential cost integrable over \(w\)). The other case is negligible for \(p^2|t|\ge1\). Thus in particular finite positive-\(s\) windows have uniformly negligible contribution from \(|t|\to\infty\). At fixed \(T_0\ne0\), in the double limit first \(\gamma\uparrow2\), then \(p\downarrow0\), the upper estimate integrand has limsup at most \[{\mathbb P}(B>0)\,\mathbb E e^{-|W_{|T_0|}|}.\] Indeed one drops \(L_\gamma^{2a/\gamma-p/2}\) with an error tending to zero by its uniform moments, and replaces the infimum by \(R_a(t)\) with vanishing error by tightness of \(w\) and Brownian increments; independence and \(\mathbb E B_\gamma^p\to\mathbb E B^p\) apply. The last Brownian factor integrates (over both half-lines) to 4 by the three-dimensional Green function. The same constant follows from the explicit density: \[\frac{p^2}{I_4}\int s^p\,dm_1 =p^2\int_0^1\int_0^\infty \frac{s^{p-1}e^{-s^2/[v(1-v)]}}{v(1-v)}\,ds\,dv \longrightarrow4.\] Hence \(B>0\) almost surely. In particular in bounded \(t\)-windows the critical approach retains tight strictly positive \(s\), under the sphere tilt too.

Convergence of the abstract matched pair.

Here are details passing the Palm cut rule itself. On compact positive \(s\)-windows its abstract pair side converges to the formula \(m_1\) with the actual limiting critical disk laws and uniform phase weld alignment. This does not require pointwise convergence of area-conditional disk laws: disintegrate the homogeneous perimeter integral by starting with independent unit disks and rescaling to \(s=(v_1+v_2)^{-1/2}\) with \(v_i\) their unit-perimeter volumes, reweighting by the power Jacobian from \(s^{-2r-3}\,ds\) and by separation. This reweighting is bounded and continuous on compact \(s\)-windows up to movable cutoff boundaries. Endpoints become area samples on opposite sides. Disk shapes/areas, added chart marks, and phase matching of boundaries by length all converge there. On the sphere side in each bounded \(t\)-window one has the joint convergence of field/area and the normalized interior Palm map just proved, independently. There is no loss escaping \(t\) or \(s\) in matching these two statements by the tightness/tail bounds above.

Convergence on the actual closed boundaries.

We spell out why the matching transfers on the actual closed boundaries (kernel convergence alone might leave a spurious shear). Restrict to bounded \(|t|\), \(w\), and \(s\) bounded above and below. Both disk maps from their canonical charts, in the \(z\) or \(1/z\) chart toward the respective separated end, are bounded univalent maps. Along the boundary, whose images are in a compact annulus about that end, (23) gives uniform equicontinuity in probability. Indeed apply the upper chains in the two independent standard boundary charts of the pair, whose laws on these tests before placement are merely boundedly weighted as just explained. For the proxy lower test only paths in a slightly larger compact parent annulus matter; costs there are uniformly positive for macroscopic motion by Section 2’s cylinder representation with tight continuous corrections and the fixed proxy theorem. Even if an upper path extends toward the end itself, it already pays for a macroscopic traversal within the annulus. We compare metrics only away from the two ends.

For precision about this comparison under re-embedding, first pull back the parent proxy by an interior-normalized map sending the separated end to the origin of the disk in inverse coordinates, with deterministic angular convention from the loop. Conditional on the Palm loop, the parent cylinder law before \(s\) cutoff is independent, a whole-plane GFF plus continuous up to change of law throughout the open punctured region as needed (via the average-process correction as in Section 2 and conformal GFF comparison). Thus fixed-map covariance applies on the punctured disk. The actual maps from twice-typical strip coordinates may also first be recentered at the separated bulk mark, now differing on the fixed punctured disk just by rotation, covered by Devlin. Their transformed field laws qualify on the punctured disk in this comparison: indeed in a standard child strip the bulk mark location \(z'\) is an area pick. At these strictly subcritical parameters, conditioning on its location, the lateral field law after subtracting the insertion singularity \(\gamma\log|\,\cdot-z'|^{-1}\) (cut off smoothly locally as desired) is absolutely continuous up to continuous corrections of a whole-plane GFF in the open strip. This follows by the subcritical Gaussian Palm/Cameron–Martin insertion formula on interior compacta (tilt by a mollified area weight and pass using local subcritical \(L^1\) for the Gaussian field); the covariance shift for \(P^\perp\) has precisely that singularity with continuous remainder, and radial, normalization, total-area and Jacobian weights in the disk pair only change the continuous part or absolute continuity. Thus, after recentering using a fixed convention given \(z'\), the charge field on the punctured disk, also with the proxy log-derivative correction, qualifies by domain comparison. Finally one can undo recentering at \(z'\) and compare to the strip metric in (23): on the whole strip both the actual unconditional field and the singularity-subtracted one qualify by the disk law and the just stated Palm calculation respectively. Away from the possibly random \(z'\) their LFPP local length elements differ just by the Weyl factor, by marginal convergence as before, while for fixed \(z'\) the adjusted field obeys ordinary covariance. Chaining these comparisons proves the derivative-weight bound for (23) on all needed path parts before approaching the marks. (Devlin 2026; Ang et al. 2023)

Consequently both maps are tight on closures (maximum principle), and cannot collapse by area and parent diffuseness. Interior field pullback convergence identifies the actual cut disks (off marks suffices for masses; distribution laws have no point-supported ambiguity, e.g. by local \(H^{-1}\)). The interior map based at the left end covers exactly the Palm CLE\(_4\) loop interior by its previously proved kernel convergence and bounded-span/Jordan limit. More explicitly the limiting conformal map from the child sends its stable area mark to the end, hence has this same kernel. Both boundary images coincide by uniform matching in the cut pair; the exterior map, taking the other end in its interior, therefore covers the complementary Jordan disk. The limiting parametrizations inherit exactly the cyclic length matching since the disk boundary measures converge to diffuse measures of full support. Thus the abstract matched critical pair law is attained by actual sphere cuts. Their critical lengths are functions of the cut field itself via disk pullback, so there is no residual choice along subsequences. Exhausting the windows in the two descriptions proves the critical Palm equality, with \[C_1=1/(2\pi^2),\] for actual cuts separating the two ends of (12).

Removing the separation marks.

Add the independent nested CLEs on both sides by the geometric conditional law if desired. Removing separation just divides by \(2v(1-v)\): by Section 2 the ends on the unmarked sphere are area samples, each loop has strictly positive volumes on both sides, and the pullback identification/lengths once valid on separation are intrinsic to field and loop. For clarity about the CLE variables here, under counting Palm on loops separating given deterministic points the ordinary inside nesting law holds by summing the nested Markov assertion (one can start descending across deterministic containing-radius cutoffs and then exhaust). By Möbius invariance the same holds from the other side viewed toward the second point; thus conditional on the chosen loop the two side CLEs indeed have the independent nested-domain laws. Independent cylinder/field sampling retains this rule and end separation exhausts intensity upon area re-sampling as stated; the orientation/blue designation is fair. Other normalization marks for the embedded sphere may be sampled subsequently by intrinsic area. This gives the precise all-loop intensity version matching (15). By integrating over scale it gives also the measure class identity between the arbitrary-total-area sphere Palm law and two equal-perimeter welded ordinary critical disks with random perimeter of positive density. No uniqueness of conformal welding at criticality is invoked. ◻

Proof. On nonabsorbed paths the increment law is exactly iid with a likelihood ratio \(f_{S_{\rm end}}/f_{S_{\rm start}}\), requiring admissibility of the side updates, with iid weights on each side \[ w_{-j-2}=c_*^{-2} f_j,\quad w_{k-2}=c_*^{-2}\widetilde g_k/2 \quad(j\ge0,\ k\ge1), \qquad \widetilde g_k=c_*^k g_k(X). \tag{24}\] The coordinate not indicated by the side stays fixed. These are normalized (\(2\sum w=1\)): divide (17) by \(F_p\) and take \(p\to\infty\), splitting the first convolution at its midpoint (either convention for a midpoint term, of vanishing effect) and using dominated convergence by the two-sided estimates on \(f\). In the face case \(l\) and the interleaving have the tilted uniform-ring law in (17), i.e. a negative-binomial allocation tilted by \(f_l\) with mean parameter \(k\) before tilt. In particular for \(k\to\infty\), \(l/k\to1\) and the order from a fixed outer base with the fixed starting-link convention identifies equal relative length positions uniformly with error tending to zero in probability. Indeed exponential errors at any fixed fractional tolerance for totals and partial sums of the geometric variables beat the polynomial reciprocal normalizer. The equivalent order in the two-sided central ring, given \(k,l\to\infty,\ l/k\to1\), follows by sampling without replacement.

Take length unit \(P\to\infty\), measuring slots divided by \(P\), and time unit \(m_P=(\pi^2/2)P/\log P\). Under the raw weights in (24) we get \(J_1\) convergence on compacts to two independent symmetric Cauchy processes in exactly the units above (densities \((1/2)dx/|x|^2\) each). Indeed the signed truncated first moment at symmetric cutoff is bounded, by \(\sum k|\widetilde g_k-2f_k|<\infty\) as proved above. Jumps outside a neighborhood of zero converge by the iid Poisson limit; smaller compensated sums vanish uniformly in probability in the small-cutoff limit by the truncated second-moment bound, and the residual bounded drifts are negligible. Thus from any convergent positive pair of initial ordinary lengths, on survival through any compact piece the discrete limit is given by the likelihood \((S_0/S_t)^2\) and the two-coordinate positivity restriction, i.e. the critical disk rule of the preceding sections (\(S\) here in scaled units). More precisely one can test horizons with both paths bounded and bounded away from zero, by path tests with null boundaries; these restrictions are convergent finite measures, not necessarily laws of the whole path. On such restrictions the exact finite ratio is bounded and convergent by (24) and the \(f_p\) equivalent. All \(0,\pm1,\pm2\) slot conventions above matter exactly on the lattice but do not move macroscopic jump times or aperture labels in this convergence.

Here and below one can couple, along further subsequences, with a true critical path on its entire exhaustion by compacts before \(T\), matching every fixed positive compact time segment in \(J_1\) eventually. Nothing is required of the terminal discrete path duration. For detail, take horizon-and-range tests, with continuity endpoints, inset them if necessary and increase a countable family covering every compact surviving initial segment in the critical law. Convergence on such tests including their path coordinates holds also on finite intersections (use the longest surviving horizon) and on differences by subtraction. The critical law is probability, staying strictly within positive finite length ranges on compacts of \([0,T)\), and extending to side sum zero at finite \(T\) as above. Thus the localized descriptions exhaust it; elementary weak-convergence coupling on increasing finite surviving tests and diagonalization give the assertion (any leftovers of mass tending to zero can be arbitrary). Time alignment matches individual jumps, their signs and sides. Child negative perimeters are \(j/P\), positive inner perimeters \(l/P\); both converge to the jump magnitudes. The matching extends to boundary label orders in the indicated direction, including the inner bases across the face ring. It extends recursively to every fixed matched jump filling by the product Markov rule and the same convergence for arbitrary converging starts. One convenient primary experiment initially chooses an independent uniform target, and does so afresh in every jump interior based at the pinch. Degenerate lattice marks can just stop that polygon. We always compare orientations from a common cyclic traversal, so the aperture order of the persistent ordinary list corresponds to the continuum order from the target back to the active tip (and on attached disks to the order along that arc). This is the clockwise/counterclockwise choice of the discovered-loop traversal in the exploration convention; interchanging left and right everywhere just reflects both descriptions. ◻

Proof. A useful extension is to several prescribed or randomly sampled target labels from one active source. Before the loss of a target label along a reference path the prefix containing it is unchanged. For a distinct target in the open ordinary arcs initially at strictly interior height \(u\), almost surely the first descent there below \(u\) is a strict negative jump with pre-value above \(u\), and until then the length is uniformly strictly above \(u\). This uses the zero Lebesgue measure of the closed descending Cauchy ladder height range, its scaling (hence polarity of any fixed positive height difference from the start), localization, and terminal limit zero. Random labels here can, for example, be conditionally independent given starting lengths, with positions converging in those units. Thus each fork of finitely many targets is in a compact-surviving piece, with strict slot transmission into a recognized jump. The prefix is simultaneously a prefix from the source toward all those swallowed targets, until separation. At the separating bridge, continuation toward them starts in its jump disk; the new active slot at the pinch is prescribed by the rule, the same one for those strictly inside this detached list. Their lengths there converge to the transmitted labels. Unswallowed targets are treated along the continuation. The continuum common-prefix and fresh component laws follow by the branching bSLE/GFF coupling and the strict stack rule proved above; unclocked branches use independent exploration laws in newly split domains (conditional also on the field supplying the quantum endpoints and the past, which is valid by exploration Markov with this initially independent input). The clock on common portions agrees, e.g. by jump counts. Fresh abstract disks with perimeter labels are used by the marked rule at the split; any imposed transmitted targets, when also conditioning on past path/split data, do not look inside them. This proves joint convergence of finite compatible routes by induction, including their offshoots. One can continue similarly on compact pieces ending before a fork, or select finitely many further paths recursively in matched raw jump disks from their prescribed pinch or a single freshly sampled source there, provided the rules are consistent (unexplored disk when resourced). We use countably many observations of this kind by increasing finite tests. No assertion that paths from conflicting sources using the same map are conditionally independent is intended. ◻

Proof. In \(T_p\), index the vertices cyclically by boundary occurrences. The probability that two occurrences at slot difference \(d\in(0,p)\) are equal is \(f_d f_{p-d}/f_p\): cutting at the two corners of this common vertex gives exactly two independently weighted disks when this equality is specified, with ordered roots (cyclic rerooting of an original root is a bijection). This allows all further pinches inside either disk. Indeed boundary arcs between the cuts and the maps inside those arcs split at the vertex in its rotation, and conversely wedge the two roots’ corners at that vertex. Loop data do not cross the cuts. Fix two small disjoint index arcs of macroscopic separation \(I<J\) in a cyclic coordinate based off them. Expected number of equal pairs in slightly enlarged such arcs is \(O(\log p)\). If some pair in \(I,J\) is equal, choose the one with smallest \(i\in I\), then largest \(j\in J\) for that \(i\). Given that \(i,j\) are equal, this selection for a specified pair uses only the outer disk of the cut (indices outside \((i,j)\)). Indeed any contender at \(i'<i\) with \(j'<j\) crosses and hence has the same vertex, giving the exterior contender \(i',j\); the case \(j'\ge j\) is already exterior. Larger \(j\) for the same \(i\) is exterior as well. Thus the inside is fresh. Its conditional expected number of additional equal pairs at \(i+s,j-r\), \(r,s\ge1,\ r+s\le \epsilon p\), is at least \(c\log^2 p\), by summing \(f_{r+s} f_{j-i-r-s}/f_{j-i}\). Choose \(\epsilon\) small in the enlargements. The unconditional upper bound thus gives probability \(O(1/\log p)\) of any equality. Finite coverings show that maximum cyclic separation among identical boundary vertices divided by \(p\) tends to zero in probability. This is not claimed uniformly over all times of a peeling. It holds at a fixed deterministic scaled surviving time with perimeter in positive compact ranges, by freshness. It holds for the post polygon and all macroscopic jump fillings at selected large jumps there, also under step-type and jump-size selection, by the factorization. To use simultaneously over jumps above any positive cutoff on localized paths, their count is tight; one can first mark in stopped order up to a fixed count with posited compact bounds. All these estimates and the analogous countable rational-time, horizon and matched-node estimates may be imposed in the coupling by further subsequences. ◻

Proof. Within either prime side, a direct stack class has a single height, so it contributes at most one label to that side of a fixed frontier. Distinct heights can be joined only through the folds at a real trunk time. The exact contact rule permits only one such time in a physical class; that time has finitely many pre/post side copies, with at most one nontrivial linkage attached. Hence only finitely many heights can occur. The source, target, and aperture endpoints add only their prescribed finite copies. ◻

Proof. Comparisons at a common state.

First compare two occurrences at one common state tending to \(t<T\). If both heights tend to their respective tip heights there is nothing to prove. Suppose a label is a positive distance below its tip. At a regular \(t\), or at a jump approached on the post side, this occurrence (and any other strictly below tip) persists to nearby fixed rational times on the right where both coordinates have moved as little as desired. At these times the no-pinch bound applies. Two interior labels must thus have the same proposed value: their circular offsets must converge together (allowing both at height zero on opposite sides). If the other is instead near-tip, track that occurrence forward while it survives. Until a first pop below its height, the identical label persists on that side. At such a pop it either belongs exclusively to the detached disk and cannot survive at all in the target polygon, or it is exactly at new height plus one in the old indexing, hence is on the other unchanged top and can switch there (the monogon pop follows this same switch rule). Since the interior occurrence guarantees survival, one can continue this tracking, always staying within the small oscillation error of one of the two tops until the compared rational time. Its macroscopic cyclic separation from the interior occurrence (both sides initially bounded below positively) violates no-pinch there. The same argument applies at time zero.

At a jump approached in pre-states transfer to just pre-jump in the lattice, with oscillations tending to zero. Again if both labels are near tip their values agree anyway; otherwise at least one interior occurrence guarantees survival to that state, and a possibly switched near-tip occurrence there still limits to the pre-tip on one side. For a positive macroscopic jump the whole old ordinary lists and endpoints embed as inherited lists in the post polygon with asymptotic lengths \(Z(t-)\); no-pinch there forces identical images of the transferred labels, since they transmit horizontally and the two top conventions have the stated aperture endpoints. For a negative jump use disjointness of the retained and detached vertex sets. Within the latter the labels in \([L-j,L]\) (for a jump on \(L\)) have no pinch at macroscopic cyclic separation, so limits agree or are the two aperture endpoints, of common image. In the retained polygon the old heights up to \(L-j-1\) project to the corresponding post lengths, with the last of those heights at the retained bridge end; all horizontal labels through that threshold transmit, the limiting threshold itself mapping to the pinch. No-pinch in the retained polygon again suffices. Here \(L,R\) in those interval descriptions are integer pre-state coordinates. This proves the same-state claim at every possible compact limit.

Completed switches have negligible total duration.

For a longer comparison follow a surviving occurrence by precisely the constant-height-until-switch rule just used, without changing occurrences at other coincidental pinches. On compact path ranges, after switching to an exact top, the total durations of completed subsequent switches before any compared later compact state are \(o(m_P)\), uniformly. We justify this uniform statement. Under raw iid steps, the time to first loss of a height starting at an exact top on one side has tail at \(n\) at most \(n^{-1/2+o(1)}\), since the coordinate walk has \(\mathbb P(Y_k\ge0)\to1/2\) starting at 0. Indeed the Sparre Andersen–Spitzer fluctuation identity (Spitzer 1956, sec. 3, Corollary 2, Equation (3.7)), with ties on the nonnegative side, gives the generating function of survival tails as \(\exp\{\sum_{k\ge1}z^k\mathbb P(Y_k\ge0)/k\}\). For every \(\eta>0\), this is \(O((1-z)^{-1/2-\eta})\) as \(z\uparrow1\); monotonicity of the survival tails and the choice \(z=1-1/n\) give the stated bound. At first loss, success in switching requires undershoot exactly one, with probability bounded above by \(1-c\) (already the first step can undershoot farther). Restart this raw rule iid from the opposite coordinate on success. A union bound over either coordinate and \(O(m_P)\) possible starting steps, and the stopping-time product estimates, give with high probability no run of more than \(C\log P\) consecutive successes, nor a run among these with at least three completed durations above \(m_P^{0.9}\): the latter costs at most \((C\log P)^3 m_P^{-1.35+o(1)}\) per start. Thus apart from at most two longest durations the total completed duration is negligible. Transfer to localized pieces by the bounded ratios. Any single completed such duration lying on these compacts is itself uniformly negligible by \(J_1\) and absence of scalar Cauchy chords: otherwise its side heights start at a top and end one below that value, sustaining it all the way before the final pop. In the limit this is a level linkage between distinct real times including the appropriate copies at endpoint jumps, which was excluded in the quotient proof. This proves the statement by another countable exhaustion and good subsequence.

Comparisons across times and at terminal.

Now track from an earlier occurrence until the later state. Survival indeed holds throughout if it appears later. If there was no pop below it, its proposed earlier value equals the one transmitted to the later state. Otherwise at the first switch, limiting at some \(r\), the original height has transmitted exactly to a tip copy (error one at the switch), of image \(P_{\rm tr}(r)\). All subsequent completed switches have total time negligible, so the final tracked height arose at a top also limiting to some copy at \(r\), then transmitted without pop through the compared time. It too therefore has that same image. Use the same-state assertion between it and any desired later occurrence. Finally, from a compact-surviving state with positive limiting height one cannot stay in polygons after every compact toward \(T\): without a switch the height contradicts \(S\to0\), hence first switch must itself occur on compact pieces after subsequence. In a neighborhood of its limit time \(r<T\) both side heights (either copy) are bounded below positively. On each longer compact horizon the remaining completed switches have negligible duration, hence the tracked height at that horizon still limits to values bounded below by this same positive bound. This again contradicts \(S\to0\). Height zero maps to the target as claimed. This argument classifies simultaneous limiting pinches without an estimate summable over every deterministic frontier. ◻

Proof. The lifting hypothesis and dense images let us follow any fixed target path inside an arbitrary open tube, with prescribed lifts near its two ends: choose a fine chain along the path, choose lifts of its intermediate points using density, and join successive lifts by (25). More generally, all inverse images of a compact path-connected target set lie in one path component of the preimage of any prescribed open neighborhood, for all sufficiently large \(n\).

Fix a fine simple 3-connected triangular grid on the target sphere. Around each vertex choose nested balls with strict radial buffers; the largest balls are disjoint and avoid unrelated edges. In the source, enclose the entire inverse image of the closed innermost ball in a compact connected polygonal subsurface \(A\), lying in the preimage of a slightly larger ball \(B_r\). This is possible by the preceding path-component statement, compactness, and a sufficiently fine source triangulation. Choose the outer vertex ball \(B_R\) with \(R>r\). Apply the same statement to \[K=S^2\setminus B_R,\qquad U=S^2\setminus\overline{B_r}.\] All inverse images of \(K\) join inside \(G_n^{-1}(U)\), which avoids \(A\). They therefore lie in one complementary component of \(A\). Fill every other hole of \(A\). The resulting closed disk still maps into \(B_R\) and contains all innermost-ball preimages in its interior. The disks for distinct grid vertices are disjoint.

Choose thin open tubes around the grid edges, intersecting one another only inside the innermost balls at common endpoints and avoiding unrelated outer balls. Lift each edge through its tube with ends in the assigned source disks. Polygonal approximation, trimming, and simplification give simple arcs between these disks, with distinct endpoints on their boundaries. Outside the disks these arcs are disjoint: any intersection of their image tubes would lie in an innermost ball, whose entire preimage is already inside the corresponding source disk. Insert disjoint stars inside the disks to complete an embedding of the grid. Whitney’s uniqueness theorem for simple 3-connected graphs identifies its faces with those of the target grid, up to reflection (Brinkmann 2021, Theorem 1.2).

For each target triangle, choose a slightly larger ball containing its triangle, edge tubes, and vertex balls. The image of the corresponding lifted face boundary lies in a smaller concentric buffer. All preimages exterior to the larger ball can be joined while avoiding this boundary, by the same buffered path argument. Some such preimages lie on non-face graph lifts and hence outside the lifted face. Consequently no point of the lifted face maps outside the larger ball. Choose the grid, vertex balls, tubes, and face buffers with diameters tending to zero. Extending the graph correspondence across each face gives homeomorphisms uniformly close to \(G_n\). A diagonal choice proves the assertion; no control of parametrizations inside individual filled triangles is required. ◻

Proof. Positions from finite truncations.

Here are details. Use increasing finite truncations sufficiently slowly and further coupling subsequences: compare on surviving compact initial pieces tending to \([0,T)\) at each paired node, with uniform path errors tending to zero after alignment there. Match finitely many jumps and their ancestry, eventually processing each fixed matched jump strictly before \(T\) in this countable hierarchy; any macroscopic limiting jump on such compacts is separately recognized. One can now assign positions (possibly several per site) in the continuum sphere.

• At a compared frontier assign vertex occurrences by \(B\) on their side at the aligned time using scaled heights clamped to the side range, in particular on original lists. States approaching a matched large jump from just before its step use aligning times approaching the pre-copy, those just after approach the post-copy. Thus these assignments have the subsequential interpretation already proved at converging states of each fixed chart.

• Assign any bridge of such a step to the trunk there. For a matched positive jump put ring cells at corresponding positions on the matched jump boundary in relative-length order; at an unmatched positive jump assign them to the trunk. Put all unmatched/unexamined jump interiors (positive or negative) likewise at the trunk of the event. Here matched but not yet processed child contents can instead always be put arbitrarily within their continuum closure. Anything past the compact piece in this chart, including a possible unused final polygon, not addressed by a specific compared jump is assigned to the target. At the central ring use its own boundary alignment.

• Allow assignments of corners and whole flags through their own Tutte triangles as well as shared-site assignments; one can use a common cell position on each individual triangle through its own piece, up to vanishing error. This still refers to the actual polyhedral \(X_n\), with monochromatic bridges and other polygon boundary slots accessible as paths there.

Within a fixed paired parent, at an unmatched jump with alignment limit strictly before \(T\), its scaled outer magnitude tends to zero by \(J_1\) and eventual matching of large jumps; boundary labels there thus limit to a top. For stable matched events child-parent orders on slots agree asymptotically: for negative compare \([L-j,L]\) on the indicated pre-frontier with the rooted detached boundary, and for positive use the positive post aperture, matching the child across the radial links. The boundedly many omitted endpoint slots in the ordinary-list convention limit to the pinch. At a varying sequence of distinct matched child events all positions in descendant closures converge toward their trunk or target limit by the continuum shrinkage. This applies also if they accumulate at a large jump time, keeping that one stable jump itself separate.

Agreement of competing incidences.

All simultaneous limiting positions through competing incidences of a vertex agree; on flags the position diameter including corner locations tends uniformly to zero. To see this, argue by subsequential extraction. Within one fixed parent, compare frontier occurrences of a common vertex by the previous consistency assertion. It covers also equality to a site still present beyond every compact piece (terminal comparison). Shared sites of fillings with other pieces go only through their boundary occurrences. Bridge ends compared on compact times are old tip incidences. Thus whenever a negative filling or outer side of a ring has to be reconciled with other positions in this parent, use the same comparison at its aperture and tip states. Inner vertices of a positive ring (only shared with its child) use the relative-order alignment to the child’s own boundary, not a parent-frontier equality across that ring. All competing positions of such a vertex reduce to comparisons within the child and its outer boundary list, or the whole ring/child already limits to the same point if it varies/shrinks or is discarded. The two disks of the central ring similarly use their own outer boundary comparisons. More formally descend through stable paired nodes when following two competing addresses. If they lie in a common child there is nothing to compare before descent. At their separating level (including a branch with no fixed next child) introduce extra occurrences of the given vertex on any involved stable child’s boundary if sharing outside that child is required, and extract their limiting labels. The parent comparison just given identifies limits there, leaving only comparisons within the children with their own outer occurrences. Descend again; any unfinished descending entry lies in an arbitrarily small closed cell, by nested mesh. Thus all comparisons hold in the limit (after separation any further recursion is a comparison to a boundary representative at a specified point, transmitted by equality or contained along with the other position in arbitrarily small cells). Every actual Tutte face is on a ring or in a discarded portion and its vertices admit its own position up to the stated errors. This proves uniform incidence consistency by contradiction.

Joining every pair of lifts of one limiting point.

Choose single values and interpolate continuously to maps \(G_n:X_n\to{\mathbb S}^2\) on the flags, e.g. by spherical chord interpolation of the close triples. Their images get dense, by accessible contour positions and nested mesh. We claim also \[ G_n(x_n),G_n(y_n)\longrightarrow z\quad\Longrightarrow\quad x_n,y_n\ \text{can be joined with images tending uniformly to }z . \tag{26}\] Move first to incident vertices. Any lifts of the same outer or aperture label can be joined on those lists along arcs of vanishing relative-label width, including the cyclic endpoint conventions (this refers to actual boundary-slot lifts; to compare other lifts see below). At a negative pinch one can cross the bridge and new active edge; at a positive pinch likewise the active edges connect the old and new tip without needing to round a macroscopic loop. Inner-to-outer boundary transfer at a ring uses the matched radial links. Tip lifts at times tending to one \(t<T\) connect by the consecutive active edges (which share ends at the updated tips), all intermediate steps having projected endpoint limits at \(P_{\rm tr}(t)\) by continuity including the folds. At the source one uses the first active edge. At terminal, sites in all sufficiently late portions are inside the same connected polygon at any fixed earlier compact state; the double-limit supremum of its image distance to target vanishes as the state tends to \(T\) by the upper assignments and shrinkage. This joins tail lifts to target endpoints which themselves have the target edge.

Horizontal linkages transfer by lowering height. Given a linked pair on a side at height \(h\), for an outgoing open positive-aperture label take the post frontier, for an incoming open negative-aperture label take the pre frontier, and for an outer label take the initial list. For a real-time occurrence take the copy at its linkage end, using the paired jump if applicable or approximating by nearby states. Top coordinates there can be joined to the desired tip by the preceding observations also across copies. With this choice the intervening states sustain height \(h\). The lattice comparison exactly transmits a lowered height \(\lfloor P h'\rfloor\), \(h'<h\) sufficiently nearby, by \(J_1\) and the step rules on compacts, or height zero without lowering. Join along each observing frontier back to the desired label, at upper-image error vanishing as \(h'\to h\), by continuity of \(B\). This permits the lowered height to be outside an aperture if needed at an endpoint; hole-boundary lifts at the exact endpoint itself freely reach the tip by the preceding pinch joins. Nontrivial linkages occur strictly before \(T\). Together these transfers and same-label/fold joins implement the exact quotient contact rule. An ordinary prefix \(B_v(h)\) itself, if observed, can likewise reach its backward linked contour occurrence.

Apply this recursively after piece-address extraction. In the strict interior of a child only sequences there can limit to the point by the upper comparisons. If sequences separate at a fixed parent, any involved stable child position must be on its boundary, so introduce a prescribed lift by boundary slots of that child. A varying/shrinking child’s interior sites can move to its attachment with all locations still tending to \(z\), by the uniform upper comparison there and connectedness including the bridge or ring. This leaves outer/aperture or tip/target representations of \(z\) at the separating level, joined by the parent rules just proved by absence of extra contacts. The central ring case only needs its own length-aligned boundary joins. Descend to join original lifts to newly prescribed boundary lifts within the stable children if necessary, also descending if the two lifts remain in common cells. This recursion suffices for any finite accuracy: stop along unfinished branches with arbitrarily small image closures and join freely within those connected fillings, using nested mesh and a diagonal choice, with finite joins at each fixed truncation. This proves (26) along further subsequences and hence in the stated sense.

Promotion to orientation-preserving homeomorphisms.

The sequential assertion (26) implies the uniform hypothesis (25). Otherwise, for some fixed \(\epsilon>0\) one could choose \(n_j\to\infty\) and pairs whose image distances tend to zero but which have no joining path in the corresponding \(\epsilon\)-balls. Compactness gives a common limiting image \(z\), and (26) contradicts that choice. Dense images were proved when constructing \(G_n\). Lemma 52 therefore gives homeomorphisms uniformly close to \(G_n\).

Their orientation is pinned by the central interface traversal once around its simple limiting curve, with the specified disk side on its left in both pictures. Interior lifts on the two sides approach points separated from the limiting curve on the corresponding sides. The uniform ordered comparison of that traversal preserves its winding, so the nearby homeomorphisms preserve orientation. Since the images of whole flags under \(G_n\) already have uniformly vanishing diameter, the same is true under these homeomorphisms.

Complete loop tours and exclusion of extra loops.

Each fixed positive jump loop, like the central one, traces its matched loop once in relative label order with sign/orientation as observed. Both its outer and inner scaled degrees converge to the same quantum length. Conversely an unnamed loop remains wholly in a yet-unexamined polygon or is itself on a discarded ring; by extraction and shrinkage of varying children, terminal parts and deep stable cells no extras of positive macroscopic diameter occur. This proves the proposition. ◻

Proof. Take \(P=B_n\) and normalize triangle counts by \(2n\), at first even without requiring the total. At perimeters tending to \(s>0\) in these units the volume law tends by (21) to \(d_s\). Jointly with the coupled path data in an unconditioned pair, matched cell volumes converge to the areas of the corresponding field disks. Indeed conditional on parent paths/perimeters the jump fillings are fresh. Through any fixed finite depth first keep the paths down to the previous depth, and also volumes of cells born at that depth; (21) and the product conditional laws give the stated comparison for the latter jointly by finite tests and exhaustion. Limiting ancestor masses must dominate the sums of descendant masses by actual containment. Those sums in the corresponding full continuum generations have exactly the ancestor-area law by area additivity in the physical model; the lattice ancestor marginal already has that same limit by freshness at its own birth and (21). Therefore domination is equality (condition on/exhaust ancestor birth tests when selecting ancestors). Increasing finite depths proves the full joint comparison. In detail take a subsequential limit with all matched lengths/paths and cell masses jointly (each fixed mass label tight by the perimeter and freshness estimates). Conditional on paths through any fixed preceding generation the born masses at the next depth then have the continuum product marginals by testing on earlier observation cylinders only; all labels can first be selected on compact ancestry/birth events which do not test their own subsequent interiors. Thus for any fixed ancestor the sum of these next-generation masses below it already has marginal equal to the true ancestor-mass law by area additivity there, conditional at least on its earlier birth data. The limiting ancestor variable has this same marginal and dominates the sum, forcing equality a.s., simultaneously through countable levels. Given any finite joint test keeping also later paths, pass to depth strictly beyond those viewed paths, where the mass marginals conditional on earlier paths are known as above and all earlier masses are the a.s. sums of the indicated descendants. This is why no independence between mass and the future exploration of its own cell was required. This works for compatible routings under these area-additive and marked Markov hypotheses. Consequently normalized flag-area measures converge weakly under the homeomorphisms: distribute according to matched cells of small limiting mesh capturing arbitrarily much of total mass. Each Tutte triangle accounts for two flags and the initial ring mass is negligible. ◻

Proof. We detail this step since absolute continuity of a macroscopic finite sphere relative to a pair of Boltzmann disks should not be asserted after exact volume conditioning. Use counting intensity with a central marked loop and restrict \(k/P\in[\alpha,\beta]\Subset(0,\infty)\). Initially on the unconditioned pair fillings put base mass per \((k,l)\) \[b_n(k,l)=\frac{4}{k+l} \frac{\binom{k+l}k 2^{-k-l} f_k f_l}{X^{2n}Z_n(4)} .\] Use a distinguished link (uniform in the ring in the loop-marked formulation). To obtain the actual restricted marked intensity multiply by \(n{\bf1}_{\{N_k+N_l+k+l=2n\}}\), as in (15); one may restore an ordinary FK root uniformly with the corresponding occurrence symmetry. Before multiplication these are finite measures with the continuum abstract limit \[\frac{1}{2\pi^2}s^{-5}{\bf1}_{[\alpha,\beta]}(s)\,ds\ \mathrm{Pair}^{\rm prob}(s)\] including common-typical boundary alignment, by the same Riemann-sum estimates in (15), (24) and the volume comparisons. Here the pair uses the two critical probability disks at perimeter \(s\), with recursive independent signed exploration in the nested coupling, and starts without imposing a total volume. We need only abstract path/volume labels in this convergence first. The views after volume conditioning will be coupled to actual embedded placements from the continuum Palm law by disintegration, not by assuming conformal-welding uniqueness.

Integrating out one unexamined filling.

Fix a finite compatible path test and finitely many cell-volume probes. Choose an additional jump filling below the interiors inspected by these tests. The choice uses only its birth path, perimeter, and attachment labels. In particular it does not inspect the filling itself. Localize the choice to a success event on which its degree \(p_n\) satisfies \[u_n:=p_n/P\in[u_-,u_+]\Subset(0,\infty).\] Compact survival, strict-margin jump matching, and jump density provide a countable family of such success events exhausting the limiting pair law. One obtains them by continuing below the finitely many viewed depths until an additional matched positive-size jump is born. The selection can use lengths and times relative to the central perimeter \(s\), so the same exhaustion applies after the scale-integrated area-one conditioning.

Let \(\mathcal F_{\rm out}^{(n)}\) contain the selected birth data and all discrete data outside this filling. Volume probes of ancestors containing it are not included as extra conditioning: write each such volume as its outside contribution plus the reservoir volume, and make the substitution below. By the exact peeling factorization, conditional on \(\mathcal F_{\rm out}^{(n)}\) the reservoir is a fresh Boltzmann disk \(T_{p_n}\). Write \(N_{\rm res}\) for its triangle count and \(N_{\rm out}\) for all other triangles, including the central ring. Put \[A_{\rm out}^{(n)}=N_{\rm out}/(2n).\] The conditional expectation of the exact-volume multiplier is therefore \[\begin{align*} &\mathbb E\left[n\mathbf 1_{\{N_{\rm out}+N_{\rm res}=2n\}} \mid\mathcal F_{\rm out}^{(n)}\right]\\ &\hspace{2cm}=n\mathbb P\{N_{p_n}=2n(1-A_{\rm out}^{(n)})\} =d_{u_n}(1-A_{\rm out}^{(n)})+o(1). \end{align*}\] The last equality is uniform on the success event by Lemma 27. Its parity condition is automatic: the outside surface with its degree-\(p_n\) hole has precisely the complementary triangle parity. The density is zero for nonpositive residual area and is uniformly bounded when \(u_n\in[u_-,u_+]\).

All outside quantities converge jointly before conditioning, by Proposition 54; for example \(A_{\rm out}^{(n)}\) is the total mass minus the reservoir mass. In a tested ancestor volume, replace the reservoir contribution by \(1-A_{\rm out}^{(n)}\). Unrelated volume probes are unchanged. The exact conditional identity above now proves convergence of every bounded continuous test of the retained path data and substituted volumes on this success event. It proves the same assertion with additional bounded outside readouts whenever their unconditioned joint convergence is available. It does not require a limit theorem for the interior shape of a disk conditioned on one exact triangle count.

Identifying the area-one version.

The resulting density selects the area-one Palm law of Proposition 46, with its specified disintegration. To verify this without choosing a conditional law at a null event, start from the unit-perimeter pair and integrate the central perimeter \(s\). Use bars for areas in the unit-perimeter pair. The reservoir has relative perimeter \(u/s\) and, conditional on its outside data, area density \(d_{u/s}\). Scaling the pair to total area one and changing the reservoir-area variable back to \(s\) gives the factor \[s^{-2}d_{u/s}(s^{-2}-\bar A_{\rm out}) =d_u(1-s^2\bar A_{\rm out})\] against the central perimeter measure \(s^{-5}\,ds\). This is exactly the limiting multiplier above. Volume entries containing the reservoir undergo the same substitution. The calculation remains valid for Borel success tests chosen from the outside data.

Under this scale-integrated area-one law the unit-perimeter pair variables are absolutely continuously weighted. They therefore retain the almost-sure cutting, contact, and mesh properties of the unconditioned pair. Finally Proposition 22 gives convergence of the whole restricted counting mass. Since the reservoir success events exhaust the candidate area-one law, none of that mass can escape their union. Refining finite continuity partitions and then increasing the family of tests proves the joint assertion for all countably encoded cutting data. A different, deeper reservoir may be used for each finite test; no filling is required to remain independent of the entire limiting tree.

The pinching and tracking estimates used in the spatial comparison also survive exact volume conditioning. For a test in one initial disk, keep the opposite initial disk unexamined and use it as the reservoir. Both initial perimeters lie in compact positive ranges, apart from the negligible ring tails already controlled in Proposition 22. The uniformly bounded local-limit multiplier then preserves every vanishing error probability needed on that test. The good coupling subsequences required for Proposition 53 are consequently available under the exact fixed-size law.

Actual continuum placements and measurable comparisons.

In particular in this restricted counting intensity we have the topological area, nested-loop and length-label comparisons to actual unit sphere cuts. Conditional on the full matched abstract path and volume data in a subsequential limit one can sample remaining continuum geometry independently of any further lattice readouts by the regular conditional law of the actual field sphere/cuts given those data. The vertex/incidence consistency and (26) did not require determinism of that conformal sphere from the cutting hierarchy. All variables used in the coupling may be encoded by countable measurable charts, paths, labels, and probability fields on the standard spherical/disk charts. Approximate maps and paths, if used probabilistically, can either be chosen measurably by finite triangulation searches up to arbitrarily small additional errors, or just used through the resulting closed sequential comparison assertions. ◻

Proof. To remove Palm bias additionally require the \(m_n\) side fractions of the central loop both to lie above \(\delta>0\), at continuity cutoffs. In the marked comparison the entire number of qualifying loops then converges. Indeed weak convergence of area to a diffuse measure under the comparison homeomorphisms prevents an arbitrarily small loop from carrying both side fractions bounded below. All positive-diameter loops are matched one-to-one, with their two degree labels and their volume fractions converging since true loop boundaries have zero area. Window boundaries can be avoided for the entire countable ensemble, by (the continuum side of) the Palm identity and absolute continuity when re-marking by counting. The limiting count is finite and at least one on the further restricted law. The critical lengths for other loops that enter this comparison are the actual intrinsic cut lengths established in the continuum counting Palm identity (hence a.s. for all loops also under central counting restrictions). For a discovered jump loop its interior chart has precisely the actual pulled-back field there, so by length-as-field-function it has that length with the same units; the two discrete circumference labels agree asymptotically by the ring comparison. Dividing by the count on both sides now yields the ordinary rooted-map probability restricted to having at least one such loop, coupled to the ordinary unit-area prescribed probability with the same nonemptiness restriction, with convergence of the masses too. This retains a central choice uniformly among qualifying loops when desired. Letting the windows increase and \(\delta\) decrease exhausts the latter ordinary probability by positivity/finite length and nonzero side areas of true loops; hence there is no leftover mass in the ordinary map passage either. One can for example partition by the first nonempty such window. Signs, if used, correspond to the fair bSLE orientation assignment, while blue and red represent opposite parities from the specified side of the central loop. Three independent \(m_n\) samples can jointly approach fresh independent area samples by the weak area comparison. One may postcompose the topological placements accordingly to use the specified orientation-preserving three-mark convention in the continuum. This proves at this stage the area/CLE/mesh assertions in coupled topological positions; replacing these by the actual \(\phi_n\), and comparing intrinsic distances, still requires separate arguments. ◻

Proof. Target-independence in the level-line coupling gives branching up to separation; here the total force weight is \(-2=4-6\). After separation, branches remain in the distinct target components and do not revisit their shared past. The open original boundary intervals toward the targets are disjoint. Simple-curve separation then gives the meeting and monotone-contact assertions. We use ordinary targets only, so no rule is needed at a target lying on another path.

Along a trunk use the full MSW exploration, tracing the encountered \(c\pm2\) CLE loops. When a target is cut into a negative domain, continue from its pinch by the corresponding exploration there. Proposition 44 gives constant-\(c\) fresh domains for this full signed exploration. The trunk alone need not have constant data in every domain it cuts. Splitting toward two targets and then exploring their disjoint fresh components gives the same branching coupling, with targets on transmitted open arcs.

To prove macroscopic excursion finiteness, equip \(C\) in conformal coordinates with an independent auxiliary critical quantum disk. Initially take a typical length source and target and draw one full MSW branch. In each negative child draw another such branch from its pinch toward an independent uniform length target; repeat only in negatives. The one-branch marked rule, negative moment contraction and composed diameter estimates in Lemma 45 imply summable large-domain bounds after a fixed parent lower-proxy cutoff, and uniform diameter shrinkage down these negative generations.

An original typical target is reached by following helper prefixes and switching to its negative child at separation. The subsequent continuation lies in that child’s closure. There are no corner ambiguities for countably many typical length tests. Original-boundary contacts are linked outer records; running-record sets and jump endpoints have zero length by Proposition 40. Open original arcs transmit isometrically, and loss and aperture boundaries are the prescribed jump boundaries. Thus a target entering a negative retains an open original boundary interval. It cannot enter a positive disk, since positive loops do not touch the true boundary of the current negative. One may equivalently route a fresh independent target after conditioning on the auxiliary surface and helpers, or sample in surviving intervals.

Only finitely many helper cells matter above a prescribed diameter. Once the first sufficiently small descendant is entered, its entire remaining continuation is small; before then there are finitely many cells by generation and sibling shrinkage and containment. Every macroscopic excursion therefore contains an interior subarc of definite positive diameter on one of finitely many helper trunks: truncate its small terminal tail and use the finite number of preceding switches. Distinct excursions have disjoint interiors, so uniform continuity and injectivity on the finite helper list give finiteness. The same construction shows that each countably tested interior passage belongs to a full excursion, with no extra interior limiting trace.

Individual paths have no boundary arcs; for example their positive jumps are dense and have interior tips by the exact stack rule. Their hits on either open boundary flank have harmonic measure zero. To see that every prescribed point other than the endpoints has zero hitting probability, first use the auxiliary length experiment: linked original-arc real-time heights lie in closed descending record sets, of length zero. Conditional on endpoints and surface the chordal law is conformally invariant; for fixed endpoints its one-point hitting probability is constant on either open flank. The diffuse full-support length measure forces that constant to vanish. Boundary contacts occur arbitrarily near the target on both flanks before terminal time: both coordinate records take positive values tending to zero at times before termination, by compact positivity, the càdlàg rule, and first entries below vanishing thresholds. Their linkage to the original arcs is exact. A dense target in a given open boundary interval and late gaps in its contact-time set consequently provide arbitrarily small excursions with both endpoints there. The contact-time set has empty interior, and simplicity makes those late excursions small.

These conclusions transfer to fixed \(x\) by conformal invariance, or by rotating the independent auxiliary chart. An additional ordinary target contributes no new excursion: an excursion on its path remains on paths toward an open target interval, and a dense test already meets that interval by target-independence. Thus dense tests can be changed, and the conclusions transport to closed Jordan disks.

We identify the complementary components and the field. A null sequence of disjoint full crosscuts, with disjoint open portions, has Jordan complementary components, each touching the outer boundary. For a point outside their closure, remove the shadows cut off from it by each crosscut. These shadows are nested or disjoint. Every positive-sized shadow is behind a maximal one: nested enlargement uses only finitely many non-small crosscuts, whereas small crosscuts away from the point have small shadows in Jordan coordinates. Replace the corresponding disjoint open intervals of the outer circle by their maximal crosscuts in cyclic order. Their diameters tend to zero, so this is a simple closed boundary with the expected interior. Coincident endpoints merely make successive arcs abut; if both complementary intervals between two endpoints are replaced, one obtains the two-sided Jordan lens. No other crosscuts accumulate locally on the open interior of a fixed one, so it has a complementary component on each side. The uncut case is included.

Finite branch unions are thin GFF local sets with conditional mean in \([c-1,c+1]\): every continuation occurs only in its remaining component, with untouched data \(c\) and matching slit data. Martingale convergence and Dirichlet covariance exhaustion give the prescription off the closed union. It is thin with bounded mean. On an interior compact set only finitely many excursions occur, lying on countably many chordal SLE traces of dimension strictly below two; alternatively, bounded finite-stage means converge against every compact test by dominated convergence off the zero-area closed union, while Green covariances decrease to the component covariances by boundary regularity. All interior boundary arcs facing a given component carry the same value \(c+1\) or \(c-1\). Two supporting paths after divergence cannot both face that component, and on one path a connected complementary region lies on one side.

A component’s contacts with \(\partial C\) lie among the boundary points hit by the countable paths, with at most one additional point. Otherwise a tested path with endpoints alternating with two unhit points would separate nearby interior points of the component. These contacts are harmonic-null both in \(C\) and within the component. Brownian exit and local agreement on the crosscuts therefore identify its bounded harmonic mean with the indicated constant. Boundary connection and BTLS uniqueness yield \(\mathcal A(C)\). Uniqueness makes its interior crosscut set source-independent; adjacent signs fix crosscut directions. Finally rotational invariance and harmonic-null contacts, summed over the cells, show that a specified prime point lies on no half-cell closure. This applies conditionally to countably many points selected before sampling a fresh half-layer. ◻

Proof. Work first in a small disk \(B\Subset U\) about \(z\). If \(z\) lies in the compactly contained bounded side of a complete \(O\) wall, that wall and side are common by Proposition 57. Its side is explored inward, since the starting ring cannot be within it; ancestor gasket sweeps do not enter it. Its entire canonical interior hierarchy is common.

Otherwise only finitely many nonlocal \(O\) arc pieces can approach \(z\). Pieces crossing a collar are finite by loop local finiteness and Jordan continuity; all other nearby walls are complete within \(B\). Unequal \(O\) walls are disjoint. If \(z\) is off the walls, it lies in a base gasket away from each individual child closure. Infinite nesting at \(z\) would have put it inside a complete small wall, so such a base exists. Its boundary lies outside a smaller neighborhood. The only child boundaries nearby are known complete loops, namely maximal small loops on the outward side toward \(z\). Nonlocal arcs remain away and cannot separate nearby wall points from \(z\); a complete loop compactly inside \(B\) has its bounded side inward. Thus the omitted open holes are known locally. If \(z\) is on a wall \(W\), shrink to a Jordan box where only its arc, among non-small \(O\) walls, approaches \(z\). On each side use the adjacent base gasket, where \(W\) is the base rim or a direct child. Other nearby holes are complete and local. One may first handle larger loops in a larger box and require each loop called small to have its bounded closure inside \(B\).

The relevant actual mean \(c\) is fixed by a common \(O\) side trace: by \(W\) in the on-wall case, or an exposed nearby child arc in the off-wall gasket case. Such arcs approach gasket points by filling and disjoint Jordan nesting. Therefore \(J-c\) agrees as a real field, rather than merely after independent choices of representatives modulo the period. The restricted graph is also common. An \(E\) arc facing the open base belongs according to its side relative to the known holes; noncrossing prevents a remote continuation into a different side. Read contact-only points by continuity along arcs, since opposite parities share no open subarc. Zero-cost movement follows actual local arc connections and does not assume an outside grouping of separated pieces.

Construct the shield off walls by starting with a small surrounding path and detouring along hole rims. Large holes stay away from \(z\) and remaining holes are arbitrarily small, so the detours exist arbitrarily close to \(z\), remain at positive distance from it, and retain winding by homotopy off \(z\). Jordan continuity controls accumulating gaps. Lemma 65 approximates this path by finitely many wall arcs in one sample, preserving winding about a still smaller neighborhood. These arcs are shared in every sample. At a wall \(W\), use a shield arch from \(W\) to \(W\) through the chosen side and away from \(z\), obtained from a semicircle in Jordan coordinates with the same detours. Close it along \(W\) through \(z\). Approximate the arch with a margin; the closed path has nonzero winding throughout a sufficiently small open half-neighborhood. Include the depth of \(W\) itself among the side data.

Consecutive shield arcs attach, so their graph depths differ by a bounded amount, uniformly over exterior samples. This bounds all differences \(d_i-d_1\) within a finite integer set. For the lower bound in (30), realize a shortest full graph route to \(w\) as a continuous finite-arc path. Its base rim has a point outside the comparison box; a route on \(W\) itself already has its recorded depth. Winding forces the route to have a contact with a shield arc on the very same wall vertex. To check this equality-contact point, suppose there were no such contact. A switch between route segments cannot occur at a shield intersection, as it would give a same-kind equality. On each route segment, touching shield arcs on walls of the other kind stay on the closure of one side of its Jordan wall by noncrossing. Flatten the wall and perturb those arcs slightly toward that side, avoiding the segment’s endpoints and switches. Compactness permits this locally for each segment. The perturbation changes neither endpoint index, so the winding index cannot change along the route, a contradiction. After the last passage into the common box, retain the suffix from such an equality contact. It gives the lower bound by the definition of \(\ell_{\mathrm{loc}}\). The upper bound is immediate by concatenating a shortest route to the shield with a local route. At contact occurrences on \(W\), use the same direct traversal or nearby open-side points. Finally retain parity, so an unrecorded common shift can be taken even. ◻

Proof. In the neighborhoods of Lemma 66, very low relative levels produce no new local trace, except possibly the already recorded side of \(W\). Empty ancestor sweeps covering the patch are irrelevant. At very high relative levels there are no active cells passing from an inner buffer out of the comparison box. Indeed choose a sufficiently late layer in one representative with tiny active mesh, and use (30) and the bounded relative alternatives to include that layer locally in every case. Direct child boundaries already reached there cannot be entered by active cells.

Only boundedly many intermediate relative levels remain. Their local half-layer sets, holes, and sides are known, for each of finitely many alternatives. Outside grouping of components may differ, but only finitely many pieces cross a fixed positive buffer. In a representative at any of these levels there are finitely many macroscopic cells, each Jordan. Their boundaries have finitely many arc portions crossing a positive-width buffer, by uniform continuity of their Jordan parametrizations. The same holds for open component passages: a traversing component either has an adjacent boundary passage across a smaller buffer or contains an interior ball of a fixed positive size. Equivalently use a closed-disk Jordan parametrization and uniform local joinability of nearby interior points. Thus finitely many external groupings suffice. Smaller wholly internal pieces have no outside grouping, and very deep internal cells are components of the known local depth pattern with determined subsequent hierarchy.

On \(W\), use the two side arrays, starting its interior base afresh when \(W\) is a child, or both sides when it is central. Treat complete nearby child interiors similarly. A covering cell with no boundary or layer trace present locally has no tip or welding operation there. Taking a finite subcover of \(K\), and if useful of an intermediate enlargement, gives finite lists for every caller crossing the outer buffer with an operation near \(K\). Group matching instances and boundary-arc orders across the cover by finite tags. Except for finitely many larger crossing cases, a child meeting \(K\) is in a common internal subtree or is a child with locally known boundary in one such caller; the Lebesgue covering margin handles small children in one member. Buffers may be chosen from countable bases with strict margins. This proves the stated quantifiers. ◻

Proof. We first prove a protected domain-restriction statement for a single path, then use it to compare two paths. Choose a stopping cut \(\sigma\) strictly inside a shared incoming marker, or use the actual start. Its individual conditional field and future path have the usual level-line prescription in the true remaining disk: constant \(c\) on unhit true boundary, oriented incoming slit values when present, and insertion values \(c-1,c+1\) on a drawn future prefix. Cuts can be selected from a countable stopping library of ordered small-circle hits after crossings, with a strict shared past segment. They are taken separately in each path’s valid marginal law.

Proof. In one comparison chart take an inner marked region, a larger local-target region, and still larger buffers. By Lemma 68, choose finitely many oriented anchors intercepting both passages from the inner region out of the local-target region and passages from the closed local-target region through the next buffer. Take an additional finite outer anchor set when classifying earlier boundary visits. A simple path uses each anchor at most once, and all paths using an oriented anchor in one fan have the same prefix to it. Between consecutive anchors, or from a shared local start to an anchor, portions visiting the inner region remain buffered. Proposition 69 gives finitely many exact possibilities; source choices are already finite. Which of the finitely many endpoint slots give buffered or relevant portions is itself a finite tag. Paths ending outside the local-target region contribute only such anchored portions near the observation.

For an inner anchor \(A\), the set of ordinary targets whose paths use it is one open interval \((l_A,r_A)\) in linear boundary order away from \(x\), ignoring target-null endpoints. Preceding contacts bound that untouched interval; target change preserves the prefix within it, and every path through \(A\) must share this prefix. Two separated target intervals would alternate with two prior contacts and be separated by the trace. We must classify these intervals jointly over all local targets.

Cover the local-target boundary by finitely many buffered arc portions. The chronological visits there before \(A\) have finitely many possible ordered descriptions: insert the additional outer anchors on its unique prefix, and use local uniqueness on the intervening start, anchor and end portions that visit these arcs. Retain their circular order and the position of \(x\), tagging exterior gaps coarsely. For any target \(y_A\) using \(A\), visits below \(y_A\) in the linear order progress increasingly in time, whereas visits above it progress decreasingly. For a fixed ordered contact description, two choices of the split cannot move two distinct contacts from one class to the other: those contacts would have to be both increasingly and decreasingly ordered. Thus the split has only finitely many possibilities, with finite tags for intervening exterior gaps relative to \(x\). Within the local arcs, nearest contacts fix \(l_A,r_A\) exactly; an exterior endpoint needs only its finite gap tag. This classifies the anchor’s entire target interval at once.

For a shared ordinary local target \(y\), its ordered inner anchors are now known. The path from its last anchor to \(y\) stays in the larger chart, or it would meet another anchor. It is common by local uniqueness and the tagged boundary order toward \(y\). With no anchor, the whole path stays there. No tag is added for \(y\). For exterior targets retain the finitely many possible ordered sequences of buffered anchor pieces. Their entire local family, with exact shared portions and path order, has finitely many joint possibilities.

For several charts choose anchors in each. External-only transits contribute only a finite ordering and branching grouping of this finite list; local terminal portions are handled in their matching chart. The countable dense tests and open target-interval agreement give the same assertion for length-almost every ordinary target. Increasing or reordering the tests changes neither the unranked rooted traversal tree nor these common open-interval prefixes. ◻

Proof. Add the positive loops hit along the matched paths. Loops crossing buffers are macroscopic and form a finite list with finite grouping choices. Whether one was discovered on an earlier exterior prefix is a finite tag: the record on an anchored prefix is shared by every path through its terminal anchor, while a terminal portion toward a local target remains local. New contacts on a local segment are visible there. Small loops meeting the inner observation region have all their first contacts visible on these chronological prefixes or terminal portions.

A negative pocket wholly local on a matched full branch is read as a complementary component of the trunk, previously hit positive loops and true boundary. Its pinch is the branch’s last trunk contact before leaving it. After a negative cut the branch has no later tip contact with that disk, by the exact loss-arc and no-two-real-times rules: loss interiors do not transmit forward. Similarly a positive loop is first met at its attachment. With additional observation buffers, complementary-domain identifications for any finite matched path/order list have only finitely many crossing groupings, by Jordan continuity, macroscopic-piece finiteness and the common local obstacles. The accessible pinch side follows the direction or is a finite crossing-jump tag.

For an internal return \(R\), choose a branch witnessing its birth. By then its entire boundary is drawn on the trunk and earlier discovered loop arcs. On the matching branch those obstacles have also been drawn by that time, including crossing loops whose prior discoveries are tagged, and \(R\)’s interior is again avoided. Thus \(R\) is already disconnected and the pinch still visited on the trunk is its cut point. It is not a cut for following original targets into its interior, so it is left for a fresh sweep. A small internal positive child has the corresponding common first contact. Branches reaching the same child cut have not yet split: until discovery its interior stays in a single component at every earlier split. These assertions hold simultaneously for tiny children in a covering chart; the finitely many larger crossing children require only the previous finite grouping/order tests.

An intermediate negative carrying original target labels continues from its pinch toward their untouched original open interval, up to a boundary-length null set. Earlier contacts bound a single such arc. Non-strict identifications outside untouched open arcs carry no additional interval or positive length by running-record polarity and length nullity. An empty inherited interval for a complete negative birth is precisely the stabilized return of Proposition 63.

Lengths on shared boundary arcs with known side agree by arc covariance. Frontier lengths can be recovered by summing inherited arcs: at ordinary pre- and post-states, almost every height on either side lies in a strictly inherited open interval from the original outer boundary or a positive circle, by the scalar stack rule. Exceptional running-record heights, including real-contact labels, have zero Lebesgue measure. At jumps use the same pre/post conventions, including the aperture. Away from finitely many designated crossing jumps, swept changes are the symmetric Cauchy sums on each directed side, by localized absolute continuity. Elapsed local clock is recovered from small-jump counts, for instance positive jumps first contacted on the segment, as in the intrinsic-clock argument in the proof of Proposition 40. Omitting finitely many crossing jumps does not affect that clock limit. Constant Weyl scaling multiplies length and Cauchy time by the same factor. Together with the shared internal sources this identifies the stated block and its future tree, while leaving its exterior-origin offset unspecified. The separate finite-chart and integer-shift bookkeeping will retain that distinction in Section 7. ◻

Proof. For countable encoding it suffices in a continuum caller, and in joint lattice comparisons, to test a dense ordinary sequence of target fractions (e.g. a randomized uniform sequence, or fixed ordinary fractions when admissible). Use the first target still eligible each time on entering a negative with nonempty inherited interval, and restart lists in newly born complete callers. Finite such routes, tested on compact survival through matched pieces, converge jointly by (24) and the compatible-route comparison of Proposition 48. At negative cuts along them an empty limiting inherited interval means a canonical return stabilized by Proposition 63: its closure is disjoint even from the original caller rim. Thus it also has empty inherited list eventually under the topological upper comparisons on the path; alternatively the strict stack inequalities give the stability, since limiting equality of an original rim label with a loss boundary would be an actual contact. Nonempty open intervals and switches of ordinary targets transmit stably by the strict Cauchy record rule. Every further ordinary target prefix through strict preterminal observations lies also on paths to an open interval of targets, and is read by common-prefix transmission from the dense lists. Thus the resulting measured tree need not keep any priority of those lists. In particular auxiliary ordinary targets added later by continuous fraction densities do not call for a new continuum disintegration.

The mesh for the one-branch-at-a-time countable encoding still shrinks. An infinite negative chain staying in one caller misses at least the first \(N\) list targets after \(N\) switches (the first eligible is strictly lost at each switch); all macroscopically passing leftover components in that caller stabilize by finite targets as in Proposition 63. Infinite chains containing repeated canonical restarts instead shrink by half-layer active mesh or by nested \(O\) mesh of Lemma 64 and compact disk containment. Siblings at any one path shrink by the single-branch rule. This gives the full uniform shrinkage. Volume and spatial coupling for these routes now use Proposition 53 and the finite-depth area dominance, since offspring interiors are area-additive and the false-pinch and joining comparisons are along countably many compatible cut paths with this mesh. One can accordingly extract universal observations using this countable primary encoding and then sample the remaining embedded field/tree data from their actual continuum conditional law given \({\cal T}\), independently of extra lattice readouts given that data input. This does not ask for determinism of the fields or embedding from \({\cal T}\). ◻

Proof. An equivalent countable navigation refined within matched raw pieces may be used to recover these buffered spatial readings. Indeed any two finite systems with sufficiently small upper-image mesh in a common coupled realization impose the same inclusions from strict compact constraints to enlarged open constraints by all-lift coverage under the spatial comparison. This works for paths, curves and all flags meeting a prescribed protected part, and then for comparisons of budgets/variational costs by inclusions; it never requires exact membership agreement at cover boundaries. Thus one can first extract intrinsic closed passage profiles/witnesses with a primary alphabet, apply the conditional continuum realization as above, and identify their spatial regularizations also with extra refining arrays jointly extracted when doing a local calculation. Subsequence witnesses/attainability can equivalently diagonalize all positive margins. Only such compatible localized observations, not conditional independence of arbitrary exterior-indexed microscopic cover tests, are intended. ◻

Proof. For separated compact observations retain whole small terminal disks and whole ordinary sweeps (including all their nondesignated jump contents) when needed there, and retain ring chunks and designated point gluings locally. Pieces retained for one patch are entirely within its larger private buffer. Other ordinary sweeps are away from the inner tests and all their stars with margin there. This can be arranged on a given drawing: reach each of the finitely many canonical macroscopic cells still requiring traversal using finite branching stabilization and recursion; choose sufficiently many further targets that uncharted pockets no longer pass macroscopically across the buffers. Truncate the tested paths sufficiently late before each target after all needed forks/discoveries, designate all crossing jumps on them, and take short time cuts between, using continuity of trunks and vanishing diameters of other jumps/tails. These are positive-support choices, with open target intervals to capture any demanded preterminal path prefix or fork. For two-target forks into both sides require interior representatives there. Unseen macroscopic positive-volume pieces away from tested data can be reserved separately for total-volume smoothing. No deterministic bound on a chart size in this first simultaneous separated test is necessary. ◻

Proof.

Proof.

Proof.

Proof. Before the separate projections are revealed, the conditional law given \(S\) is \(\prod_b\mu_b(\mathrm du_b\mid S_b)\). Reweighting by a function of \(S\) and adding the stated nuisance observations do not change it. For fixed \(S\), each \(G_b\) is a function of \(U_b\) alone, with fixed parameter \(I_b\). Disintegrating over these separate functions gives the assertion. Measurable versions in \(I_b\) can be obtained by countable refining partitions of the projection range, as in the application below. ◻

Proof.

Proof.

Proof. Let \(S\) contain the first-scale full chart parameters and sweep paths. The preceding construction and ratios leave \(U_b^*\) product given \(S\), with separate kernels just at \(S_b\). The negligible-scale nuisance variables appearing in the ratios can equivalently be integrated out. The actual nuisance continuation trees in uncharted entries, conditional on \(S\), add no dependence on the \(U_b^*\)’s: given lattice axis paths and sizes these are fresh nuisance fillings; any fixed system of compact ordinary observations with volumes there converges to its disk kernels for converging positive perimeters and actual compatible transmitted ranges. Indeed positive macroscopic ordinary steps have \(l/k\to 1\); use (21), (24) and compatible path/volume convergence in each fresh disk. When continuing further given an initial eligible interval, a countable strictly internal target test suffices, negative loss thresholds avoid exact ends/minima by jump counting, and the child kernels iterate. This proves the claimed conditional convergence on actual supported first-scale data by finite matched tests and exhaustion. Ancestor volumes not themselves sampled fresh at this stage are functions by area additivity in the true limit. Thus microscopic perturbations of the axis parameters from other blocks do not condition these nuisance macroscopic trees.

Local actual projections, by contrast, can of course condition \(U_b^*\); they are exactly the separate \(g_b(U_b^*,I_b)\) given \(S\), by the comparison just described. It follows by disintegration that given \(({\cal T},\text{chart})\), the augmented retained outputs in the unconditioned chart use the product kernels (32), where \(k_b\) is the reference disintegration given \(S_b\), additionally conditioned on the indicated projection with parameter \(I_b\). Entry modes, color tags, matrix lattice types etc. in this formula are as specified for the local reference type. One can disintegrate measurably also in the interval parameter: after fixing the kernel given \(S_b\), use countable refining Borel partitions of the projection range with ordinary conditional differentiation for each pushed-forward law, defining fallback versions on its null exceptional data. This works for \(S\)-determined interval parameters without an assertion of continuity of (32).

Apply Lemma 84 to \((S_b,U_b^*)\). The completed chart parameters give \(S_0\), and the actual nuisance continuation trees have the required kernels depending only on \(S\). Lemma 86 identifies the sigma-field generated by these data and the separate \(G_b\) with that generated by \((\mathcal T,\text{chart})\). This yields (32) without continuity of its disintegrations. ◻

Proof. For clarity, chart choices themselves can be included without altering a primary limiting law conditionally on its input. Use countable finite patterns with fresh cuts/extra marks sampled with positive smooth densities, and specified jumps by ordered counting on compact surviving portions. In the original extraction these marked data have exactly their auxiliary limiting selection law given \({\cal T}\), even jointly with primary readouts: normalized fraction and time marks are fresh; path prefixes to new ordinary marks use open-prefix agreement with the primary tests and \(J_1\) jump matching; volume/perimeter labels and positions of cuts compare as above. Localize counts/cutoffs at continuity tests with bounded multiplicity. Counting tilts and these restrictions are by input/mark data only. Thus (32) can be applied on chart successes chosen by limiting data from this library (or absolutely continuous mark choices conditional on that data). Extra continuum placements and field data are sampled by the actual conditional law given \({\cal T}\), independently of additional readouts then, and so do not reweight (32). This sampling can be done after chart extraction as well; the extra chart cuts/tests are inputs as just described, and the same joint primary law results.

Explicitly, let \(R\) be the primary readout array and let \(\Xi\) contain the neutral chart library and its selection data. Write \(C=C(\mathcal X,\Xi)\) for the selected chart. If \(V\) is any spatial profile array, a measurable function of \((R,\mathcal E)\), neutrality gives \[\operatorname{Law}(V\mid\mathcal X,\Xi) =\operatorname{Law}(V\mid\mathcal X,C) =\operatorname{Law}(V\mid\mathcal X).\] On successful charts the middle law is the product law already proved. The joint law and its marginals are therefore unchanged by removing the chart, so the product assertion holds given \(\mathcal X\). This uses chart neutrality; an arbitrary mixture of product measures would not suffice. ◻

Proof. The product rule persists at fixed sphere volume, with the same reference and kernel versions in matched local inputs. On such charts leave at least one positive-size unexamined nuisance terminal (a jump interior will do).

For every cylinder comparison of \(({\cal T},\text{chart})\) we can, within it, further select a fresh jump-filling reservoir unseen by the comparison in its interior. This is the finite-observation exhaustion of Proposition 55: after any finite compatible paths and partially observed depths there, continue if needed to select another matched macroscopic birth by strict continuity tests. Existence and interior compact cutoffs exhaust the unit-area Palm data. Local retained lattice payloads are never tested in this nuisance reservoir. Integrating it out multiplies the unconditioned base limit by \(d_u(1-A_{\rm rest})\) by (22) on the corresponding successes, substituting that volume into ancestor volume entries; \(u\) is the matched perimeter in \(P=B_n\) units. Joint convergence of the rest volume and labels before substitution is the unconditioned area comparison above, applicable jointly with the chart limits and the extra birth selection.

To see why the reference versions remain unchanged, let \(B\) collect the macroscopic input outside the reservoir interior, including the local actual projections \(G_b\) and their volume labels. Put \[r(B)=d_u(1-A_{\rm rest}),\qquad k_b f_b=\int f_b(v)\,k_b(dv\mid S_b,I_b,G_b).\] Let \(\tau(B)\) substitute \(1-A_{\rm rest}\) into the observed reservoir and ancestor-volume entries. The unconditioned product rule gives \[\mathbb E_{\rm pair}\!\left[r(B)\Phi(\tau(B))\prod_b f_b(U_b^*)\right] =\mathbb E_{\rm pair}\!\left[r(B)\Phi(\tau(B))\prod_b k_bf_b\right].\] Here \(A_{\rm rest}\) is measurable from the macroscopic conditioning data: local block volumes are included in \(G_b\), and ancestor volumes are their area-additive sums. The reservoir is a nuisance filling, so neither \(r\) nor \(\tau\) changes \((S_b,I_b,G_b)\).

The reservoir local limit theorem identifies the left side with the fixed-volume joint limit on the chosen cylinder. The input comparison identifies the right side with the unit-area input law using the same reference kernels. One first uses continuous determining probes of the payload arrays; equality of the resulting finite measures then gives the identity for Borel probes. This is not a limit through discontinuous kernel evaluations. The perimeter integration and area change of variables in Proposition 55 identify the prescribed unit-area Palm version. Different cylinder tests can use deeper reservoirs in the same nuisance piece, and success tests partition/exhaust; there is no need for a conditioned shape convergence inside a reservoir at an exact specified volume. Convergence of the actual marked input without passage readouts was already proved by (15), (22) and exhaustion, also for these auxiliary chart patterns. We conclude by cylinder determination that given the full unit input and chart the product rule is (32). Dividing out Palm bias and restrictions by limiting input weights as before does not alter it. ◻

Proof. Here and below chart payload formulas are used to localize spatial observations as follows. Inside the retained pieces one can refine the compatible parts of the augmentation to arbitrarily small meshes (also partition a sweep further in time, its large attached pieces, and retained rings in length). Use only routes in the actual projection tree for this purpose. Their labels on the raw maps agree by transmission/open prefixes with primary comparisons. Thus in a common coupled realization they have the stated upper images at continuum locations, including their incidence stars; compact constraints pass into the covering pieces and those meeting a prescribed compact set have all images in any strict open enlargement eventually. Together with the exact local gluings this lets either the primary cover arrays or the localized arrays evaluate the same spatial regularizations.

More explicitly keep attainability and exclusion with compact ports/ranges versus open enlargements (and similarly ordered/subpath marks and budget slack); between any two strict margins a finite cover by either refining alphabet gives intermediate tests for all paths. For universal endpoint tests apply both containments also to all lifts/vertices, with the corresponding opposite margins for allowed connections. Curve-family costs such as polyhedral extremal lengths with compact versus open passage constraints are squeezed by inclusion, using the true oriented flags near the confinement; finite ordered tubes/ports can enforce robust crossings and winding. A family confined there uses no exterior conformal metric densities in the extremal-length supremum. Taking one-sided envelopes over margins or testing at continuity contours gives determinate readings from either array (subsequence upper versus strict-lower comparisons likewise use the corresponding margins). This explains why exact boundary-equality readouts in an exterior alphabet are not needed.

Arbitrarily many countable such tests strictly in an inner protection can be treated together. Their conditional laws use (32) and the actual protected positions of those compatible refined pieces, and in particular are independent across patches given \(\mathcal X\), by the neutral-chart identity. Such array extractions can always compactify costs or use countable hyperspace tests; path tests need not yet have a vanishing endpoint-cost modulus. ◻

Proof. These comparisons of versions are intrinsic to the same lattice experiments. They allow replacing the perimeter scales by others in fixed positive limiting ratio with the macro data converted accordingly: \(m_{\lambda P}/m_P\to\lambda\), and the triangle count units (using asymptotic inverses of \(B_n\)) transform by \(\lambda ^2\). Reference experiments with those normalizations are smooth lattice changes of density of the same experiments after converting the coordinates, with ratios depending only on the conditioned parameters on compact ranges; raw words, runs and terminal laws at exact sizes are identical. Readout cost units of course have to be used commonly or separately converted. Thus constant local field/coordinate comparisons below involving fixed unit conversions do not assume passage-law scale invariance. ◻

Proof. Use extra independent local random pools for the source positions. These can be chosen as functions of local data and iid uniform seeds. For detail, use a still buffered region inside field agreement, large enough for all local observations below. A caller whose boundary meets it has a key consisting of the caller side/open domain and boundary there (not its remote portions), and the side length measure there. Conditional key types for boundaries reaching that region can be enumerated measurably as atoms. Indeed for callers of diameter at least any fixed positive bound there are only finitely many such restricted keys over countably many conditional copies, by the depth/cell comparison in a finite cover: use small balls relative to the bound with enlargement, so that a boundary there is a local operation in a caller extending across the small ball’s comparison collars. Away from boundary the open-domain bit only has local whole/empty choices. Group the restricted pieces by finite tags. Arc lengths agree by the boundary comparison. Thus conditional iid sampling already gives finite support at each diameter threshold; enumerate atoms, for example using a countable weighted sum law of keys at that threshold. Smaller whole internal hierarchies can equally be matched by their own domains.

Two distinct caller keys within a copy (with boundary strictly here) need not share a uniform seed: the open-domain portions distinguish disjoint interiors, and strictly nested callers have disjoint rims. Use a separate key for a central-ring phase if it uses a position there.

Couple by choosing whether a length source lies on the observed boundary portion by its actual conditional probability and using the pool point sampled on that portion if so, otherwise using an exterior point. Caller geometry uses the unrouted canonical hierarchy, not earlier source choices. Thus after averaging the pools the sources are correctly conditionally independent. This works also for many disjoint patches with private pools, choosing among disjoint boundary categories over those patches by the actual full-field probabilities. Extra pools can be conditionally neutral for passage draws given the full input and actual sources. For the following comparisons we can add dense local target pools too, with private completions; positive joint length-density marginal choices in any finite chart suffice. ◻

Proof.

Proof. Combining (31)–(32) and this construction, the local reference readout instructions, lattice and residue types, conditioned values and embedded comparison alphabets, as needed to compute the robust profile kernel, have only finitely many possibilities over the copies. For this bounded-alternatives comparison one may work in arbitrary-area pair law (no specified nuisance volume reservoir is needed). Omitted remote histories can change the matrix but the kernel uses only its local projected type as proved above. All countable common unit/cost conversions use the same finite geometry/order choice. Hence conditional on the local fields and independent source pools the corresponding vector of conditional laws is finitely supported. This conditional-support argument can use conditional iid exterior samples and the matching extra target/cut libraries just described: all marginal (32) identities still hold, and the original conditional law given full input does not depend on those extra library draws. Finite support for the marginal vector then follows already by finiteness over the countably many iid extensions. Feasibility can always mean positive conditional weight under the particular common reference measure class.

To identify the common chart choices, record the geometry and order choice of Proposition 94, the effective local reference type of Lemma 83, and the conditioned local macroscopic values from that proposition. Whole retained words and fillings are the reference payloads; every refining readout for every \(\alpha\) is extracted from those same payloads. Duration and microstep counts belong to the sampled parameters and raw words, not to a new extraction-dependent constant term in (31). All closed viable cases have already been retained in Proposition 80; their exact microscopic pass/fail results are payload readouts. Neither changing a budget nor changing units adds a new geometric choice. Thus, after a single countable diagonal extraction of the references, \(F_\alpha\) may vary with \(\alpha\) but the argument encoding these choices ranges over the same finite set in the coupled comparison. The conditional-support conclusion below then permits \(T\) to be chosen simply as a measurable index of the atoms of the vector law given \(L\).

For completeness, the conditional iid argument proves support finiteness, not just finiteness within a selected list of examples. A probability measure on a standard Borel space that is not supported on finitely many points gives an iid sequence infinitely many distinct values almost surely. This follows either by repeatedly sampling outside any finite set of previously seen atoms, or from its non-atomic part. Apply this fact conditionally to the countable product of the kernel spaces. The matched charts above bound the distinct vector values among all the conditional iid exterior copies by a finite number. Hence the conditional vector law itself has finite support. ◻

Proof. One consequence for high-probability uses is that the number of candidate kernels and their smallest positive conditional weight can be truncated in probability in a fixed prototype law, simultaneously over the labels in the vector. Thus high probability with arbitrary slack can be required for every candidate’s draw, at a fixed label (uniform loss bound across labels), using the local fields, pools and auxiliary sampling. Under absolutely continuous transport the actual choice stays among the candidates. Across disjoint patches with pools as above one can have independent arrays of candidate draws conditionally on input and pools (at respective tested labels), whose distribution then needs just the separate local fields/pools, with actual profile draws occupying the selected slots by the product rule. Exterior variables can select alternatives but are not used in the simultaneous all-candidates success test. These are statements at the limiting kernel level.

Indeed let the conditional vector law have atoms \(\kappa_1,\ldots,\kappa_N\) and positive weights \(p_1,\ldots,p_N\), and put \(p_* =\min_j p_j>0\). For every \(\eta>0\), choose \(N_0<\infty\) and \(\delta>0\), using this one vector law, so that \[\mathbb P\{N>N_0\text{ or }p_*<\delta\}<\eta.\] If \(q_{j,\alpha}\) is candidate \(j\)’s conditional failure probability at label \(\alpha\), then on the complementary event \[\sum_{j=1}^{N}q_{j,\alpha} \le \delta^{-1}\sum_{j=1}^{N}p_jq_{j,\alpha}.\] The union bound gives an unconditional all-candidate failure probability at most \(\eta+\delta^{-1}q_\alpha\), where \(q_\alpha\) is the marginal prototype failure probability. The same \(N_0,\delta\) work for all labels, although the success event is tested at one label at a time. Absolute continuity transports the supported alternatives; density truncation transports the probability estimate when needed. For separated patches, first draw the arrays independently given their private local data and pools, then identify the externally selected slots. The product rule supplies the joint law of those selected entries. ◻

Proof. Here also are exact comparison conventions between field classes. Randomize area and embedding (e.g. cylinder translation and rotations preserving two marked ends, and further independent Möbius changes with positive densities if needed). By the Palm split already proved, a positive-density mixture of perimeters and pair bases can be transported to the arbitrary-total-area sphere class with a separating central loop under positive counting weights; give every feasible separating-loop choice positive support when comparing classes. Color parity, length phases and internal source data use the conditional rules above. Bias/unbias by full limiting inputs is harmless for (32).

If two full inputs have common \(J\), ends and loop choices and their real fields match on observation neighborhoods (or differ by exact constants there with matching unit conversions), differing by a bounded smooth Weyl change elsewhere, successful same finite unclocked charts give exact comparison of kernels on those observations. This assertion is for coupled typical marginal inputs in the supported classes: caller sources can be shared with absolutely continuous marginal choices. Indeed only finitely many noninternal callers influencing a strictly protected chart have to be serviced (all required ancestors included); all further sources relevant inside retained blocks are in whole constant-comparison neighborhoods. Unrelated changed subtrees need not be coupled.

Boundary lengths multiply by the continuous boundary factor, by the disk-field/arc comparisons above. Clock cut measures on common unclocked paths are equivalent by the positive-jump count formula and uniform upper/lower bounds on that factor; on retained segments elapsed clocks compare exactly in converted units. Thus the needed finite outer choices have equivalent supports with positive densities given the fields; inside subtrees one can share the source law exactly.

The very same affine histories/thresholds can be used: unchanged-slot viable transfers compare by their interval placements along the actual cut path, which do not change under remeasuring arcs, and near protected points their required shift values and binding arc lengths are the same in matched units. More explicitly the interval arrows carry actual arc labels together with their closure endpoints; remeasuring by a continuous positive position weight maps these monotone parametrizations homeomorphically onto the new ones, consistently on unchanged intervals and at every specified prime-end threshold. Cut clipping can shrink with room in the common local coordinates. Protected \(I_b,G_b\) and local subdivisions remain matched. A whole finite chart may be shared this way (include a nuisance reservoir when needed), so no assumption of uniqueness of an exterior-order choice from local data is required for this comparison. ◻

Proof. For local almost sure transfers to the area-one version work before three-point normalization, off the marked cylinder ends, in the representation (12) with a random translation. There is a useful exact way to match to inputs absolutely continuous in the arbitrary-area class while preserving the real field on any compact interior slab. Choose a nonzero smooth Cameron–Martin function \(g\) of the lateral field with zero angular mean, supported away from that slab. Write the lateral Gaussian as an independent Gaussian scalar \(u\) times \(g\), plus its complementary part. Conditional on the latter and on average-process data, the normalizing area \(A(u)\) before constant subtraction is strictly positive analytic and strictly convex as a function of \(u\): it integrates \(e^{2u g}\) against the other positive finite area by continuous chaos correction, with full support. Hence \(-(\log A(u))/2\) has a density.

Resample the coefficient of \(g\) independently but keep this old normalizing constant. The output equals the old field throughout the slab and has full-field law absolutely continuous in the arbitrary-area class (conditional on the complementary variables its coefficient and constant now have a joint density). Keep \(J\) and separating-loop geometry, using the positive-choice and source comparisons just explained.

This transports supported local kernel assertions to area one by the fixed-volume rule (32) on its own charts and the exact matching comparison; one does not condition a generic full-input almost-sure metric assertion on a null area event. Similar local comparisons on separated inner tests can use wide encompassing slabs. The sampled embedding marks can finally be restored using area marking.

The density assertion used here follows directly from one-variable calculus. Conditional on the complementary field, write \[A(u)=\int e^{2u g(z)}\,\mu_0(\mathrm dz).\] The bounded smooth function \(g\) and finite measure \(\mu_0\) make this function real analytic on \(\mathbb R\), by termwise differentiation on every compact \(u\)-interval. Moreover \(A''(u)=4\int g(z)^2e^{2u g(z)}\,\mu_0(\mathrm dz)>0\), since \(g\) is nonzero and \(\mu_0\) has full support. Thus \(A'\) vanishes at at most one point. The map \(u\mapsto-\frac12\log A(u)\) has nonzero derivative off that point and is locally a smooth change of variable there. A nondegenerate Gaussian coefficient consequently pushes forward to an absolutely continuous law for the retained constant. The resampled coefficient is independent of this old constant conditionally on the complementary variables, so their pair has a joint density. This is the precise density mechanism; no conditioning of an arbitrary-area null event is involved. ◻

Proof. Uniform candidate probabilities and changes of density.

All-order success suffices. Apply the common finite-index and independent-draw conclusions of Section 7, in particular Theorem 95 and Corollary 96. For labels in a common extraction it turns arbitrarily high prototype probabilities into arbitrarily high local simultaneous-candidate probabilities uniformly by truncating smallest weights. If considering successive different extractions/probability choices one can equally include them in a countable common sequence of tests/extractions on the same prototype inputs: the finite-variant bound is by the common fields/order comparison, using common chart choices, not by the value of the distance or EL units. For fixed comparable reference types the functions may be different universal disintegrations; the alternative used is indexed by the same finite choices over the extensions for the whole vector. Joint diagonal countersequence extraction suffices for uniform probability-loss assertions.

Under Gaussian Dirichlet trials on \(\Omega\) with real-height and \(J\) harmonic corrections bounded on a slightly smaller region (imaginary constant taken modulo period), this high probability transfers uniformly with strict buffers: cutoff Cameron–Martin shifts for harmonic data bounded smoothly there have uniformly bounded finite-moment likelihoods, and fixed restricted Gaussian domain/background comparisons use absolute continuity as above. Pools remain private.

For any one deterministic ancestry of disjoint trial shells in a limiting experiment, the actual selected kernels are among the feasible prototype choices at the separately zoomed labels (the input pushforward is absolutely continuous in that global class, and the same conditional chart rule applies). Simultaneous-candidate arrays can be sampled as above including the selected observations by disjoint conditional locality.

To bound a union over such ancestries it suffices to make this comparison for each path separately conditional on enclosing local fields. Truncate once the field-density change on the enclosing ordinary patch from a pure log Gaussian experiment; multiply the path probability bounds by that truncation. Thus there is no density factor per spatial grid point from supplying the sphere fields. One can for instance restrict to ends outside fixed enclosing neighborhoods, or exhaust that event. Feasibility and conditional draws in these arguments are only asserted for typical data in the indicated measure classes.

The grid and the critical change of measure.

Use nested square grids of radii \(r_j=m^{-j}\), \(T=\log m\) with a large fixed integer \(m\), cell side \(c_*' r_j\) very fine relative to the hole. Around the cell center \(z_j\) let \(H_j\) be the real average on radius \(R r_j\), with \(R\gg P_0+p_+\) fixed. Ancestor shells are disjoint and these circles are off the shells on that ancestry, increasing \(m\) as needed. Use \[X_j=2Tj-H_j .\] Take \(Tj_0\sim aN\) for very small fixed \(a>0\), and \(J_N=\lceil\exp (A N)\rceil\) as last index, for a sufficiently large constant \(A\). In the enclosing pure Gaussian calculation use covariance \(-\log |u-v|+\mathrm{const}\) on a containing disk for the real field, and the same modulo constants plus a phase for \(J\). With probability tending to one all starting \(X_{j_0}\)’s are below the target range with any desired fixed clearance (Gaussian grid union bound, \(4a<\log 2\)). Along each path stop at the first \(K\) with \(X_K\ge 3N\log 2\).

For a fixed grid path to \(k\), tilt by \(\exp (2H_k-2\operatorname{Var}H_k)\). By nested-circle harmonic averaging the increments of \(X_j\) to \(k\) now have independent centered Gaussian laws of variance \(T\); \(X_{j_0}\) has bounded mean and variance \(O(N)\). The number of grid cells at level \(k\) is \(O(e^{2Tk})\). Since \(\operatorname{Var}(H_k)=Tk+O(1)\), multiplication by the inverse tilt gives \[O(e^{2Tk})\exp\{-2H_k+2\operatorname{Var}(H_k)\} =O(e^{2X_k}).\] Thus the grid cardinality is absorbed into the terminal value of the centered tilted walk.

Paths not stopped by \(k=J_N\) are negligible after this union: under tilt the probability of staying below the barrier is at most \(C(N+\sqrt {\log J_N})/\sqrt {J_N}\), by reflection for the Brownian interpolation allowing the maximum intersample oscillation (add a negligible tail), and averaging the start. Choose \(A\) accordingly.

For first-stop paths to \(k\le J_N\) with start good, \(e^{2X_k}\le \exp (6N\log 2+2|\Delta X_k|)\) with \(\Delta X_j=X_j-X_{j-1}\). Thus tilted bad-on-stopping probabilities \(\exp (-B N)\) at arbitrary large required \(B\) suffice by Cauchy–Schwarz, including the sum over \(k\).

Bounded overshoots and clumping.

Assign tested labels to first sampled hits \(j\) of \(X_j\ge \tau _i\). Clump those at the same \(j\). Except at the above arbitrarily small exponential probability, a positive fraction of labels occur with last increment \(|\Delta X_j|\le U\) for fixed large \(U\); then each such clump has bounded size. Indeed one can continue the tilted Brownian walk beyond \(k\) in estimates. At the first passage of a next level from below, the last increment has a uniform sub-Gaussian upper tail. To see this at cutoff \(u\) first wait if necessary for the continuous interpolation by Brownian motion to hit the level minus \(u/3\). Immediate crossing by the next sample then costs \(\exp (-c u^2)\). Otherwise wait up to \(\exp (c_0u^2)\) samples, bounding large increments by a union with small \(c_0>0\), and longer delay by reflection and intersample oscillations as before. This works conditionally when proceeding from clump to clump.

Labels lost at a large-increment clump number at most \(1+|\Delta X_j|/\log 2\); conditional exponential moments of this loss restricted to \(|\Delta X_j|>U\) tend to one at each fixed exponent. Iteration proves the estimate. Keep a representative per remaining clump.

Harmonic data at the selected levels.

The harmonic data off the disjoint shells on this path have bounded needed norms for most representatives, and differences of \(H_j\) with the corresponding same-radius averages centered at all grid neighbors within \((3P_0+1)r_j\) are bounded there as well, with arbitrarily small exponential failure.

Here are details to account for random hit times. Measure harmonic profiles minus own outer circle values in \(L^2\) on a slightly inset annulus enough for all cutoff comparisons, and include the neighbor differences. These Gaussians are zero-mass boundary charges with uniformly bounded kernel variances at each scale (annular Poisson densities and uniform circle charges). Across deterministic levels their cross covariances decay geometrically by fine-charge cancellation and separation. Thus covariance on any level sublist has bounded operator norm and trace \(O(\#\mathrm{levels})\), also after conditioning on the \(X\) path. Imaginary profiles use an arbitrary lift with constant removed here.

Conditional real means at \(j>j_0\), including under the terminal tilt, are bounded in profile norm by \[C\Big (1+|\Delta X_j|+\sum _{l=j+1}^k m^{-(l-j-1)}|\Delta X_l|\Big ).\] Indeed covariance with coarser path-circle values vanishes exactly; with finer ones successive differences decay by smoothness away from the charges. Use independent circle increments/regression. At the selected clumps with previous increment bounded these squared-mean bounds sum to \(O_B(N)\) off an \(\exp (-B N)\) failure. For each offset \(s\ge 0\), increments one step after the distinct hit indices plus \(s\) satisfy a sum-of-squares bound \(C_B(s+1)^2N\) simultaneously with this error, by conditional Gaussian exponential moments at those sequential stopping times (use the extended walk and all clump leaders).

Now condition on the path; the covariance bound and Gaussian squared-norm exponential estimate finish the profile assertion, by taking norm cutoffs large and discarding just a small fraction. Harmonic interiors then have the requisite smooth bounds, with real constant aligned up to \(U+O(1)\).

Delayed hits, independent trials, and overlap.

Let \(W\) exceed the retained neighbor-error bound with slack. We can further keep hits \(j\) for which all \(X_{j'}<\tau _i-W\) for \(j_0\le j'<j-L\), for large fixed \(L\), still losing only a small fraction. The delay from first lower-barrier hit to target hit exceeds \(L\) with conditional probability \(O((W+\sqrt {\log L})/\sqrt L)\). Split labels into finitely many spacing classes greater than \(W\), giving sequentially bounded delay failures and hence again arbitrary exponential estimates.

Condition for actual trials off the shells (including the lifted exterior data); the stopping, harmonic and label-hit data needed to run trials are exterior, and the tilt changes no Dirichlet interiors. Neighbor errors if needed can just be discarded by the preceding bounds separately without conditioning interior trials on them. Conditional independence and the uniformly high all-candidate trial probabilities now lose only a small fraction of the many representative hits, again except for arbitrary exponential failure. This proves the union estimate and all-point assertion.

If enlarged successful disks for one label intersect at indices \(j<l\) with \(l-j>L\), the ancestor center of the fine one at \(j\) is a tested neighbor of the coarse one, contradicting the lower-delay requirement. This gives bounded overlap per label. In later single-shell Weyl changes supported within \(P_0 r_j\), the own and these neighbor outer circle values and all coarser delay checks on that ancestry are unaffected; the overlap criterion for eligible outputs therefore still holds. ◻

Proof. Almost surely \(x\) avoids ends and the chosen central seam and has arbitrarily small whole \(O\) loops about it with Jordan interiors \(D\) containing it. In any larger field buffer choose such a loop entirely inside with room. It is shared, together with both-side labels, in two conditional extensions agreeing on that field buffer by wall locality; given two extensions one may choose it small enough to avoid both central choices (and have its interior toward \(x\) in both rooted hierarchies). The signed/colored system uses \(J\) as above. This disk is a complete positive birth, with a new canonical source and an actual fresh subtree.

Strictly interior readings there have the disk kernel, given the disk’s path/volume input, its perimeter, and its protected placements, irrespective of caller-phase, ancestors and the boundary point where the loop was found. To see this in (32), chart through that designated birth, retain the initial unexamined internal filling, and use only tests protected in its interior; no exterior slot tests are needed there. Its size \(l\) is a full free lattice coordinate (outer total on this designated face is free, \(l\) equals that total plus the discrepancy); there is no projected lattice restriction or residue left. Thus the reference rule here is just smooth size mixing followed by the fresh colored \(T_l\) and its compatible source/tree rule, and disintegration on the interior subtree data and size.

Reroot the slots at its first fresh uniform source before calculating that rule. Cyclic rerooting preserves the entire conditional map law at exact size (boundary edge occurrences with fixed boundary color), and the remaining fresh cyclic offset conveys no shape information. In particular where the parent matched a circumference point is merely a label rotation when expressing the all-target and embedded internal data from the source. This argument uses no shape-independence from the field after disintegration.

At fixed volume reserve the nuisance disk away from the protected birth, as in the finite-cylinder proof of (32) in Section 7. Choices of birth charts can use successes on open intervals and positive-support cuts; the versions hold for all actual interior disks needed here.

Placements and length parametrization, full canonical hierarchy and level traversals within the disk are local to the fields containing it, using its own source family. Integrate the sources there by their actual fresh independent conditional law from the continuum coupling. Thus two conditional extensions can share these data in sufficiently small common \(D\)’s and have identical integrated kernels for germ readings; end choices and exterior sources do not change the result.

Two distinct test points use disjoint such disks and the product rule, including independence of their internal source choices. This proves the claims (one may compare conditional iid full-input copies for each fixed larger buffer and then shrink the buffer; simultaneous product uses full inputs first).

By spatial measurability, integrating bounded functions of the variables on rational balls now has zero variance conditional on the fields. Differentiation shows the variables themselves agree almost surely at area-almost every point with their field-conditional versions (area here is ordinary planar area).

Ordinary GFF germs are trivial at such deterministic points, also for the independent pair of fields here. For a Dirichlet field the decreasing Gaussian spaces shrink trivially by density of Dirichlet Cameron–Martin functions vanishing near the point (zero point capacity), hence the same for generated sigma fields by Gaussian chaos. For the density statement, use logarithmic cutoffs which vanish in a disk of radius \(\varepsilon^2\), equal one outside the disk of radius \(\varepsilon\), and have Dirichlet energy \(O(1/\log(1/\varepsilon))\). Multiplying smooth Cameron–Martin functions by these cutoffs and then smoothing proves the required approximation.

Modulo-period field restrictions locally are covered by ordinary actual lifts. Actual real-height restrictions off ends use Gaussian absolute continuity as above, also at area one by Lemma 98. If needed restrict to events of ends outside deterministic neighborhoods with probabilities tending to one: under domination on one such neighborhood triviality of the inner completed tail follows by conditional-density martingales, and then by exhaustion.

Hence at almost every fixed \(x\) such a variable is deterministic. Probability-one or zero statuses transport under rotations about \(x\). Indeed the full input classes with free embedding have equivalence under those similarities, and all readings are intrinsic from the primary cover arrays, with conditional placements from the continuum law; under embedding rotation the arrays and path input are unchanged and the spatial tests rotate exactly (or via the same strict margins). This applies before any claimed invariance of passage profiles, and one can unrestrict any fixed prototype cutoff first. Countably many rotations suffice. ◻

Proof. A countersequence and one common finite index.

If false for a positive quantile, there are failures down a sequence of quantile probabilities tending to zero (one can take monotone lower quantiles). For each extract at new lows \(j\to\infty,\ b_j\to0\). Then \(b_{j-i}\ge b_j\) for fixed positive offsets eventually. Work in EL units \(b=b_j\). Choose one of these probability levels sufficiently small for the high-probability shell test with a fixed band, slack and alignment constants, taking joint diagonal subextractions.

Here the all-candidate probability loss can indeed be taken uniformly: put the countably many vectors of converted prototype conditional kernels on the same full prototype fields and source choices. Finiteness over conditional extensions with pools in (31)–(32) was by common local chart/order comparisons for the whole countable list, not with a variation bound depending on EL units or on the individual subsequence likelihood of a budget value. Truncate smallest positive weights of alternatives once for this list. Field density and harmonic cutoff changes in Proposition 99 are fixed as well. The remaining upper (small-EL) witness probability at the chosen quantile is strictly positive, not needed uniformly.

Charging densities from the aligned shells.

Work on ordinary planar compacts in this last extraction. With probability tending to one as \(N\to\infty\), good holes at aligned labels in \([N,2N]\) cover each point with multiplicity at least \(\beta N\), their radii \(r\le e^{-aN}\), with margins and bounded overlap of enlarged disks per label. This is Proposition 99 with the eventual lower barrier and all offsets available. At each such annulus, for the discrete index sufficiently large, there is a length density supported inside its buffered disk charging hole-to-exterior traversal by \(r\), with squared density norm at most \(C r^2/b\).

For a finite open union \(U\) of ordinary rational boxes in the compact region of interest, retain the holes needed over \(U\), and sum these densities divided by \(N\) to get \(g\). For each fixed \(N\) passing, eventually \[b\int g^2\ \le\ C\,|U^{+C e^{-aN}}|,\] where the right uses Euclidean area and enlargement. Indeed per label the sum of squared radii is bounded by packing there and density supports have bounded overlap (one can also use the uniform small-error homeomorphisms and an enlarged buffer factor). Use Cauchy–Schwarz over labels, not over the potentially much larger number of Euclidean size levels.

Every path piece in projected \(U\) of diameter \(l\) has \(g\)-length at least \(c l-O(e^{-aN})\). For \(l\) much larger than the maximal radius, each used hole meeting the path with central clearance thus forces a traversal. On a coordinate projection interval covered by the path integrate the central-hole multiplicities at path points; each such interval contribution from one hole is at most a constant times its radius. All comparisons persist for all lifts eventually at this fixed \(N\) by strict slack. Charges add on disjoint path subintervals. Use countably many rational tests and an almost surely passing sequence of \(N\)’s by diagonalization.

Compactness and arclength of small-energy flows.

Keep the event on which small measurements occur along a further subsequence with the successful compact comparisons; it has positive probability by the quantile inequality in the prototype coupling and the almost sure eventual budget statements. All conclusions about limiting local data can now be proved pathwise there. Use the above unit flows, with mean length traffic of squared norm at most \(2b\), confined to a fixed ordinary compact and making fixed positive advance in projection.

By Cauchy–Schwarz with the \(g\)-bound, for each fixed \(s>0\) the expected number of disjoint advances by at least \(s\) is bounded uniformly eventually (take \(N\) sufficiently large depending on \(s\)). Thus projected path probabilities can be extracted, up to increasing time changes, on continuous paths.

For detail, mark each first move of size \(2^{-k}\) from the preceding mark at that resolution, beginning from the start. Eventually the counts are tight at any finite collection of resolutions. At resolution \(k\), distribute clock mass \(c\,2^{-k}\) equally among the successive first-exit intervals, continuously over each interval, and add a strictly increasing continuous base clock of unit mass. Normalize the total clock. An advance larger than a fixed multiple of \(2^{-k}\) contains a complete first-exit interval at the next finer resolution. On an event with bounded exit count at that resolution, it therefore consumes a uniformly positive amount of clock time. Tightness of these counts gives the required fine-scale modulus in probability and continuous parametrizations.

Keep a limiting probability with positive advance.

For each finite union \(U\) as above its expected arclength in \(U\) is at most \(C\sqrt{|U|}\).

Indeed take sums of projected advances on finitely many disjoint intervals confined compactly inside \(U\); with their number bounded, the same bound with enlargement and additive errors \(O(e^{-aN})\) times the number follows for the expected supremum from \(g\) and the traffic norm. Strict lower tests on those sums pass under uniform convergence; then increase \(N\), and exhaust interval numbers and partitions. In particular paths are rectifiable almost surely, and the integrated arclength measure on this band enlargement is absolutely continuous and nonzero.

Transverse tangent families.

Ask for path families at every sufficiently small dyadic radius \(s\) about \(x\), crossing from projection less than \(-s\) to more than \(s\) in some fixed unit axis about the center, with transverse deviation \(<s/4\), longitudinal extent within \((-3s,3s)\). Allow reversal and require strict clearance, witnessed with intermediate covers forcing such passages whose limiting EL upper bound is finite in units \(b\) (possibly radius-dependent). This is a germ reading. Use a countable fine net of axes. For arclength-almost every interior point of almost every path from the limiting flow there is a straight nonzero arclength tangent, hence for a sufficiently good axis all these requirements on the path hold with extra slack.

Paths in the support realizing strict passage give positive mass of paths making the required subtransit at each such fixed radius. This transfers to the finite flows, hence finite normalized EL by restriction to that mass and the traffic bound (any charging density applied to those subtransits charges at least as much along the full paths). Choose covers intermediate within the slack so these bounds register along the extraction.

The integrated arclength is absolutely continuous and nonzero. Hence, on the positive-probability flow event, some axis in the countable net has its germ assertion on a set of positive planar area. Countability therefore supplies one fixed axis for which this happens with positive probability. Fubini gives a positive-measure set of deterministic points where that assertion has positive probability. Lemma 100 makes the assertion probability one at almost every such point; its rotation conclusion does the same for the perpendicular axis. At one common sufficiently small fixed radius the two tested families both force crossings of a common topological square in transverse directions with clearance, by the homeomorphism comparison. The two quadrilateral ELs are each bounded above by \(O(b)\) eventually by mandatory subcrossings, contradicting product one. This proves (33). ◻

Proof. In raw EL units ordinary planar compact tests in arbitrary diverging perimeter extractions enjoy the same eventual density budgets with \(b\) replaced by 1 and fixed positive constants. For powers of two use (33), taking the quantile probability sufficiently small for the common candidate weights and shell tests. For general units compare simultaneously to nearest powers of two and their fixed offsets by (32), after extracting the bounded ratio (convergent unit conversions, using the limiting shift).

All constants in the resulting advance and norm inequalities can be fixed across ordinary compact locations and countable circular-ring tests in that planar chart; local density truncations there only affect the success probabilities tending to one. These are localized almost sure limiting profile assertions, using intermediate compact/open margins where required. Thus they hold also at exact unit sphere volume via (32) and the area-one field matching, on compacta off marked ends before three-point normalization. The point at infinity can equivalently use the inverted chart.

For the first assertion, every curve making the prescribed advance makes a fixed compactly confined advance in one of finitely many ordinary tests avoiding the isolated exceptional points. Sum the corresponding densities. Countable compact exhaustions also cover random end locations.

For the second assertion, use traversals in a domain of area \(O(r^2)\) at radius \(r\), and take \(N\) sufficiently large for each fixed test. A radius ratio of two suffices, and comparable buffered versions follow from the same estimate. It is enough that the traversed subbands avoid the ends, by exhausting them compactly within the punctured chart. Thus the estimate also covers shrinking rings centered at a missing point. Countably many strict tests give the conclusion for every center and radius with comparable margins. For any fixed finite collection of rings, take the largest of their eventual indices. This is the order used below when stacking finitely many rings and then increasing the number of rings. ◻

Proof. Equicontinuity and exclusion of collapsed fibers.

Corollary 102 concerns actual discrete flag-surface extremal length. Include three fresh independent area marks in the comparison (use the normalized measures), which tend to three distinct continuum area points by the area coupling. Write \(f_n\) for their normalized flag uniformization expressed by the projection homeomorphism in reference spherical coordinates. These homeomorphisms are equicontinuous along the good subsequences in the limit: at any hypothetical accumulation location of large oscillations on shrinking scales stack disjoint comparable-radius circular rings toward it using Corollary 102(2), making the through EL arbitrarily large by the series inequality. Yet the inner and outer complementary closed continua in image then have diameters bounded below, the latter containing at least two separated normalized marks for small outer radius, contradicting the planar connection bound. Thus subextract to continuous surjective \(f\). Each fiber is connected by uniform convergence of sphere homeomorphisms (use inverse images of small closed balls).

No nontrivial such continuum can collapse: take shrinking neighborhoods of it and inside them simple arcs of uniformly positive advance with Jordan disk thickenings, possible by connectedness. On the other side fix a nontrivial closed disk with image separated. The connection EL between the two disks in the discrete image is bounded above uniformly by annular duality, since all essential ring curves in projection make a fixed positive advance (otherwise both disks reach outside one ball containing the curve, excluding essential winding), and Corollary 102(1) is uniform for them independently of the thinnings.

Collapse instead forces this connection EL to infinity by uniform convergence, taking the limits successively. Hence \(f\) is a homeomorphism.

Quasiconformality.

On ordinary compacts \(f\) is quasiconformal. Concentric rings in its image (meaning images by \(f\) of concentric test rings) with radius ratio, say, 4 in the source have the uniform positive through EL lower bound by using nested subbands for \(f_n\) and uniform convergence. Therefore locally in planar charts \[\max_{|y-x|\le r}|f(y)-f(x)|\ \le K\min_{|y-x|=4r}|f(y)-f(x)|.\] Indeed a diverging ratio gives arbitrarily many radial log-scales around the image center with both complementary continua hitting each circle, bounding the connection EL to zero by integrating along those circles. Put \[H_{1/4}(x,f)=\limsup_{r\downarrow0} \frac{\max_{|y-x|=r/4}|f(y)-f(x)|} {\min_{|y-x|=r}|f(y)-f(x)|}.\] The preceding estimate gives \(H_{1/4}(x,f)\le K\) at every point of an ordinary compact patch. Applying (Cristea 1989, Theorem 1, with \(\alpha=1/4\) and any \(H>K\)) on an open disk compactly contained in that patch proves local quasiconformality. In particular \(f\in W^{1,2}_{\rm loc}\), and its Jacobian is positive almost everywhere (Astala et al. 2009, Theorem 3.1.2). The normalized differential tensor is therefore defined almost everywhere off the marks.

The normalized differential tensor.

Finally \((Df)^t Df/|\det Df|\) in local conformal coordinates at its almost everywhere nondegenerate differential is itself a germ reading (take bounded injective coordinates for it). Indeed through moduli of every fixed small rectangle in its image can be read by varying strict endpoint ports and transverse confinement. Intermediate families encompassing through paths of the rectangle and forcing subtransits of a slightly wider-transverse shorter-longitudinal rectangle squeeze costs from below as the margins shrink. Explicitly, allow confinement up to a small outward error, with the two ports neighborhoods of the complete two end sides. Choose cover tests enveloping the exact quadrilateral passages yet still inside these port/confinement allowances by the all-lift comparison. Their EL is bounded above by the through EL for the rectangle, and below by the through EL in the wider-shorter one lying with transverse/end clearance (every allowed path pays for a subcrossing). In these inequalities the exact rectangles at finite stages use the spatial homeomorphisms; their conformal images use the actual flag uniformization. Uniform convergence of the homeomorphisms and inverses gives the Jordan quadrilateral modulus limits used here (classical continuity under uniform convergence of parametrized Jordan boundaries and sides, or use further strictly nested crossing envelopes). Thus the values can equally be taken from the local variational arrays, regardless of other information used to extract \(f\).

Shrinking the rectangles reads the corresponding moduli under the differential by the same continuity. For example for side vectors \(Lv,w\) along orthogonal unit vectors, the through EL in the long direction after an invertible linear map \(A\), divided by \(L\), tends to \(|Av|^2/|\det A|\): compare uniform parallel lines and the constant area density. Countable axes and aspect ratios recover the tensor.

Apply Lemma 100 to bounded injective coordinates of this tensor. At almost every deterministic point \(x\), its value is a deterministic positive symmetric matrix \(T_x\). Rotating the source coordinate about \(x\) by a quarter turn \(R\) rotates every buffered rectangle test and changes its tensor to \(R^tT_xR\). The input law after this rotation is equivalent to the original free-embedding law, and the readings transform by exactly these rotated tests. The probability-one assertion that the tensor equals \(T_x\) must therefore also hold for \(R^tT_xR\). Consequently \[R^tT_xR=T_x.\] A symmetric matrix commuting with a quarter turn is scalar. Since \(\det T_x=1\), it is the identity. Thus \(f\) is conformal away from the finitely many marks by the distortion-one case of the analytic criterion (Astala et al. 2009, Theorem 2.5.4 and the following paragraph). Only now do we remove those points: continuity of the sphere homeomorphism makes it bounded near each omitted point in a target coordinate chart, so the holomorphic removable-singularity theorem extends it conformally there. It is therefore the prescribed orientation-preserving Möbius normalization.

The argument applies also in the unrestricted area classes (normalizing by area probability samples).

Transfer of the joint observations.

In particular the topological comparisons for area, the full nested loops (also with their auxiliary signs and colors), and maximal flag mesh now hold under the actual flag uniformizations with common joint laws as identified before, since every such uniformly convergent subsequence has this normalization. Removing central bias on the fixed finite sphere by (15) and the count exhaustion gives the demanded area and CLE law with independence in the surface sense and vanishing mesh there. No metric convergence is used for this step. ◻

Proof. For \(J\) this is the whole-plane reconstruction of Section 6, including the parity convention. For the real field we use the log-mass reconstruction strategy of Berestycki–Sheffield–Sun (Berestycki et al. 2023, sec. 2); critical reconstruction under Gaussian covariance regularity hypotheses was established by Vihko (Vihko 2024, Theorem A). We give the local argument needed for our critical-area normalization and comparison classes. In a patch of an ordinary pinned log GFF with covariance \(\log(1/|v-w|)+\mathrm{const}\), at an inset \(x\) subtract its circle average \(h_\eta(x)\) and zoom by \(\eta\). On the unit open disk the law of the result \(Y_{\eta,x}\) (scalar pullback here) is fixed with pure log covariance. For two distinct inset points these distributions jointly converge to independent copies by Gaussian covariances. With \(\mu=\mu_h^{\rm crit}\), field and scale covariance give \[\tfrac12\log\mu(B(x,\eta/4))+2\log(1/\eta) =h_\eta(x)+\tfrac12\log\mu_{Y_{\eta,x}}^{\rm crit}(B(0,1/4)).\] Write \(M\) for the unit-disk mass in the last term. Its logarithm has finite moments of all orders. For the upper tail, local chaos convergence and Fatou’s lemma transfer the positive fractional moment of Lemma 7 to the critical Dirichlet mass. A Markov decomposition on a larger disk adds a harmonic field whose supremum on \(B(0,1/4)\) has Gaussian tails. Hölder’s inequality, with a smaller positive exponent if necessary, therefore gives \(\mathbb EM^s<\infty\) for some \(s>0\), and hence \[\mathbb P(\log M>m)\le C e^{-sm}.\]

For the lower tail, pack \(N_m\ge c_0e^{2cm}\) disjoint balls of radius comparable to \(e^{-cm}\) inside \(B(0,1/4)\), where \(c>0\) will be small. Decompose into independent zero-Dirichlet fields inside them and harmonic parts. Gaussian harmonic estimates and the grid union bound give an event of probability at least \(1-Ce^{-cm}\) on which all the harmonic parts on the half-balls are at least \(-C_0cm\). Critical scaling then gives, on that event, \[M\ge e^{-(4+2C_0)cm}\max_{1\le j\le N_m} Z_j\] after adjusting fixed constants, where the \(Z_j\) are independent copies of a positive unit Dirichlet inner-ball mass. Choose \(z_0>0\) with \(p_0=\mathbb P(Z_j\ge z_0)>0\), and then choose \(c\) so that \((4+2C_0)c<1\). For all sufficiently large \(m\), \[\mathbb P(\log M<-m) \le Ce^{-cm}+(1-p_0)^{N_m}.\] This proves the asserted log-moment bounds.

By fixed marginal laws and measurability, the two log masses at distinct points converge jointly to independent copies, by Lusin approximation. The log-moment bounds make their covariance tend to zero. Put \[m_0=\tfrac12\mathbb E\log M,\qquad R_\eta(x)=\tfrac12\log\mu(B(x,\eta/4)) +2\log(1/\eta)-m_0.\] The log-mass identity shows that \(R_\eta-h_\eta\) has mean zero. Integrating its covariance against any two smooth test functions on the inset patch and using the uniform second-moment bound gives convergence to zero in \(L^2\). Since \(h_\eta\to h\) in tested \(L^2\), the measure-based functions \(R_\eta\) reconstruct \(h\). A subsequence and a countable determining family of tests give almost-sure measurable reconstruction.

Local actual-height absolute continuity off ends transfers this in ordinary spherical charts of the unpinned experiments (express \(h\) in planar area coordinates). One need not know end locations for determination: two conditional copies given the mass agree throughout the complement of both finite end sets by such patches, hence agree everywhere by \(H^{-1}\). Indeed a distribution supported on finitely many planar points is a finite linear combination of delta masses and their derivatives, and none of these nonzero distributions belongs to \(H^{-1}\) in dimension two. ◻

Proof. We use a conditional-sampling principle for the true profile array \(Z\). If an auxiliary choice \(C\), sampled after the finite map, has in the limit a conditional law given \((h,J,Z)\) that depends only on \((h,J)\), then \(C\) and \(Z\) are conditionally independent given \((h,J)\). We verify this first for the central choices and then, round by round, for the source family. The spatial product kernels are used with the sampled field and embedding retained in the conditioning; they are not obtained by averaging a shared embedding out of the conditioning variables.

The central loop, ends, and phase.

Use \(b_n\)-type unconditioned bases, with \(B_n\) nearest the desired units, localized by first-perimeter windows and further cuts with both central side fractions positive and total area in compact positive ranges. These exhaust the arbitrary-area Palm measure class and agree on overlaps up to input weights. Conditional on the finite map and its orientations beforehand, the joint mass is exactly loop counting with the indicated restrictions, up to factors depending only on the full map: this is the same \(4/(k+l)\) ring calculation in (15) with the total not fixed (on maps of \(m\) edges the change from the root contributes a factor common at that \(m\)).

We can take two additional end samples by area, requiring separation, and restore any equivalent positive counting weights by limiting changes of density. These are not required to be the three conformal normalization samples. Interior continuity-window cutoffs exhaust.

The entire counting and end-sampling conditional rule now passes to the limit conditional on map outputs (including true profile arrays): diffuse weak area convergence excludes small loops under the side cutoff and samples their positions, and all relevant loops, their perimeter labels and side measures converge, with null boundary area. Equivalently, under central counting within such restrictions one averages uniformly over all qualifying loops given the map, then end samples on opposite sides by their area laws (or biases by the separation probability). This conditional sampler converges and depends only on \(h,J\) in the limit. The common typical central phase has the same assertion by length-label convergence.

All available loop choices and opposite end samplings are covered by increasing these restrictions. Conditional profile versions on different windows come from the same extractions/formulas, so this proves independence from these choices given \(h,J\), in the full supported class (each two choices occur eventually in overlapping experiments). Independent normalization marks before the gauge change used map area; no loop selection or source operation is used to generate that initial true profile array.

A finite-round survey of caller births.

We detail similarly the source assertion, since disk identities might be reached adaptively. Run a countable survey in rounds starting at the two central sides. At a complete caller birth being entered, draw its source freshly before testing there. In previously entered callers test paths to a dense sequence of ordinary length targets, on compact surviving prefixes, using preassigned loop orientations. Other original-target branches use the same caller start and the side-dependent pinch switch rule, so testing whole individual paths here requires no new independent source along an inherited-target portion. List macroscopic full births discovered, including a negative at which the inherited interval is empty, and append new callers to the entered list, drawing sources there for the further tests.

Repeatedly increase target lists, time cutoffs and resolution of jump sizes and of minimum positive surviving side length, also in old callers. Target seeds and continuity cutoffs can be diffuse fresh choices; every round uses finitely many tests/births almost surely. For example use increasing clock bounds and shrinking lower bounds before the first small length, taking only jumps with strict compactly surviving pre/post comparisons; order listed births by the path test and chronology, skipping duplicates.

At each fixed round the decisions/callers and slot measures at their births are matched stably along good subextractions by the \(J_1\), fork/empty-interval, shared-prefix and length-label spatial comparisons. Bounds can avoid discontinuities by the fresh cutoffs. Shared births met in finitely many paths are along their common prefixes with strict original target eligibility.

The countable survey enters every complete macroscopic caller: every next birth is on an ordinary-target open interval in its parent with a compact surviving jump observation, and depth can be taken successively. Use a fair schedule on all previously entered callers.

All these finite tests can be applied to the same primary source assignments on the lattice. Only actually needed independent slots have to be drawn as of a query (extra not-yet-examined calls deferred); one can skip a proposed fresh draw if it is not a valid new birth, checking this by the prior histories. No primary absolute priority among targets is imposed.

Passage of the conditional samplers to the limit.

Thus conditionally on the map, marks and the survey past, every listed fresh slot has its uniform distribution at that birth. Passing this equality against bounded continuous tests to the limit uses the matched uniform boundary-slot measure convergence in the true conformal coordinates for that birth (negative as well as positive). Failures/cutoff tails in any fixed round can first be truncated.

Hence given the limiting map outputs, central choices and past, the new source law is the corresponding normalized disk length law. Those outputs include enough to give \(h,J\) and the true regularized profiles by the preceding determination; survey seeds may likewise be drawn progressively independently of them.

Given fields, earlier choices and tested seeds, the limiting history deciding the next draw uses exactly the canonical level paths and their physical lengths/clocks determined above, without a further profile variable. Iteration and then round exhaustion proves the assertion for the whole source family, including with its intrinsic caller identities in that exploration.

All these independence assertions concern profile readings which could themselves have been subextracted from countable true map tests before choosing loops/sources; no independence of a microscopic exploration array from the map is being claimed.

Consequently conditional profile kernels on disjoint strictly buffered observations multiply given just \(h,J\). Indeed they did so given the whole input by (32), the component kernels are now unchanged after dropping the source/loop/phase/end information, and the embedded primary path input itself is given by the fields and these choices. All-candidate bounds over field extensions persist. ◻

Proof. Localization inside a whole loop.

In a planar patch containing a whole \(O\) loop whose bounded-side Jordan disk \(D\) strictly contains the protected test, the field kernel for that test is local up to this field patch. One can use the isolated disk calculation in the germ proof, rerooting if necessary so that a central choice exterior to \(D\) and ends there are used and \(D\) is a positive birth. Such choices have positive conditional support by nested CLE and positive area. Bounded planar tests have this determination with probability arbitrarily near one by increasing a finite surrounding field buffer, since bounded-side \(O\) disks enclose each given compact (use nesting toward infinity).

Smooth field changes and simultaneous observations.

For \(h'\) related to \(h\) by a deterministic bounded smooth Weyl change in typical supported inputs with the same \(J\), suppose the two real fields agree in a strict neighborhood of one observation. Their kernels there are identical. This also compares \(h+f\) to \(h+b\) when \(f\) is constant \(b\) there. Use the exact same-chart comparison in Lemma 97.

The common central loop/order and the needed caller sources can be shared when making it: globally large callers (and required ancestors) are finite, with mutually absolutely continuous length-sampling laws; smaller ones required throughout whole matching neighborhoods have the identical internal laws, and unrelated subtrees entering no retained observation need not be shared. Alternatively couple all callers above a fixed small diameter bound and all smaller ones meeting a slightly inset comparison region. Boundary measures change by a continuous positive weight. By the source/loop independence just proved, this establishes equality of the field kernels themselves almost surely without a likelihood price per crossing caller.

When several disjoint patches are considered, use conditional product also to sample auxiliary kernels there independently (including at specified constant shifts); one can thus keep designated inner constant-comparison and exterior unchanged readings in the modified law together. All such statements use joint extractions with fixed deterministic units; they are not Weyl scaling for graph costs. ◻

Proof. A bridge-and-tail description of the measure class.

Randomize the cylinder maximum position and height with positive densities, with two ends and full-support spherical re-embedding; positive global Palm and mark weights do not change this arbitrary-area class.

For cutoffs \(m,K>0\) use comparison laws with full-density endpoint heights at \(\pm m\) below \(K\), a Brownian bridge of circle means inside \([-m,m]\), and independent tails of \(K-{\rm BES}_3\) from those endpoints; use the same independent lateral field. Each such law is dominated in measure class by the arbitrary-height/position maximum law, which conversely is dominated by countable mixtures of them.

Indeed in a longer interval \([-T,T]\) localize the maximum construction to maximum attained there with height in \((-B,B)\). Its interior path then uses the Brownian bridge class with those endpoint heights of positive densities and the given maximum/time (both have positive joint conditional density on the admissible range); the two independent Bessel pieces are precisely the Brownian maximum decomposition bridges (Denisov decomposition, or the killed Brownian bridge density differentiated in the maximum).

The tails are killed Brownian height paths conditioned by the linear avoid-the-maximum transform. To compare barriers, start at \(x<M_-<M_+\). For the height process \(M_+-{\rm BES}_3\), the probability of staying below \(M_-\) forever is \[\frac{M_--x}{M_+-x}>0.\] Conditioning on this event gives exactly the \(M_--{\rm BES}_3\) tail law started at \(x\). Indeed the two Doob transforms have harmonic functions \(M_+-x\) and \(M_--x\), whose ratio tends to one at minus infinity. Thus a lower-barrier tail is an absolutely continuous restriction of a higher-barrier tail, with the displayed normalizing factor. Apply this with the actual maximum as the lower barrier and, for example, \(B+1\) as the upper barrier. Thus on these restrictions the maximum law has the same class as Brownian interior/full-density endpoints and higher-barrier Bessel tails, restricted to inside maximum dominating both tails and in the stated height window. This proves one domination.

Conversely any of the comparison laws clipped to these maximum/time restrictions at larger \(T,B\) (\(B>K\)) is dominated by the same class: finite segments of its stopped-height tails are ordinary killed Brownian paths with endpoint tilts and the residual tails use a lower barrier. Exhaust using convergence of tails to minus infinity. Apply these comparisons also with full positive re-embedding densities, conditioning on such end-coordinate parameters when convenient.

Quantitative changes within the slab.

On an observation patch inside such a Brownian slab, compactly supported deterministic Cameron–Martin changes therefore preserve the full-field measure class.

For quantitative changes one can actually use the bridge comparison laws with a large enough slab, clipping to inputs of interest beforehand. The likelihood for a smooth compact shift there only uses the ordinary Dirichlet Gaussian pairing and energy of the shift (conditional Brownian linear mean gives no term), retaining the exterior, and the independent lateral part with the same Dirichlet normalization. Conformal chart expression just includes the known background transform.

Thus for finite recipe lists in small similar shells off ends one can impose bounded likelihood tests of arbitrarily high marginal probability by Gaussian local cutoffs in their aligned shell coordinates. The comparison is made within the chosen Brownian-slab law, with its exterior retained. It does not require a uniform bound on the original sphere’s global area-tilt likelihood at each shrinking shell. ◻

Proof. Apply the aligned all-point shell estimate of Proposition 99. Include fixed reference-metric tests in the annular buffers. For upper chaining, every path from the central hole, with its fixed clearance, to the first possible circuit radius must pay at least \(a_0\) in prototype reference units; a strict subband gives this charge. For lower chaining, take a round circle past the charged traversal band and bound its internal \(D\)-diameter by \(b_0\), using paths in a slightly larger annular buffer. These tests can have arbitrarily high probability after shrinking \(a_0\) and enlarging \(b_0\), by the ordinary topology and local length structure. The fixed annular trial region can be made large enough to contain all the required buffers. On an aligned shell at label \(i\), these charges and sizes are respectively \(a_0w_i\) and \(b_0w_i\).

With probability tending to one, the successful central holes give a finite cover of each ordinary compact set. Choose an increasing sequence of \(N\)’s along which these covers succeed almost surely. The covers are finite at each \(N\); fix one such cover before sending the discrete index to infinity. All realization slack on the good couplings can be absorbed into \(K\) and \(k\).

For an upper comparison, begin at a point of the reference path in a good hole and follow the path until its first exit from the filled disk bounded by the associated simple circuit, or until the path ends. Every complete step consumes at least \(a_0w_i\) of reference length. At this fixed finite mesh the minimum relevant \(w_i\) is positive, so the procedure terminates. Consecutive filled disks overlap. When two adjacent disks are nested, compress their block to the containing representative, assigning to it the reference steps of that block; all constituent disks stay inside that representative. The remaining successive boundaries intersect. One can therefore walk along the chain, traversing each boundary at most once. The total normalized graph cost is at most \[K\sum_{\text{complete steps}}w_i+O(K\max w_i) \le \frac K{a_0}\operatorname{len}_{D_h}(\pi)+O(K\max w_i).\] The compressed blocks still follow the original path in its prescribed order, up to the uniformly shrinking diameters. The end locations have the same approximation property. Passing through the finite discrete witnesses and then sending the shell mesh to zero proves the upper comparison. Applying the same cover to a single point supplies surrounding simple circuits of arbitrarily small cost about every ordinary point.

For a lower comparison, follow a tested graph path from a hole to its first exit on the indicated round circle, and repeat. Each complete step costs at least \(kw_i\) in normalized graph units. A fixed bounded total budget bounds the number of these steps at a fixed mesh; marked-subpath extraction handles any variation in the witnesses. The limiting circles, after the same containment reduction, form a chain connecting arbitrarily close to both endpoints. Its reference cost is at most \[b_0\sum_{\text{complete steps}}w_i+O(b_0\max w_i).\] The last incomplete step is included in the error. Reference continuity removes the remaining endpoint errors. This uses no bound on the diameter of a shell’s interior in that shell’s units. It follows that the lower comparison is at least \(k/b_0\). The same argument with arbitrarily large \(k\) gives infinite lower comparison. In bounded raw graph units, infinite lower comparison already follows from vanishing primal-edge mesh. ◻

Proof. Starting at \(j_0\), test lower traversal at the raw threshold \(2^{(\xi-\delta)(j-j_0)}\). As \(j_0\to\infty\), vanishing mesh supplies every fixed finite initial run of successful tests. If arbitrarily large starts had a later first failure, extract at those failures with \(T\) equal to the proposed threshold. Their successful preceding runs have diverging length, and every fixed preceding offset has a high barrier of size \[\text{constant}\cdot Tw_i2^{\delta i}.\] For \(i\in[N,2N]\) this gives lower factors tending to infinity as \(N\to\infty\). Lemma 109 gives infinite lower comparison and hence strictly re-passes the allegedly failed test. Thus all sufficiently large starts have no first exit.

Fix such a start and extract at arbitrary \(j\to\infty\) in its proposed threshold units. The same argument gives infinite lower comparison. In particular, winding circuits cannot have bounded cost in those units with positive limiting probability. The upper probability in (34) proves the asserted lower growth of \(A_j\).

To obtain records, choose exponents tending to \(\xi\) from below. For an exponent \(\xi-\delta\), the preceding lower growth at the slightly larger exponent \(\xi-\delta/2\) shows that \(A_j2^{-(\xi-\delta)j}\) has arbitrarily late records. At each record, \(A_{j-i}/A_j\le2^{-(\xi-\delta)i}\) for preceding indices. Let \(\delta\downarrow0\) and the record index tend to infinity diagonally. This gives (35) simultaneously at all fixed positive offsets. ◻

Proof. Follow a reference path and then take its infimal length to obtain (36). To join two nonconstant certificates at a common endpoint, choose a surrounding simple primal circuit in \(U\) of arbitrarily small diameter and cost, supplied by Lemma 109. Cut the two certificates at this circuit and join along it. At each fixed circuit and protection, use simultaneous path witnesses with slack. Even a separately obtained circuit forces the required intersections, by its central-hole clearance. Compact confinement is preserved. Let its cost and diameter tend to zero to obtain the triangle inequality. The upper bound and the reference length topology then give continuity in each endpoint, and hence jointly on local compact sets. Ordered ports survive the same construction when their margins are fixed. ◻

Proof. Otherwise, on a fixed positive fraction of labels along arbitrarily large intervals, absence of all these shortcuts has arbitrarily high probability. Use this absence as a shell test. Include the upper circle-diameter cutoff \(b_0\) and a uniform positive lower bound \(a_1\) for reference internal distances between opposite faces of the candidate rectangles. These are compact parameter tests and can be imposed with countable spatial refinements.

In every complete lower-chain step, mark a rectangular subtraversal as follows. In the middle band start a small centered square, follow the path to its first face exit, and retain the segment after its last crossing of the parallel midline. The resulting half-square is a permitted \(R\). By compact marked extraction, absence of a shortcut forces this portion to pay more than \(c_1D_h(q_0,q_1;\Omega)\), with a slightly diminished strict threshold if necessary. Since \(D_h(q_0,q_1;\Omega)\) dominates the unrestricted distance, the excess over the original \(cD_h\) endpoint charge is at least \((c_1-c)a_1w_i\) before the strict-loss adjustment. Fix, for example, \(0<\delta_1<(c_1-c)a_1\), and absorb that adjustment into the remaining gap. The extra charge is then at least \(\delta_1w_i\). In particular, a bounded budget bounds the number of complete steps at a fixed mesh even when \(c=0\).

Retain the original lower comparison on all intervening portions. Summing their endpoint distances and using the triangle inequality gives a total charge at least \(cD_h(x,y)+\delta_1\sum w_i\), up to the last incomplete-step error. The circle chain gives \(D_h(x,y)\le b_0\sum w_i+o(1)\). Hence the lower factor improves by a fixed positive amount, contradicting the definition of \(c\). Fixed label shifts discard only boundedly many labels and are handled by the same unit conversion; the argument is unchanged. All constants in the strict gap \(C-c_1\) are fixed before any later near-saturation tolerance. ◻

Proof. Suppose \(c<C\), and retain the shortcuts, probability tolerances, inner offset \(m\), and finite recipes constructed above. All of these choices are fixed before choosing the small near-saturation tolerance \(\eta\) below.

A near-saturating compact experiment.

For arbitrarily small \(\eta>0\), optimality of \(C\) supplies deterministic ordinary patches \(U\Subset W\), a compact protection \(K\Subset U\), fixed anchors \(p_0,p_1\in K\), and a positive-probability event \(E\) on which \[ \begin{gathered} A<L<A_1,\qquad A>0,\quad A<A_1\le2A,\qquad L=D_h(p_0,p_1;W),\\ \min_{j=0,1}D_h(p_j,W\setminus K;W)>s_0L,\qquad F_U(p_0,p_1)>(C-\eta)L, \end{gathered} \tag{42}\] where \(s_0>1\) has fixed slack. To justify this localization, if there were no such witnesses, subdivide any reference path and its open following instructions into sufficiently short subarcs in small ordinary domains having that exit safety. Compactness, the length topology, and positivity of exit costs permit arbitrarily fine subdivision in reference length. Apply the resulting improved \(F_U\) bound there, using endpoint continuity to reduce to countably many anchor tests, and concatenate with shrinking-cost circuit joins inside the prescribed tubes. This would improve upper followability strictly below \(C\), a contradiction. Restrict the length to one dyadic window and adjust thresholds slightly to make them continuity values.

The domains can be kept compactly away from the cylinder ends. The Brownian-slab comparison of Proposition 107 gives a slab law retaining positive probability for the witness, with \(\overline W\) in the slab interior, independent \(J\), and the same field kernels. Restrict the end-coordinate parameters only as needed for this room. Refine the thresholds to continuity values in that law, and write \(p_E>0\) for its probability. Although \(E,A,U,W\) may depend on \(\eta\), the previously chosen local recipe constants do not.

The exit safety implies that minimizing paths for these anchors are attained in \(K\), by compactness and the length topology. Such a geodesic is almost surely unique up to parametrization. Here is an elementary argument sufficient for these fixed anchors. Distinct geodesics can be distinguished by a smooth nonnegative bump compactly supported in \(W\), missed by one and visited in its positive set by the other. Under addition of \(t\) times that bump, the distance is nondecreasing in \(t\). At a parameter with these two minimizers it is constant to the right, because the avoiding minimizer is unchanged, and strictly smaller at every smaller parameter, because the other minimizer has positive length in the bump’s positive set. There is at most one such parameter for a fixed field and bump. A countable family of bumps distinguishes all distinct traces. Local Weyl scaling, Brownian-slab shift equivalence, and Fubini then show that the exceptional event has probability zero. The whole distance family is obtained from the original metric by length reweighting, so this parameter argument uses no additional metric construction or confluence assertion.

The input weight of successful visits.

Integrate reference arclength along the middle half of this geodesic on \(E\). The aligned-shell estimate, including the successful shortcut proposals and all the cutoffs above, gives order \(N\) successful holes around every point, with probability tending to one. For sufficiently large \(N\), the anchors lie outside every visited shell’s enlarged disk, and each disk has room inside \(U\). Indeed the middle half is a fixed positive reference distance from either anchor, while all spatial radii tend uniformly to zero; use reference continuity, geodesicity, and the compact safety condition.

Time spent in one hole along the middle half is at most a constant times its length unit \(w\). Replace the portion between first and last encounters with the attachment circle by a path of cost at most \(K_0w\) in its annular buffer, and use geodesicity. Integrating with multiplicity therefore gives the expected weight bound \[ \mathbb E\!\left[\mathbf 1_E\sum_{\text{successful hits}}w\right] \ge c_0 A p_E N \tag{43}\] for all sufficiently large \(N\), with \(c_0>0\) independent of \(\eta\) and \(E\). Ordinary compact success events whose probabilities tend to one may be imposed first. Their error is harmless even when \(p_E\) is small, since \(E\) and its positive probability are fixed before \(N\to\infty\).

Domination for one transformed hit.

For a successful hit, take a usable recipe from the finite list in Lemma 113. One may count all usable bin and recipe choices instead of prescribing a choice at an exact endpoint. For each fixed recipe and shell entry, its hit submeasure, together with the retained inner and old exterior data, is bounded by \(\Lambda\) times a single common output law, where \(\Lambda<\infty\) is independent of \(\eta,E,N\).

To prove this domination, first condition on the slab coordinate parameters and fields, before imposing the hit. Keep the selected independently proposed inner profile at \(h+b\), and the old profile readings beyond a strictly enlarged comparison disk. The inner preservation region has \(h+f=h+b\); the outer region has \(h+f=h\). Source elimination and the field-kernel comparison give exactly the corresponding two kernels of the output law at \(h'=h+f\). They have the conditional product law, by (32) after source elimination. Retain jointly all countable regularized exterior readings with compact confinement in \(U\) beyond a strictly containing comparison circle. Each finite collection is separated from the inner protection and lies in the field-agreement region, so compact exhaustion gives the whole collection. These kernel identities hold for typical supported fields, including the deterministic shifts in the finite list, and are independent of the end and source choices.

Now apply Cameron–Martin to the fixed shift in the Brownian slab. Its energy and pairing have the ordinary Dirichlet expression in the local planar chart: the cylinder coordinate charge is harmonic on the small support, and the conditional Brownian linear mean contributes no term. The bounded-likelihood cutoff was part of shell success before selecting anchors or a hit. Its bound is uniform in the slab endpoints and the parameters restricted to have the required interior room. Imposing the hit, choosing a covering bin, or imposing further restrictions only decreases the resulting submeasure. Complete the output profile over its retained readings using its usual conditional law; the same domination then applies to all additional output variables used below. This is a marginal comparison for one recipe and one entry. It makes no simultaneous claim about transporting the auxiliary proposals for different entries to one output realization. Gain and existence requirements can be expressed by measurable countable tests with slightly weakened slack.

Stability of the near-saturation event.

The output satisfies the inequalities defining \(E\) with errors tending uniformly to zero over the eligible hits, after a negligible input restriction if needed. First, anchor distances and exit-safety distances change by \(o(1)\): the shift is bounded and supported in one shrinking interior ball. Cut a path from its first entrance to its last exit and splice across that ball. Ordinary continuity of the old internal metric and the bounded Weyl multiplier control both directions of this comparison uniformly on an interior compact.

The certificate cost cannot decrease by a nonvanishing amount either. Suppose an output certificate is cheaper. Mark its first entrance and last exit on a circle farther out than the transferred-reading threshold, still with the ball and buffers compactly inside \(U\). If the certificate stays exterior, its reading transfers directly. Otherwise approximate it with losses tending to zero and extract the two approach points and subbudgets. Its two exterior pieces have compact confinement exterior to the threshold and are among the retained readings. The old \(F_U\) can use them and join the two approach points across the ball at cost \(o(1)\), by its triangle inequality and upper bound, together with the uniform continuity of old internal costs on neighborhoods of the hit region. Take infima over certificates, using countable intermediate protections if necessary. Each candidate only needs its own compact confinement in \(U\); no common predetermined distance from \(\partial U\) is required. This proves \[F'_U(p_0,p_1)\ge F_U(p_0,p_1)-o(1).\] This argument concerns open endpoint losses, and does not attach an individual graph vertex to a small ball.

Old modulus and clearance cutoffs make all these errors deterministic and uniformly vanishing over the shrinking hits, while preserving (43). Hence the relaxed output event has common-law probability \(p_E+o(1)\), by continuity of the thresholds. Its safety still gives minimizing paths in \(K\) for large \(N\).

A bound on the total output weight.

For each eligible transformed hit, the unique output geodesic has the strict-gain interval and the in-disk traversal supplied by Lemma 113. The per-label overlap tests of the aligned-shell construction are unchanged by this one shift: its own large outer circle average, the required neighbor averages, and the coarser delay checks are all outside the support. Thus eligibility can be defined using only necessary output properties: these unchanged alignment tests, the relaxed event, and existence of the weakened gain and traversal requirements. On every realization of the common output law, eligible enlarged disks have bounded overlap for each label, including the fixed finite recipe multiplicity.

Their marked intervals also have bounded overlap in geodesic time for each label. All intervals containing a specified time lie in a span of length at most \(2B_0\), since each has length at most \(B_0\). Each spends at least \(a'u/2\) in its own enlarged disk. If the disks have overlap at most \(M\) at this label and \(n_t\) intervals contain a given time \(t\), summing these in-disk durations gives \[n_t\,\frac{a'u}{2}\le 2M B_0, \qquad n_t\le\frac{4M B_0}{a'u}.\] The right-hand side is fixed because \(B_0/u\) is fixed. Partition the intervals into a bounded number of families of disjoint intervals. Within a family, use the triangle inequality for \(F'_U\) to splice the improved pieces with the intervening geodesic portions, charged at upper factor \(C\). The total gain satisfies \[\sum\delta_*u\le C L'-F'_U(p_0,p_1) \le O(\eta A)+o(1),\] where \(L'=D_{h'}(p_0,p_1;W)\). The geodesic is compactly in \(K\Subset U\), so its subpaths are admissible for every such splice. Since \(w=2^{\xi m}u\) and \(m\) is fixed, summing the bounded number of families, then the labels in \([N,2N]\), gives \[ \sum_{\text{eligible output hits}}w\le C_2\eta A N \tag{44}\] on the relaxed event, for all sufficiently large \(N\). Here \(C_2\) is independent of \(\eta,E,p_E\). Countable interval tests and slightly weaker gains can be used throughout to make this bound measurable.

Apply the per-hit domination and integrate (44). Together with (43) it yields \[c_0A p_E N\le \Lambda C_2\eta A\bigl(p_E+o(1)\bigr)N.\] Choose \(\eta>0\) sufficiently small first, obtaining its fixed event of probability \(p_E>0\), and then take \(N\) sufficiently large. This is a contradiction. Thus \(c=C\), as claimed. ◻

Proof. Use the transported prototype law at offset \(r\). Upper following of finitely many reference winding instructions with strict clearance and forced transverse stitching gives an upper probability cutoff for the circuit cost at a fixed multiple of \(C\). Lower comparison charges a nontrivial compact subtraversal of every winding path, giving a strictly positive lower probability cutoff at a fixed multiple of \(C\). The reference lengths in question have the same finite upper and positive lower probability cutoffs at every offset. Together with the two quantile inequalities (34), these bounds give (45). All margins are taken strict before returning to the measurement band.

At a quantile-normalized label with finite upper comparison, the second inequality in (34) already forces \(C>0\) by upper following, so Proposition 114 applies. Putting \(r=0\) and \(T=A_j\) in (45) gives \(1/D_0\le C\le1/d_0\). Finally, lower comparison by \(C\) and the common positive reference traversal cutoff give high-probability lower success at a fixed small multiple of \(CTw_r\). The upper bound in (45) makes \(d_1A_{j-r}\) strictly smaller than this threshold for sufficiently small fixed \(d_1\). This gives the last assertion with its probability and geometric slack. ◻

Proof. Fix \(\sigma\in(0,\xi)\). At every sufficiently large index test the following three assertions:

1. \(M^{-1}<A_j/A_{j-1}<M\), for a fixed large \(M\);

2. \(A_{j-m_0}/A_j<2^{-\sigma m_0}\), for a fixed large integer \(m_0\);

3. lower traversal success at threshold \(d_1A_j\).

Choose these constants so that rigid comparisons strictly re-pass all three tests. Such a choice is possible by Lemma 115. In particular, choose \(M\) larger than the uniform bounds for the adjacent ratios. Then take \(m_0\) so large that the fixed multiplicative loss in (45), times \(2^{-\xi m_0}\), is strictly smaller than \(2^{-\sigma m_0}\). The lower test uses the fixed \(d_1\) of that lemma, decreased if necessary to retain strict slack.

Every fixed finite success run eventually holds about the record starts of Lemma 110. Indeed, every extraction at such starts has the upper ratio bound (35) at all fixed preceding offsets. Upper chaining gives a finite upper constant, and quantile normalization makes it positive. Proposition 114 and Equation (45) then apply at all fixed offsets, including forward ones. A failure in a fixed finite run would contradict their strict retests after further joint extraction.

Suppose nevertheless that there are later first exits after arbitrarily long successful runs. Extract at the index \(j\) immediately before such a first failure. All fixed backward offsets, including zero, now pass the three tests. The adjacent bounds permit a further extraction with \[ r_i=\lim\frac{A_{j-i}}{A_jw_i}\in(0,\infty) \quad(i\ge0),\qquad r_0=1. \tag{47}\] We claim that these \(r_i\) are uniformly comparable.

Fix a starting offset \(i\). In \(A_jw_i\) units, the farther tests at relative offset \(l\) give high upper and lower factors of order \(r_{i+l}\) in their exact-power units. If at least a fraction \(1/10\) of the offsets \(l\in[N,2N]\) had \(r_{i+l}\ge L_0r_i\) along arbitrarily large \(N\), lower chaining would give a lower factor of order \(L_0r_i\) at \(i\). For a sufficiently large absolute \(L_0\), this contradicts the winding upper test there, whose scale is \(r_i\). If instead the same fraction had \(r_{i+l}\le r_i/L_0\), upper chaining would give an upper factor of order \(r_i/L_0\), contradicting the second quantile inequality at \(i\). The chain constants and reference probability cutoffs are uniform, as proved above. Thus, for each fixed \(i\), a large majority of all sufficiently long such intervals satisfy \[r_i/L_0<r_{i+l}<L_0r_i.\] The large-majority sets for any two fixed starting indices overlap after translating their intervals by the bounded difference of these indices. A common member shows that their \(r_i\)’s differ by at most \(L_0^2\). Comparing with \(r_0=1\) proves the claim.

At the index before failure, (47) now gives an upper ratio bound by a fixed multiple of \(w_i\) at every fixed preceding offset. Upper chaining again gives finite upper comparison; normalization gives positivity; and rigidity follows. Equation (45) and its lower traversal retest strictly re-pass all the tests at the alleged failure index, a fixed forward offset. This is a contradiction. Consequently all three tests hold eventually.

The same majority argument applies to any extraction at later indices: the adjacent tests give positive finite fixed-offset limits, the upper and lower tests make them uniformly comparable, and chaining followed by rigidity gives equality of the comparison factors in \(A_j\) units. Lemma 115 bounds these factors uniformly above and away from zero.

Iterate the second eventual test in blocks of length \(m_0\) and use the adjacent test for the fewer than \(m_0\) remaining steps. After enlarging a deterministic constant to include the finitely many small indices, this gives \[\frac{A_{j'}}{A_j}\le K\,2^{-\sigma(j-j')},\qquad j'\le j.\] Converting dyadic indices to \(P',P\) costs only a fixed factor and proves (46). In particular \(\mathcal B(P)\to\infty\).

For arbitrary diverging perimeter units, their ratio to the lower dyadic unit lies in \([1,2)\). Pass to a subsequence where it has a positive limiting ratio, and apply the joint fixed-ratio unit conversion of Lemma 92. This transfers the same local statements, with a deterministic common factor in a universal compact positive range, after further extraction. Only local internal reference length data occur in these assertions. The fixed-volume product rule (32), retained by Proposition 90, and the field matching and area-one transfer of Lemmas 106 and 98 therefore transport them to ordinary patches of the unit-area sphere. ◻

Proof. We use the critical exponent characterization in the Ding–Gwynne LFPP results (Ding and Gwynne 2023, sec. 1.2, Equations (1.4)–(1.7)) (whole-plane pinned GFF, heat regularization at length \(\epsilon\), median internal square crossing \(\epsilon^{1-2\xi+o(1)}\)). Indeed for any \(\sqrt2<t<2\), all but \(o(\epsilon^{-1})\) of the \(\epsilon\)-squares in that square have regularized field everywhere at least \(-t\log(1/\epsilon)\), in the sense that the number of exceptions is \(o(\epsilon^{-1})\) in probability. This is by Gaussian tails and first moment (use an intermediate exponent); unit-\(\epsilon\)-box field oscillations have uniformly Gaussian supremum tails by the heat-kernel covariance derivative bounds after scaling (center variance \(\log(1/\epsilon)+O(1)\)). Hence order one of projected crossing length must use this lower height, giving \(1-2\xi\le t\xi\).

Since \(t<2\), the inequality gives \(\xi\ge 1/(2+t)>1/4\). ◻

Proof. Consecutive active edges have common or adjoining tip vertices: the nearest retained slot shares the corresponding endpoint, and a bridge joins the retained and detached endpoints. These incidences join successive active midpoints, including the final absorption, at bounded Tutte cost. For a ring child, start at the corresponding first inner base. It is reachable by boundedly many Tutte links, since consecutive outer bases with no intervening inner run adjoin the same inner vertex. Choose the root of each recursive child at this transition site.

Every exposed site lies on a tip trajectory, on an adjoining ring or child boundary, or in a singleton attachment. A site not yet exposed remains in one of the children with its boundary occurrences retained. Iterating these alternatives gives the asserted access to the ancestral trajectories. The finite recursion exhausts the map by the exact peeling decomposition (17). ◻

Proof. Before stopping, the word of signed two-coordinate increments uses (24), with negative magnitude restricted to keeping the larger (\(j\le(S-2)/2\)) and the tie factor, and overall likelihood \(f_{S_{\rm stop}}/f_p\); overshoot down is no smaller than \((S-2)/2\), overshoot up unrestricted. In particular this whole stopped segment word for large \(p\) is boundedly dominated by the raw iid experiment to time \(T\) with the same stopping (invalid negative steps need not be used). Fillings are conditionally the marked products as before.

The bound is uniform in the stopped word. Before exit, the perimeter is comparable to \(p\). A downward exit keeps at least \((S-2)/2\) and hence at least a fixed positive multiple of \(p\); an upward exit can be arbitrarily large, but the perimeter weight decreases at the polynomial rate already established. The ratio \(f_{S_{\rm stop}}/f_p\) is therefore bounded in both cases. The restriction on negative jumps and the tie convention can only remove raw words or decrease their multiplicity. ◻

Proof. In (18) the perimeter moment’s \(c\)-derivative equals \[p F_{p-1}+\tfrac14\,\mathbb E[(c-U)^p(U-\eta/U)]\] by the differentiation above. At \(c\downarrow c_*\) the \(\eta/U\) term is bias converging to an atom of mass \(c_*\) at zero (\(\eta=c/m_{-1}\)). Consequently the derivative divided by \(c_*^p\), after this limit then \(p\to\infty\), tends to \(-c_*/4\) (the other terms vanish by the moment equivalent and the endpoint gap). Thus \(V(p)\sim c_*^2/(4 f_p)\asymp p^2/\log p\), finite positive for \(p>0\), zero for \(p=0\).

Use the substochastic spine kernel on positive perimeters \[{\cal K}(p,q)=\frac{V(q)}{V(p)} \mathbb E\,\#\{\text{segment children of size }q\},\] using volume additivity/domination. Its large-\(p\) masses tend to one, log ratio drift to zero, variance stays between positive constants, with uniformly bounded two-sided small exponential moments of log ratio. Here is the stopped-segment calculation. Let \(\tau\) be the stopping step, \(S_r\) the retained perimeter just before step \(r\), and \(Q_r\) the perimeter of the child detached at that step (the inner perimeter at a ring step). The final retained disk is itself a child. For \(z\) near \(2\), with \(0^z=0\), the exact telescoping identity is \[\mathbb E\sum_{q\text{ child}}(q/p)^z-1 =\mathbb E\sum_{r<\tau}p^{-z} \bigl(S_{r+1}^{\,z}+Q_r^{\,z}-S_r^{\,z}\bigr).\] The first and second derivatives in \(z\) give the corresponding logarithmic-ratio moments. We estimate the one-step summands before summing them over the stopped path. For one step from \(s\to\infty\), the scaled drift (including the detached child) of perimeter power \(z\) relative to \(s^z\), times \(m_s\), tends near \(z=2\), with first two derivatives, to \[G(z)=\mathrm{PV}\int_{-1/2}^{\infty} \frac{(1+v)^z-1+|v|^z}{(1+v)^2}\frac{dv}{v^2}.\] Indeed use the (24) equivalents, \(f_{s+d}/f_s\) likelihood, and the positive-face inner/outer ratio tending to one with geometric-run and smooth tilt tail bounds. Truncated signed first moment of the unrestricted raw increments is bounded; at small \(|d|/s\) use \(|f_{s+d}/f_s-1|\le C|d|/s\), second-order cancellation for the survivor term and absolute power integrability for the detached term. For the latter all positive outer sizes \(k\) have inner powers up to slightly past 2 bounded in expectation by \(C(1+k)^z\) from the geometric sum tilt; use nearby powers also for log factors. At the upper tail the additional likelihood decays as at least power \(2-\delta\); these bounds give dominated truncations.

For the small-step likelihood bound used here one may take the limiting positive moment law from (18), normalized in \(c_*\) units, whose negative endpoint has modulus \(<1\). On \(y\in[0,1]\), \(|y^{s+d}-y^s|\le C (|d|/s)y^{s/4}\) for \(|d|\le s/2\); moments there at comparable indices are bounded by \(O(f_s)\) by the equivalent and the exponentially small negative-cut contribution. The latter itself causes no problem for nonzero integer \(d\). The two coordinates together have the stated unit Lévy coefficient.

Now \(G(2)=0\) by integrating \(2/[v(1+v)]\). For \(G'(2)\) the first integral \(\mathrm{PV}\int \log(1+v)dv/v^2=2\log2\); the second is \(\int \log|v|\,dv/(1+v)^2=-2\log2\) by substituting \(v/(1+v)\). Also \(G''(2)>0\).

Sum these drift estimates over pre-stopping states (all comparable to \(p\)), multiplying by \((s/p)^z\) and including derivative terms; expected step count divided by \(m_p\) is bounded and bounded below (a small initial time with small raw oscillation has positive probability by Cauchy convergence and the bounded positive likelihood there). This gives the claimed weighted estimates first with \((q/p)^2\) weights. Replacing by \(V(q)/V(p)\) is harmless by the nearby power bounds and the equivalent. Indeed the ratio is comparable on compact positive ratio ranges and controlled off them by arbitrarily small positive/negative power losses relative to \((q/p)^2\) from the \(\asymp (1+p)^2/\log(2+p)\) estimates on positive perimeters, with relative multiplier tending to one on the compact ranges. ◻

Proof. Start at \(p_{\rm init}\asymp B_n\) and truncate on rising above \(p_*=C_* B_n\), which somewhere in the tree has probability bounded by \(V(p_{\rm init})/\inf_{p>p_*}V(p)\) by the spine/first-hit sum; \(C_*\) can be chosen large. Put \(k=\lfloor\log(p_*/p)\rfloor\ge0\) for bands. For each small \(\delta_1>0\) the expected number of nodes in band \(k\) before such crossings is \(\le C\exp((2+\delta_1)k)\). Moreover with arbitrarily high probability, simultaneously over all ancestry lines and \(k\), at most \(C\exp(\delta_1 k)\) such nodes occur per line. To see it on the killed/upper-stopped spine, tilt steps at large \(p\) by \(e^{\theta\log(q/p)}\) and normalize for fixed sufficiently small \(\theta>0\). Normalizers then exceed one, tilted drift is bounded below positively, and a small negative exponential of the increment has tilted expectation \(<1\), uniformly beyond a fixed cutoff, by the preceding moment bounds. From a band visit sufficiently high above that cutoff, with uniformly positive tilted probability the chain stays above cutoff and crosses the upper barrier in \(O_\theta(k+1)\) steps with log total rise \(\le C_0(k+1)\) at crossing (absolute coefficient for large \(k\)). Indeed use the negative-exponential supermartingale for avoiding cutoff, the positive drift and bounded variances, and the step tail for the overshoot. Thus before tilting a kill/crossing within that many steps still has probability at least \(c_\theta e^{-C_0\theta k}\). On the remaining fixed bounded perimeters there is uniform direct kill probability (positive expected exposed ring volume). Let \(N_k\) be the number of visits to band \(k\) by this killed, upper-stopped spine. Restarting at later visits gives \[\mathbb P_{\rm spine}(N_k\ge m) \le \exp\!\left[-q_k \left\lfloor\frac{m}{C_\theta(k+1)}\right\rfloor\right], \qquad q_k\ge c_\theta e^{-C_0\theta k}.\] The volume bias converts this to both asserted tree estimates. Indeed, \[\mathbb E\sum_{v:\,p_v\text{ in band }k} \frac{V(p_v)}{V(p_{\rm init})} =\mathbb E_{\rm spine}N_k.\] On this band, \(V(p_{\rm init})/V(p_v)\le Ce^{2k}\). The same identity at the stopping line of the \(m\)th band visit gives \[\mathbb P\{\text{some ancestry line has at least }m\text{ band visits}\} \le Ce^{2k}\mathbb P_{\rm spine}(N_k\ge m).\] Choose \(\theta\) so small that \(C_0\theta<\delta_1\) with strict slack. Summing the geometric tail proves the expected-node bound. Substituting \(m=Ce^{\delta_1 k}\) and then summing over \(k\) proves the simultaneous ancestry bound, with arbitrarily high probability as \(C\) increases. Fresh disk tests at a node can then use its unconditional \(T_p\) probabilities. ◻

Proof. Both initial disks of the restricted central comparison obey these bounds with bounded domination in the unit finite sphere: leave the opposite disk for volume smoothing by (22), localizing both initial perimeters in positive compact ranges. Also the full sphere has vertex and face degrees \(O(\log n)\) with probability tending to one under the rooted law. Indeed at a vertex of root degree \(d\) split its list from the root into \(j+1\) nonempty consecutive groups in order, replacing the vertex by a path with \(j\) occupied connecting edges and placing the groups along one flank at the successive vertices with the other flank free. Given \(j\) invert by tracing that empty flank starting adjacent to the root half-edge in the specified rotation and contracting; the FK exponent is unchanged. Thus a root-origin degree tail costs at most \(Z_{n+j}/(Z_n\binom{d-1}{j})\); use (15) and \(j\) a sufficiently small fixed fraction of the threshold for exponential decay. Use duality and uniform rerooting. This high-probability bound may be intersected after transferring estimates back from central intensity by dividing by counts.

On the degree event, incident primal vertices around a face have primal joining distance \(O(\log n)\). Within a disk we use genuine primal edge portions, including the possible blue half-diagonals up to a boundary red base; the costs of these internal primal walks carry to the joined sphere. ◻

Proof. Intersecting good intervals and the recurrence.

Indeed overlapping good component intervals with neither contained in the other have intersecting real boundary paths. For two capped paths, if cap endpoints alternate, the real paths already have to meet by planar order (the root face gives the occurrence order even when the boundary is pinched); otherwise one can cap disjointly when testing that pair. Coincident real endpoints already give contact. Nonintersecting boundaries would then be nested if their open chosen sides overlap, since both avoid the common start point also in their closures, and the overlapping component intervals would nest. Internal capped/full contacts suffice anywhere on the real portions. Thus one can prune containments in each intersecting chain and join along actual paths paying at most their total real lengths and the local incidence corrections. The endpoints at uncovered gaps are actual hits on those paths. Uncovered gaps across a tested interval require in sum \(O(1+\#\mathrm{bad})\) intervals of size \(O(T e^{-bN})\), since every non-end good grid time covers a spacing on both sides. More explicitly gaps of even shorter length must have separating bad centers between the good centers of their two chains. Endpoint remainders caused by using intersecting intervals, or by skipping the two full-segment end strips, cost only \(O(1)\) intervals of size \(O(T e^{-aN})\).

Consequently if \(\Omega(N)\) bounds joining costs for tip-point pairs within \(T e^{-N}\) (using adjoining blue representatives in the joined sphere), it suffices recursively to pay \[\begin{aligned} &O(1)\Omega(\lfloor aN\rfloor) +O(e^{\tau(b-1)N})\Omega(\lfloor bN\rfloor)\\ &\qquad+O(e^{(b-1)N}) \bigl[\mathcal B(p_*)e^{-s(k+N)}+O(\log n)\bigr]. \end{aligned}\] The second line accounts for circuits, capped paths, and incidence transitions. Fixed size factors can instead shift log indices by bounded amounts. Start this on all integer levels \(N\ge\max(M_0,c_2 k)\) with small \(c_2>0\), stopping recursion at \(k+N\ge(1-\delta)\log p_*\). In the terminal range reached from below (\(k+N\le(1-\delta/2)\log p_*\) after choices), trivial chronological costs \(O((1+T e^{-N})\log n)\) are negligible relative to \(\mathcal B(p_*)e^{-\varrho(k+N)}\). For completeness, divide each recursive term by the proposed bound \(Q(N)=\mathcal B(p_*)e^{-\varrho(k+N)}\). Apart from bounded shifts of the indices, the two recursive coefficients are bounded by \[C e^{-\varrho(a-1)N} \quad\text{and}\quad C e^{-(\varrho-\tau)(b-1)N},\] and the circuit-cost ratio is at most \[C e^{-(s-\varrho)k-[s-\varrho-(b-1)]N}.\] Each tends to zero by the strict exponent choices. The incidence errors are absorbed by \(b-1\ll\delta\ll\sigma-\varrho\). At the terminal levels the chronological time is at most a constant times \(p_*^\delta\), up to logarithmic factors, whereas (46) makes the proposed bound a larger positive power of \(p_*\); the same margin absorbs the terminal \(\log n\) costs. Choose \(M_0\) so that the sum of the recursive coefficients is less than one, and use the negligible terminal costs to initialize downward induction. This gives the bound \(Q(N)\) throughout the recursion. Bands starting already too small need only trivial whole-trunk costs, \(\ll\mathcal B(p_*)e^{-\eta_0 k}\) for some \(\eta_0>0\) by taking \(c_2,\delta\) small. Summing starting-mesh intervals on a trunk similarly yields \(\ll_{M_0}\mathcal B(p_*)e^{-\eta_0 k}\), as well as normalized chronological equicontinuity on bounded \(k\). By the spine bounds and fresh-disk estimates with \(B\) large, \(M_0\) can give arbitrarily high joint success probability over nodes and levels. Sum along each ancestry using \(\delta_1\ll\eta_0\), also absorbing \(O(\log n)\) attachment errors per node. This implies uniform approximation of vertices, at normalized cost error tending to zero with band cutoff, by points on the finitely many macroscopic tip trajectories (use the last sufficiently large ancestor). Central ring sites are covered on the initial disks. In these high-probability arguments \(\mathcal B(p_*)/\mathcal B(B_n)\) is bounded for fixed truncation by the adjacent-ratio bounds. One may first spend an arbitrarily small fixed probability error on the perimeter/degree truncations and the constants in the visit and trunk bounds, then test a putative failure in a further coupled limit on the remaining events (with bounded-band node counts tight by the spine estimate, or truncated simultaneously by growing band-dependent cutoffs). Thus no almost sure constant bound for untruncated maxima through all \(n\) is needed.

Nonconstant limits of macroscopic tip trajectories.

The preceding estimates approximate every vertex, with arbitrarily small normalized cost, by a point of one of finitely many macroscopic tip trajectories. We need one more property of those trajectories: a limiting trajectory cannot remain at one location throughout a nonzero time interval.

First extract the macroscopic trajectories jointly, parametrized by their rescaled chronological times. Cost equicontinuity and the positive lower comparison on ordinary compact advances give spatial equicontinuity. Indeed, a fixed spherical advance contains a compact subadvance avoiding the finitely many excluded points, and that subadvance has a positive normalized cost. The duration of a macroscopic segment does not collapse in probability: its stopped word has bounded density relative to the raw word, and the Cauchy limit stays in the stopping range for a positive initial time.

On every nonzero time interval before stopping, positive jumps of the raw Cauchy coordinates are dense. Choose two distinct positive jumps in that interval. Their limiting sizes are positive, and the ring comparison aligns their inner and outer perimeters. By the disk-volume limit (21) and conditional freshness, both born disks have positive limiting area. Each also leaves positive exterior area, supplied, for example, by the other disk. The area comparison and diffuseness then prevent either corresponding FK loop from shrinking to a point. The tip is incident to each loop at its birth. Distinct macroscopic limiting CLE loops are disjoint, so the tip limit cannot be constant on this interval.

These statements are first made for finitely many strict surviving tests. Exhaustion over such tests, the bounded-band spine counts, and domination by the unobserved opposite initial disk make them simultaneous for the finitely many trajectories retained in any fixed truncation. Taking rational time intervals proves nonconstancy on every nonzero subinterval of each retained limit trajectory.

Short surrounding circuits and the endpoint contradiction.

There are surrounding primal circuits of arbitrarily small normalized cost with strict spatial clearance about every sphere point. For ordinary points, use the aligned estimate with reference-metric tests: the length-space topology supplies finite-length winding and transverse crossing paths in a prototype annular buffer, and the unit actual-height comparison supplies shrinking successful annuli. To include the two marked ends, resample the area-marked representation. Almost surely the old ends are ordinary points in the new representation. At each fixed test the reference paths can avoid all excluded points, so the open-port upper comparison produces the required primal circuits. The same construction gives short reference-metric bypasses of the excluded points.

We prove the asserted endpoint modulus by contradiction, making the order of the truncations explicit. Suppose there are \(\varepsilon_0,\eta>0\), a sequence of map sizes, and vertex pairs \((u_n,v_n)\) such that \[\rho(\phi_n(u_n),\phi_n(v_n))\longrightarrow0, \qquad \mathbb P\{d_n(u_n,v_n)>\varepsilon_0\mathcal B(B_n)\}>\eta.\] Measurable choices are available since each vertex set is finite. Fix the perimeter and degree truncations and the spine/trunk constants so that their total failure probability is less than \(\eta/2\). Next choose a band cutoff so that, on these events, each vertex can reach one of the retained macroscopic trajectories at normalized cost less than \(\varepsilon_0/8\). Include the failure indicators, the chosen trajectory indices and joining times, and the connection costs in this extraction. Pass to a further joint coupling of these variables, the finite trajectory family, and the chosen endpoint locations. The intersection of failure with the retained truncation events has limiting probability at least \(\eta/2\). On that limiting event the endpoint images approach the same point \(z\), and the failure indicators are eventually one.

Consider the low-cost connection from \(u_n\) to its approximating trajectory. If its terminal point has limiting location different from \(z\), this connection already reaches away from \(z\). If its terminal point limits to \(z\), move a sufficiently small fixed chronological distance along that trajectory. Nonconstancy supplies a different limiting location, and cost equicontinuity makes the added normalized cost less than \(\varepsilon_0/8\). Apply the same argument to \(v_n\). We obtain two primal connections, each of normalized cost less than \(\varepsilon_0/4\), from the approaching endpoints to points whose limiting locations differ from \(z\).

Choose a sufficiently small surrounding circuit about \(z\) which excludes both of these latter locations and has normalized cost less than \(\varepsilon_0/4\). Its strict clearances force both connections to meet its real primal path for all sufficiently large \(n\). Joining at those intersections gives \[d_n(u_n,v_n)<\tfrac34\varepsilon_0\mathcal B(B_n),\] contradicting the failure. All choices were made after fixed truncations losing less than the putative failure probability. Letting that truncation error decrease proves the endpoint modulus in probability. This argument uses no almost-sure bound on untruncated maxima through all map sizes.

Diameter tightness and identification of the whole distance graph.

The endpoint modulus, finite spatial chaining, and vanishing flag mesh give tightness of normalized diameters. The open-port upper comparison follows finite-length reference paths away from the excluded points. Endpoint continuity attaches the specified vertices, while the short bypasses and the reference-metric completion remove the excluded points. This gives the upper bound by the same deterministic multiple of \(D_h\) for every pair.

For the lower bound, bounded-cost graph geodesics have continuous limiting projections after extraction: every fixed spatial advance has a positive lower charge. Off the excluded points their compact subpaths obey the local lower comparison. Portions visiting a shrinking neighborhood of an excluded point can be omitted at vanishing reference-metric error. More explicitly, skip from the first to the last visit to a neighborhood of the first such point encountered, and then do the same for the other point if it is encountered later. The triangle inequality and continuity of the completed reference metric bound the resulting reference error by the diameters of these neighborhoods. Let those diameters tend to zero. The lower and upper bounds therefore agree on the entire embedded distance graph. ◻

Proof. Describe the main balanced segment’s pending ordinary list (all its boundary slots except active) by one lifted interval with successive integer vertices \([b_u,h_u]\), of width equal to perimeter minus 1; the active slot reconnects its two ends. We use one oriented order consistently. The walk is in \((h_u,-b_u)\); exactly one coordinate updates by (24). On a positive expansion step (\(k-1\) replacement slots), \(k-2\) is added on the chosen side, leaving the last slot at the other end active; \(k=1\) is the previously described deletion/reassignment. On a negative at one end the \(j\) intervening slots there pass into \(T_j\), and \(j+2\) ordinary slots disappear at that end in the main list (reverse-glue, reassign). Thus an open interval of labels strictly between both end thresholds before and after moves that do not sweep through it is unchanged, literally slot by slot. On a positive step it can arrive entirely new from a ring if strictly inside the inserted range; on a negative step it passes literally to the detached list if strictly inside the removed range. Here and below strict clearances in assertions at scale \(P_y\) absorb the bounded endpoint conventions. Along increasing labels the order relative to the ring, or to a detached list, is the same whichever end made this update: use the direction around the oriented unfilled polygon on ordinary slots. The ring inner boundary has its fixed gluing order. Color across a positive changes and into a negative does not. These conventions also apply to arcs of the initial boundary and in a survivor at segment stop. In particular gluing at consecutive ordinary slots transferred into a bridge child or survivor uses their inherited boundary order with no new phase on that arc. New ordinary outer bases at either end of a face update can be read in the same increasing frontier order along an open inserted interval, paired through binary runs to the oppositely glued inner disk. Chunks strictly inside such an interval can always take a cut at a specified one of these outer bases and the same incident inner-corner convention, irrespective of which end of the main list supplied the insertion. The iid geometric gap calculation before tilt works in this cyclic convention as well. ◻

Proof. Conflict-free main and seam observations.

Here are details to include the raw contents in that entropy argument. Treat one base, coordinate/crossing type and fixed window choices at a time. First handle the main increments and main offsets by the separated annular sublattices already described. The subsequent estimates on a given sublattice can condition on the full primitive main walk (and choices of bases) to decide which designated aperture to try by tracing the level, without conditioning yet on its contents or rings. Query on the required large-aperture ranges if outside the retained main time blocks of this test scale. Drop other queries. Resolve overlap conflicts before further comparison: an aperture step in one scale’s query can occur in a queried main block for a different scale in the spaced list, but has at most one such block index. Thus color the indices with a bounded number of colors. Indeed the directed conflict graph has at most one outgoing edge per index, so each connected component of its underlying simple graph has at most one cycle and admits a coloring with three colors. Within one color there is no such conflict. Initial/final boundary disks are treated without a step conflict. Terminal survival queries refer to the actual final polygon, separate from all the retained detached pieces. Repeated queries of the same positive seam here use disjoint chunks around the same \(H\) by sufficient sublattice spacing.

Conditional product laws and adaptive helper observations.

Conditional on the main word, detached pieces in retained main blocks have their raw product law, and ring decoration at the designated positives, independently, can use ordinary geometric sequences with the \(f_l\) tilt. This tilt has bounded log-density above per queried ring on comparable totals. Noncomparable totals for these queries cost negligible even in the positive part of the log-density expectation under (48), since queried outer sizes are at least a constant times the smallest scale and at most \(O(p)\); the untilted-law atypical estimates are exponential in outer size up to polynomial factors (under the original \(f_l\) law as well). Disjoint chunks at repeated scales thus cause only \(O(1)\) cost per index on average. Now, given the whole applicable ring data, one can handle helper outputs separately for each possible prescription index within disk. For that index, the prior sigma-field contains the primitive main word, applicable ring and seam data, and earlier helper word histories. It excludes unused map interiors and raw filling outputs of other prescription indices. Condition on these preceding operations for those disks. The starts, pending-arc sufficiency and transmissions of anchors can then be chosen before those new paths; invalid starts/anchor queries can have dummy outputs. For the same prescription in a disk the later word blocks queried are disjoint across the spaced scales and have fresh inner intervening gaps. Use bounded path-law likelihood once per such trajectory (iid extension and artificial fresh step contents after stop if necessary), conditioning detached contents of its queried blocks by their raw kernel. Thus both requested initial translations, even with shifts supplied by rings or preceding path histories, use the joint gap local-mass bound \(C/P_y^2\). They need not be centered by an exactly resampled source. Fillings inside a helper word block are not further queried for this same prescription in that disk.

Joint records and the entropy chain rule.

We combine the comparisons in their order of revelation. The needed entropy inequality allows a prior to contain more information than the retained record. Let \(R\) be a retained record, let \(Z\) be a prior with \(R\) measurable from \(Z\), and let \(U\) be newly queried outputs. Write \(Q\) for the lifted law and \(P\) for the original disk law. If \(\nu\) is the reference law of \(R\) and \(K_R\) is a reference kernel for \(U\) depending only on \(R\), convexity and the chain rule give \[\begin{align*} H\bigl(Q_{R,U}\mid\nu(dR)K_R(dU)\bigr) &\le H(Q_R\mid\nu)+\mathbb E_Q H\bigl(Q(dU\mid Z)\mid K_R\bigr)\\ &=H(Q_R\mid\nu) +\mathbb E_Q H\bigl(Q(dU\mid Z)\mid P(dU\mid Z)\bigr) +\mathbb E_Q\log\frac{dP(dU\mid Z)}{dK_R(dU)}. \end{align*}\] Thus one can forget the extra prior information after comparing the new outputs jointly. The conditional entropy term is at most the full lift entropy budget, by the chain rule. A new finite tag \(T\) selected from \(Z\) can be included by giving it an independent uniform reference; this adds at most \(\log |\mathcal T|\) per queried index. Here \(\mathcal T\) denotes its finite alphabet.

Apply this inequality first with \(R\) equal to the retained primitive main words and noisy offsets on one spaced scale list. Their product reference and \(O(N)\) entropy bound were established before revealing the full main word. At this stage \(Z\) contains that full primitive word and the base choices. The designated aperture and the conflict color are determined from \(Z\). The new outputs are the main-block detached contents and the queried ring chunks. Within one color, the designated aperture lies outside all its retained main blocks; repeated seam queries use disjoint chunks. The reference kernel therefore uses the original filling kernels given the retained words and ordinary geometric runs on the queried chunks. It depends only on the retained record and its finite presence/type tags. The \(f_l\) ring tilts and their atypical-tail estimates give expected logarithmic cost \(O(N)\) for this entire conditional list.

For helpers, fix one prescription index at a time. There are only boundedly many possible indices, and a complete local record uses just one of them. The common retained record \(R_0\) consists of the main and ring outputs already compared. For index \(j\), take a prior \(Z_j\) containing the full primitive main word, the applicable ring and seam data, the retained main-block contents, and earlier helper word histories. It excludes unused interiors and raw filling outputs of other helper prescriptions. In particular, we do not put earlier complete helper records into \(R_0\): such a raw filling could contain a polygon that the current helper will explore.

The new output \(U_j\) is the record for prescription \(j\), including its own detached contents. Starts and transmission anchors are chosen from \(Z_j\); at each scale include the finite indicator \(I_j\) that this prescription supplies the accessible helper, and use a dummy output when \(I_j=0\) or the query is invalid. Accessibility and preceding-path interference or availability choices made from this prior enter as finite tags here. Give these tags the independent finite reference described above. The conditional comparison is made separately for each \(j\) and then summed over this bounded set of alternatives.

For a fixed prescription its queried blocks at the spaced scales are disjoint and have fresh intervening gaps. The joint gap bound gives mass at most \(C/P_y^2\) for the two offsets, uniformly in the absolute anchor shifts supplied by \(Z_j\). Its reference kernel \(K_j\) therefore uses independent offsets on the prescribed bounded boxes, raw helper words, and their original conditional filling kernels, with parameters determined only by the retained local anchors and finite type. The sequential bounded path likelihood and gap bounds give expected logarithmic cost \(O(N)\) against the product of these new record kernels. Applying the displayed inequality to \((R_0,Z_j,U_j)\) gives a joint entropy bound \(O(N)\) for the main/ring record together with this helper. It asserts no product law between records of different helper prescriptions on the same disk. These elementary references factor over scale indices, retaining the within-record dependence of detached contents on their own raw words. Superadditivity of entropy over the factors bounds the sum of the one-record entropies for each masked record \((R_0,I_j,U_j)\) by \(O(N)\). Divide by the number of sampled scales, comparable to \(N\). At one sampled scale the actual complete record uses at most one helper prescription. An elementary-reference null event for that record therefore has lifted probability bounded by the sum of the corresponding masked-record probabilities, over the boundedly many colors and indices \(j\). Each tends to zero by the entropy bound. This proves the claimed joint null transfer without a joint law for all alternative helper payloads. No positive lower bound on the size of an individual selected list is needed. In particular, main-offset smoothing was used only in the initial term \(H(Q_R\mid\nu)\), never conditional on the full main word.

The probabilistic implication can be written explicitly. If \(Q\) is a lifted record law, \(P\) its elementary reference, and \(H(Q\mid P)\le C\), then for every event \(A\) with \(0<P(A)<1\), the binary entropy bound gives \[Q(A)\log\frac1{P(A)}\le C+\log2.\] The averaged conditional entropy bound gives the same conclusion for the uniformly sampled scale. Thus reference probabilities tending to zero force lift probabilities to tend to zero. This also justifies the helper tangent path absolute-continuity uses without raw contents. ◻

Proof. Three preparations of the distinguished aperture.

We explain the reverse comparison on the strict regular subset of records just specified. Use a synthetic ordinary paired-disks experiment in perimeter units \(P_y\), with the main side of the required color a first central-side caller, its length in a sufficiently large fixed compact positive range (smooth scale mixing), with ordinary fresh source and target. Explore its targeted branch. Put a deterministic central time in it with both blocks as above in exact step lengths, leaving preparation time before them; request positive large side lengths throughout all prescribed main operations. Below, a “large” size in this experiment can be fixed as a sufficiently large constant in viewed units. Restrict the initial split and size to appropriate large intervals with room. Simulate the relative \(h,b\) directions with the corresponding two main list coordinates; the absolute separation of those directions need not be the original one. Orders on slit arcs and the two color types agree under the following choices:

• If no aperture is needed, prepare with small relative displacements before the first window. For the initial-wall case also do this: the protected past outer-cut height exceeds the center height by the stipulated margin but a bounded amount, so \(H\) in this realization is strictly on the ordinary original arc and protected as required.

• For a past positive aperture, make a large positive jump on the synthetic protected coordinate in preparation, with magnitude much larger than the offset bound and helper window bounds, and keep other preparatory fluctuations small compared with the margin. Then the band about its \(H\) has arrived from that jump unchanged. The increasing pending order and reverse ring gluing there depend only on the aperture type, even if in the actual test the giant step reached \(H\) through the other end of the lifted list.

• For a future loss aperture or final survivor, use a negative separation after the protected outer post-cut, on that same synthetic coordinate, dropping by a large amount across \(H\) with both margins. Keep intermediate fluctuations small. Arrange by a still larger preparatory positive gain before the windows that this whole separating interval is strictly above the initial side height. It then carries no inherited caller targets, and the negative disk is reached for an actual fresh sweep. Indeed original target slots transmitted along this path remain on their original prefixes until left, and on the limit the corresponding loss circle has no contact with the original boundary by the strict stack rule (opposite side has no record linkage at that jump, also immediate by its Cauchy oscillations on both time sides simultaneously at the countably many jump times of the first coordinate under compact base-law domination). This is a compact canonical negative restart in the primary comparison. Main target positivity and sufficient size room on both lists can all be imposed. The local data at this loss use exactly the direct transfer type, regardless also of whether the actual path stopped as a survivor instead.

These preparations reproduce the retained record type, rather than the entire balanced exploration. A positive seam is read through its ordered outer bases and inner runs; that local order does not record which remote end of the main list introduced it. A loss aperture and a final survivor both transmit an unchanged interval of ordinary slots with its inherited order and color. Their retained direct-transfer data are therefore identical. Only those interval data and the stated buffers enter \(\mathfrak C(\mathcal R)\), which is why the synthetic negative restart may represent either case.

Ordinary sources, targets and exact noisy offsets.

When used, \(J\) now has its ordinary fresh source, which we restrict to a small macroscopic tolerance interval about the desired label in boundary order, and an ordinary target with enough room for its one helper block and terminal clearance outside the seam band. Use the targeted branch from that source. No earlier helper operations have to be simulated: the wide pending unaltered arc in the test is sufficient. Thus all synthetic spatial tests are on ordinary compared primary data (typical additional target choices are allowed).

The mass on synthetic completions satisfying the required omitted-piece buffers, projected to these records, dominates a fixed positive multiple of the elementary references restricted by the indicated observable bounds/dips, with tighter inset offset and accuracy bounds if needed. To verify this, all macroscopic preparatory and separating moves above can use open jump time/size intervals of positive raw probability with intervening small oscillations, by iid Cauchy convergence or truncated-jump sampling. Word likelihoods on both compared branches have the positive bounded ratios (24) on the prescribed ranges. Given sizes, seam chunks use independent geometric runs up to the smooth total tilt; complete the omitted runs with the indicated local linear accuracy with probability bounded below, including between seam anchors needed for positioning the helper start. Their chunk positions on outer bases are determined by main labels, not by conditioning on inner counts. Both main offsets in each time direction are made by exact lattice displacements in the two omitted small-gap halves. The helper offsets use two coordinate displacements in its initial omitted gap after selecting a source slot from an ordinary tolerance interval as above: in particular, any bounded shear through the seam’s omitted middle within the chosen small tolerance is absorbed. The restricted source probability is bounded below given those main/ring data; start/root occurrences can use their true bounded lattice conventions here. For offsets as inset above, net displacements required in each such gap are small compared with \(vP_y\); local masses of order at least \(P_y^{-2}\) per gap are available while keeping oscillations \(<vP_y/5\), say.

One direct constrained local bound is to reserve, in a gap of fixed positive relative length, one positive move in a small size band and one correcting negative of comparable size on each coordinate in separate time portions. The bands can be strictly inside the small-oscillation budget but larger than the required tiny net-offset and source-shift bounds (decrease those bounds beforehand). Suppress other moves of these sizes or larger and bound all remaining sums/oscillations by a sufficiently small fixed tolerance, an event of probability bounded below: truncating at a still smaller fixed relative size gives this by the truncated second moment, with negligible truncated drift, and the finite larger-jump exclusions cost a fixed positive factor. The correcting moves can now set both offsets exactly by summing over their possible locations, using (24) at each required integer in a comparable-size range; specifying each value costs order \(P_y^{-1}\) after its time sum. Distinguished moves don’t have an unbounded overcount with these exclusions. This proves the lower bound with stricter numerical margins as necessary.

In the main path, central values and thus seam positions can be computed from the earlier prepared heights, first block word and the sampled relative offsets; the past/future protective inequalities in the recipes then hold for any such recorded words. Detached raw contents inside all word blocks have exactly the same conditional laws. This proves the comparison without any lattice conditioning at zero tolerance on relative pure-field or attachment-source positions. ◻

Proof. Closed near and far sets.

Here is the certificate of success on this synthetic law portion. Along good ordinary-experiment subextractions use the homeomorphic spatial comparisons, the exact quotient contact descriptions and the open-port upper comparisons in \(\mathcal B(P_y)\) units. The main disk is denoted \(D_0\). Work on limits where the buffer conditions above still hold with slack. In particular the main protected synthetic band is on a single targeted side with absolute heights well inside the relevant positive regimes; crossings and cuts here are compact observations before terminal. Define closed near sets by including:

• the whole main gap, trunk and jumped contents born there, and the frontier labels at its two cuts on each side within \(2v\) in scaled units of that side’s central height;

• in the seam or initial-wall case the boundary band \(|\text{label}-H|\le v\) (and the paired loci on a ring, which are identified in the limit);

• in \(J\) when used, the initial helper gap and its born contents, and frontier arcs at its ends within \(2v\) of the corresponding initial tip heights.

Small roundings have no effect. The far sets include the main time portions outside the outer cuts with all their jumped contents except distinguished \(J\) (do include its main pinch); on \(J\)’s boundary include the complement of the open band of radius \(D\) about \(H\). Include also the helper portion after its post-cut through terminal with all its born contents. Include the exterior of \(D_0\) with its rim, except in the initial-wall test where we work only in \(\overline D_0\), including the rim portion outside the open band about \(H\) as far. All these definitions use closures, equivalently the unions of the respective closed trunk pieces, closed disks and arcs by the vanishing diameters at accumulation. Unknown root-face exterior information in an initial-wall test will not be needed.

Exclusion of every near–far contact.

Near and far are disjoint. Within one chart every linkage out of a central gap label or the indicated near frontier arcs would have to sustain a level within that side’s narrow indicated heights. A required dip bars this before the corresponding outer cut. On the remaining protected main direction, backward (or forward) linkage instead before that cut into far times is impossible by the strict transmission from (or to) the open seam band: no near-level operations occur in the intervening far period, and levels on the opposite side of the exterior crossing are strictly beyond the band at that crossing. In the initial-wall case the linkage is to the open ordinary original arc with the same strict protection. These uses include aperture endpoints/trunk tips: in the exact-contact description folds cannot propagate a nontrivial linkage into another direction/level across a real-time tip. For a frontier point use its pending boundary label by the same rule. Here heights of near frontier labels and of all gap apertures, on any working side required for such ties, differ by much less than the dip/protection margins from center. An aperture boundary can only use its actual side linkages including the automatic endpoint fold at its birth; a class through a real time has at most one nontrivial side linkage among its copies. Thus, for example, following a pending near height to a tip outside the gap cannot then change to a different height to continue past the corresponding barrier. The outer/target conventions give no further such contacts by the strict buffers and original target room. Far parts within either chart cannot contact the opposite chart’s near parts: such a contact on the distinguished circle could use only near-band levels there, all protected on the main side as just described (and barred by the opposite-time dip), and barred from helper post times by the two dips since the seam order is matched, the source has small offset there and the near helper arcs/gap meet that seam only well inside the band. Interior disks in each trunk description have no other contacts. This proves disjointness, also from the stated outer rim sets. The far set is connected: main earlier and later parts join to the rim by source and target, with the source and target on the connected included rim portion; the included arc on \(J\) joins at its pinch, and the helper post part at the remote target. The near set is connected as well. Its frontier arcs contain the corresponding cut tips by the small gap-oscillation bounds, and each jumped closure attaches to the central trunk. In the seam case the protected center level links this main piece to the seam band; the helper source, when present, lies in that band. These are the same exact contact and persistence rules used above. Thus the whole near set lies in one component of the complement of far.

Separating corridors, circuits and crosspaths.

In the sphere case that component is simply connected with the near set in it compactly inset. There is thus a compact Jordan annular buffer there away from both sets, with a simple circle separating the central tip from far (e.g. use Riemann disk exhaustion enclosing all near points in the component). For the initial-wall case flatten \(D_0\) to a closed round disk and double by reflection of the sets through its simple boundary. Doubled far is still connected, doubled far and near disjoint. A complement component as above either is in the open disk or touches the boundary and is symmetric. In the latter case use disk exhaustion invariant under reflection (Riemann map at a fixed point of the reflection); the circle runs transversely across the flattened rim at two points. Use just its simple crossarc on our main side, extending at both endpoint ports transversely into the exterior in the ordinary paired sphere. Endpoints have margin on the transferable part of the initial arc. All corridors can be taken arbitrarily thin with buffers; choices/perturbations can avoid the finitely excluded points for port comparison.

Ordinary metric length topology and open-port upper comparison now give respectively surrounding simple primal circuits (force stitching by transverse crossings in the annular corridor), or primal paths in the crossarc corridor passing through \(D_0\) from exterior port to exterior port, at costs bounded in \(\mathcal B(P_y)\) units along further pathwise extractions. Indeed follow finite reference paths of finite length for these open instructions approximating the requisite Euclidean passages, with endpoint and confinement clearance. Only fixed limiting tests or their countable strict approximations are used. In the latter case crop to a real subarc through the main lattice disk with endpoints on the two separated rim-port lifts. Such a crossing portion exists since central compact parts of the crossarc are in the main open disk, transitions to the complement require a rim touch, and the paired/exterior/ordinary placements have the all-lift upper comparison. Erase loops to make the arc simple; it stays in a narrow crossarc neighborhood connecting the two ends. Real arcs can terminate in the appropriate half-edges at boundary slots. Along the blue primal graph all internal intersections use its actual planar drawing.

Exact pairings on all retained stars.

The required control of a vertex star is uniform in its degree. Indeed, Proposition 53 supplies comparison maps whose maximal flag-image diameter \(\delta_n\) tends to zero. Every flag incident to a vertex \(v\) contains \(v\), so the image of its entire closed star has diameter at most \(2\delta_n\). Finitely many star enlargements therefore remain in any fixed open buffer eventually, even when the number of incident flags grows.

The finite certificate consists of the separator’s ordered edge occurrences, its endpoint sectors in the crosspath case, the full rotations of its vertices and adjoining stars, and the retained tip movements through this neighborhood. We must show that all these data can be read from \(\mathcal R\), independently of the member of \(\mathfrak C(\mathcal R)\).

We spell out why these real boundaries are transferable at exact incidence and preserve separation in every disk test with these same records and the omitted-data buffers, for sufficiently advanced successful synthetic comparisons. At a compactly buffered point of the circle/crossarc (on the main/queried side in a crossarc test), all the stars of lifted sites and boundedly adjacent incidence data stay off near and far by the all-flag upper comparisons, including the aperture, cut and frontage label comparisons. The triangle pieces there must all be from retained blocks with their raw attached disks/rings, or retained seam chunks. Iterating their pending-slot operation records confirms all actual pairings there:

• Within one raw block use its own stack operation order and comparisons. Across the main central gap, slots which pass and are needed on both sides necessarily stay outside the unknown thresholds (otherwise their synthetic points are in the near pieces or the near frontier arcs), and use the same offsets on their side. A pairing created by an omitted operation itself would lie in its omitted content/tip. Active-slot and bridge-pair assignments occurring locally on opposite sides at a retained operation use each separate end displacement; the absolute width across the far middle of the list isn’t needed.

• Comparisons with an exceptional seam through the far transmission require labels strictly inside the protected radius by far avoidance, and outside the omitted seam middle by near avoidance. They use the literal labels, contiguous ring chunks if positive, and on the helper side the two anchor offsets with an untouched sufficiently wide initial arc passed through any prior operations and the near cuts (source and omitted initial fluctuations have inner clearance). Thus both gluing ends of actual pairs at retained operations are reproduced even when comparing a loss aperture here to actual transmission into a survivor. The helper block on its own again uses just its raw word/fillings. A direct gluing through an omitted helper operation at these sites is excluded by avoidance.

• At the initial rim include actual root slots and their successive boundary sectors on the indicated open arc pieces. Pairings on the \(T\)-side and root-sector rotation incidences are all known by those same label comparisons. There is no need to determine the true neighboring triangulation exterior to \(T\) on the sphere.

Composed occurrence pairings and closure of vertex stars.

More explicitly, initial/final cuts of a retained word block may be read with arbitrary formal unfilled stacks. The only tests then needing translations outside the block to confirm a pair are tracing ordinary input/output slots by unchanged indices across intervening omitted steps or a split/ring to their actual retained origin and consumption. New/consumed actives at steps inside the block have the raw bounded neighbor conventions, and do not silently persist across omitted peeling steps. Between main blocks a pending relevant slot has index near one of their working ranges, which remain far apart in all realizations used here. If its comparison could use a gap threshold within the allowed gap fluctuations, its same synthetic pending height at a gap endpoint belongs to the enlarged near arcs (or the slot reaches a gap operation). Between a seam and block there is exact transmission as above, and across a helper start it is the assigned side of the source/range relative to its chunk that matters, fixed by the margin. A pair incident to an omitted terminal polygon, raw ring chunk or word operation would have at least one flag or peeled bridge/tip in the omitted sets. These checks exhaust the pending-slot gluings; at a protected vertex the already confirmed pairs close its entire synthetic rotation cycle (with root-face sectors in the boundary case). The raw flag/sector occurrences are distinct in every realization, so additional unknown identifications cannot change such a closed star or merge nonidentical ones.

To elaborate, along a gluing traced through several pieces, root bridges with two boundary occurrences in a raw filling are read with their full pairing from that filling. At such a trace all intermediary occurrences representing the same final edge are themselves at the protected sites. A bridge deletion by the axis word there must likewise be recorded (both its ends at bounded incidence of its tip). Unchanged-slot transfers with comparisons on the main blocks need translations only near the respective end working ranges determined by the bounded word windows; these ranges are disjoint with margin, so a transmission through the gap cannot silently exchange them. In comparing to \(J\), original slots used on the transmitting band up to the helper start are still pending consecutively in its starting polygon regardless of changes beyond; on each of the two protected band pieces the comparisons from that side’s near-cut anchor to actual helper operations now use literal increments, even across raw bridge moves. Thus it is occurrence paths of gluing, not just abstract final vertex equalities, which are confirmed.

Complete paired sides at a site trace back along these paths to their same origins: constructions glue successive unfilled slot sides, with boundary-only bridge occurrences passed between (an interior final edge has its two filled sides at the ends of the composed pairing; at the original wall include its original root-face sector if present). There is no additional operation later merging already closed edges. In the root case every root-face corner needed there is itself flanked by the indicated comparable original boundary slots by all-occurrence rim comparison. The spatial buffers allow boundedly many star enlargements before this exact confirmation. In particular simplicity along the transferred path, endpoints as occurrences, and the rotations of all adjoining paths are exact. This argument does not require a simple actual boundary at an initial wall.

No omitted visits and preservation of sides.

No omitted main tip movement can then revisit the certified boundary in the test, since making a main move there or boundedly adjoining would pair a peeled slot or introduce a ring at that position by an operation not in the data; all such incidence operations at the boundary and its adjoining stars have already been confirmed with their retained locations/orders. Within retained main blocks, visits and passage sides read by the tip path through that neighborhood agree. Thus on either side of the central gap the parity of sides switched across the boundary is the same; one may use the oriented collar of the real path and local rotations, counting changes from one open side to the other, to avoid transverse-general-position assumptions. Gap endpoint/outer-cut ambiguities are away from this collar by buffer separation. At terminal transitions of the actual balanced path outside the cuts the old active at the shared endpoint is included in the same exclusion. For a crossarc cap, attach an arc through the root face of the main disk (away from the real tip walk except possibly at its bounding endpoints); its end-sector germs and all corresponding sides can be matched by the known boundary-label rotations.

Odd side counts, including a pinched root boundary.

For the crosspath, the winding argument uses the following precise property: the comparison from the capped discrete sphere to the doubled limit disk has degree one, and the chosen near and far inset image points each have a unique preimage in the main disk. The collar and cap avoid those image points. We construct this comparison before applying parity.

For detail the side counts on both retained main blocks are odd. For a full circuit this is the surrounding/separation property on the synthetic spatial comparison. In the crossarc case it can also be checked as follows, allowing pinched root boundaries. Give the root face artificially back to the synthetic main disk, thickening its real closure by a collar in that face to obtain a simple closed topological disk, then cap (remove a smaller simple open disk from the parametrized root-face interior, using its prime-end boundary walk at the real sites). There are continuous degree-one comparisons from this sphere into the doubled \(D_0\) on the good subextractions: on the real part take the approximate spatial homeomorphisms retracted to \(\overline D_0\) from a small outer Jordan-coordinate neighborhood, identity retraction inside; over the collar interpolate to the ordered boundary parametrization using the uniform rim-label comparison (all widths here can tend to zero in the images). On the capping disk use a homeomorphic extension to the reflected disk. On a compact inset of the main open disk these comparisons have ordinary homeomorphic degree-one lifts, since the complementary parts are spatially outside with margin there. Extend the cropped real path through the collar at its two separated slot-sector ports, joining through the cap along the reflected crossarc. Its Jordan loop then projects within the annular buffer and winds once between the indicated sides (the real path lies in a narrow crossarc corridor joining the two ends; backtracking does not change this homotopy). The synthetic central time and representative times in the pre- and post-outer pieces give the separated points on the respective sides. We can take them with limits interior to the main disk: deterministic compact-surviving trunk times almost surely have no outer-rim contact by Cauchy oscillation and the quotient rule. Allow extra surviving pre/post buffer times in the preparation if needed, or use a dense deterministic sequence. Winding parity between their lifts follows from the degree-one local property there: inset image positions at those tests have unique preimages where the old homeomorphisms apply. If the Jordan loop did not separate those lifts, the opposite source disk side would contract the projected loop away from both positions, impossible. Since no omitted tip visits the real boundary or the cap between tests, this gives odd switches across each retained block just as for a full circuit. Actual capping in a test realization can use any simple root-face arc with the matched germs.

The transferred good-time witness.

It follows that in the actual \(T_p\) test one open side contains the whole grid-radius neighborhood at the center, the full tip ends are outside, and its time component ends on both sides within the outer cuts. The latter reach is smaller than the allowed reach, and the costs here are within the allowed budget. For full circuits near winding curves, use a simple separating subcircuit in the narrow buffer; at capped paths the cap contributes no cost. The certified transfers work simultaneously for any completions with the indicated margins, not by continuity of a discrete distance in abstract word displacements. ◻

Proof of Proposition 130. Consequently on the regular record subset the elementary-reference mass of possible failure (existence of a buffered test completion without this good-time witness) tends to zero. Indeed otherwise the synthetic lower comparison would put nonvanishing mass on this event with synthetic buffers. On unconditioned good synthetic subextractions with those buffered limits it is impossible by Lemma 133. Finite strict-slack choices/exhaustion suffice, and possible completions or finite witness failures at lattice indices are measurable (equivalently take the countable finite combinatorial records/maps). This reasoning is uniform in the sense needed for a varying log scale: a sequence of offending scales/addresses would have the same diverging units and synthetic-law subextractions, with cut roundings and length-unit ratios as above. By Lemma 131, the same event has vanishing lift probability. This contradicts the chosen bad bases on the stated positive-probability buffered regimes. This proves the passage assertion. ◻

Proof. Apply the typical-logarithmic-scale stationarity and absolute-continuity alternatives of Section 10, together with Proposition 130, first with all factors bounded. On coupled further extractions, the complement of the alternatives in the statement is a null event. The bounded factors may therefore increase sufficiently slowly so that the resulting tests hold simultaneously on the shifted levels assigned to a probe. This is a diagonal choice of finite witnesses. It does not condition the tangent assertion on a prescribed sparse phase: first sample a level uniformly on the whole lattice in \(\mathcal I\), and then average the sparse failure fraction over its phase. Up to the endpoint levels, this average is exactly the typical-level failure probability. Markov’s inequality gives an \(o(N/L)\) number of unsuccessful probes.

We verify the extra separation, since it is needed when several auxiliary charges occur on the same path. The unscaled base choices are independent of the sampled log level. In a typical-scale limit, the normalized separation of two distinct bases is therefore either zero or infinite. In the infinite case, requests in a common positive \(J\) are both viewed from its past, while requests in a negative or terminal \(J\) are both viewed toward its future. The clearance through far times for the later base in the former case, and for the earlier base in the latter, includes the other base’s time. At that time an actual list threshold is the other base’s label. Thus the two centers are infinitely separated in the lifted interval. Aperture-end clearances also exclude proximity through the cyclic seam. The fixed ring order and its local linear comparison transfer this separation to the circumference. Decreasing the long margins by a fixed factor gives the claimed finite separation.

All required factors can be imposed in one slow diagonal. For example, the unit-window oscillations in a near-link are finite in each bounded test; \(e^{\sqrt L}\) can be chosen as their increasing cutoff while the larger separation scale is \(e^{2L}\). Fixed spatial and slot-rounding enlargements can be included at the same time. Omitting finitely many additional levels at the probe ensures that every fixed factor \(CP(i)\) in a record requirement is at most \(P(j+1)\) on its retained assigned levels. Auxiliary raw paths can be augmented beyond their stopping times for these necessary tests.

The first-consumption window can be bounded by an absolute constant in boundary units, using the zero-distance case of the passage exclusion, the bounded endpoint conventions, and long-arc transmission. Here consumption of an original occurrence means reaching it as active or using it as a paired or deleted slot. Merely passing an untouched interval into a fresh detached child is not consumption. Once the occurrence has passed strictly into such a child, the same main trajectory in its starting polygon cannot later reach it. Therefore the ladder depth on the reaching side is its depth at the start of that auxiliary path. The array of middle labels in one \(J\) is determined by the original base heights and the fixed ring convention.

Choose actual witness times and types before the independent time-grid translation used below; dummy or unsuccessful probes can be omitted. Only finitely many tags per base and probe remain. A choice of \(J\) is recovered by the level-tracing rule from the main history, so it does not supply an additional freely chosen main time. This gives the stated tag count. ◻

Proof. The occupied boxes form an ordered forest with a bounded number of roots. At fixed node count its unlabelled shape has at most exponentially many possibilities, by parenthesis coding. Each child has at most \(R\) possible positions in its parent, contributing at most \(g\) per child. These costs give \(V+O(V/g)\).

Attach each row label to its terminal box at its probe depth, and each base label to its box at the deepest level. On the proposed bad-time grid, a base needs only \(O_R(1)\) further choices within that box. Only a bounded number of entries terminate at each level. The remaining logarithmic assignment cost is bounded by a sum of \(O_d(n_0)\) terms of the form \(\log(Cm_j)\), with bounded multiplicity. Concavity bounds this sum by \(O_d(n_0\log(1+C V/n_0))=o(N+V)\). The tags from Lemma 134 have the same small-order cost. The available number of nodes is polynomial in \(N\) for fixed \(d\), so summing over its possible integer totals adds only a logarithmic term. ◻

Proof. Partition the specified terminal-box step intervals at all their integer endpoints. For a resulting interval of length \(w\ge1\), truncate ordinary moves at \(u_w=c_0w\log(w+2)\), with \(c_0>0\) small enough that \(u_w\le P(i)\) for every box covering that interval. The bounded increments, truncated variance and drift, and tail bound in Equation (24) imply bounded small exponential moments for their maximal partial sum divided by \(u_w\). The number of ordinary moves larger than \(u_w\) has a bounded small exponential moment as well.

In summing the \(G_e\), the coefficients of the first normalized bound over all boxes covering a fixed interval sum to \(O_d(1)\). Indeed the assigned scales are geometrically separated with bounded multiplicity. A larger move of size \(x\) contributes at most \(O_d(1)\), since the relevant sum of coefficients is \(\sum_{P(i)\ge |x|}|x|/P(i)\). A forced move contributes the same bound individually. Independence on disjoint ordinary intervals therefore gives a small exponential moment of \(\sum_eG_e\) whose logarithm is \(O_d(n_0)\), uniformly after fixing the forced choices.

There are \(o(N)\) rows. By concavity, violation of Equation (54) forces \(\sum_eG_e=\omega(N+V)\), uniformly for fixed \(\varepsilon\). The exponential bound thus costs \(\exp(-\omega(N+V))\), before any of the finer tests is imposed. It absorbs arbitrarily large fixed multiples of the code entropy in Equation (52). Conversely, composition counting for nonnegative integer bins with the retained total bound gives \(\exp(o(N+V))\) choices.

For a forced group, its time has \(O(r(i_0))\) choices and its size must exceed \(P(i_1)\). The logarithmic Cauchy tail in Equation (24), together with comparability of the logarithms in our scale range, gives \(C\exp(-cg(i_0-i_1))\) after summing both time and size. Sharing partitions do not require a permutation of all row labels. Choose the first label of each group in a fixed order, then its highest probe and a bounded-size mask of probe labels at each sparse level across its span. This uses \(O_d(1+\text{span in logarithmic units}/L)\) digits per group. Part of the exponential span payment sums these masks and the possible spans, leaving a total factor \(\exp(O_d(n_0))\). Distinctness constraints may be dropped when multiplying nonnegative upper bounds. This also preserves the uniform erasure-tail estimate conditional on the forced choices.

The representative is used only as a witness for the equality. We never condition on the unknown small-step state inside its box that originally furnished the link, and we pay no sum over those states. The same argument discards events exponentially small in a positive power of the smallest time scale, which will be used for allocation imbalances below. ◻

Proof. At a fixed level, choose a sufficiently separated subcollection containing a fixed fraction of the irregular boxes. Each selected box exhibits a pair of full leaves at a distance between the two regularity cutoffs. Starting with a maximum-cardinality packing of the leaves at separation \(200r(j)\), pass to separation \(cR^{-4}r(j)\). The selection can be made sparse enough that, for each pair, at least one new point may be added while retaining the old packing. Thus the number of irregular boxes is bounded by a constant times the difference of these two packing counts. Sum over levels. The two separation scales differ by a fixed number of \(R\)-adic levels, so the differences telescope and give \(O_R(n_0)\). This also bounds the number of nonregular blocks.

First apply the random translation to the true witness times before replacement by \(\bar q\). A regular tight cluster crosses a grid division with conditional probability \(O(R^{-4})\) and then occupies at most two boxes. The clusters are determined before the translation. Their active counts are bounded by \(m_j\), and the irregular contribution has the deterministic bound just proved. Markov’s inequality gives Equation (55) for these true positions, with the weaker \(O(R^{-1})\) loss.

Now apply the packing bound again to the complete representative set. Consider a regular representative block split over two boxes. Unless the level is within a bounded distance of the encoding terminus of a replaced entry, the corresponding true positions remain close and in those same boxes. They therefore belong to one true full-gap block. The representative split is already paid by the true box-minus-block excess, because a box cannot contain anchors of two distinct full-gap blocks. Exceptional levels near encoding termini cost \(O_g(n_0)\). This proves the displayed bound for the witnessed representatives, without asserting it for unrelated forced choices. ◻

Proof. For a row attached at depth \(i\), imprecision can persist only over a logarithmic length bounded by \(C D_e+O(\sqrt L+1)\), in view of Equation (53) and the scale ratios. Sum over rows and use Equation (54). Since there are \(O_d(N/L)\) rows, their \(O(\sqrt L+1)\) contribution is \(o(N)\). This proves Equation (56).

For a precise interblock row, the true interval has the required outward wall, and replacement of its endpoint by its representative changes time by less than \(r(j+1)\). The shell slot between, for example, \(4r(j+1)\) and \(r(j)\) lies strictly between its two true endpoints. In fact precision and the \(e^{\sqrt L}\) part of the tolerance force the terminal-box length to be \(o(r(j+1))\). The regular center is within \(r(j)R^{-4}\) of the representative. Thus the wall test on that slot has at most a fixed multiple of \(P(j+1)\) additional allowance. This reasoning also applies when the representative lies immediately before or after a large move.

A covered singleton cannot have a near-link witness: its far representative would lie outside the tight block and would connect it to another node. The other alternatives of Lemma 134 give precisely the stated tests. Shifting from its base to the chosen regular center is absorbed in the inner core.

For \(gj\le N-O(1)\), all bases fit in at most one full-gap block. This contributes \(CN\) to the uncovered count. Above that range there are at most \(d\) singleton components, because each contains a base. The unprobed portions have logarithmic lengths \((a_+-1)N\) and \((b-b_-)N\). Sparse-edge omissions, the common phase, and unsuccessful probes remove only \(o(N)\) additional log length. This proves both bounds in Equation (57).

Finally, fix ordered tie-breaking conventions for the regular center, the row charged at an incident node, and the qualifying base at a singleton. These choices are determined once the tags, bins, and forced positions are specified. Codes that do not admit the necessary tests contribute zero. No arbitrary new bit assignment at every occupied node and level is included in the count. ◻

Proof. Set \(F(s)=(1+s)^{-\lambda}\) for \(s\ge0\) and \(F(s)=0\) for \(s<0\). This is eventually superharmonic for the signed walk. To verify this, divide its one-step expected increment by \(F(s)\) and accelerate time by order \(s/\log s\). The density equivalents in Equation (24) give, up to a positive constant, \[ \begin{split} \operatorname{PV}\int_{\mathbb R} \frac{(1+t)^{-\lambda}\mathbf 1_{\{1+t>0\}}-1}{t^2}\,\mathrm dt &=-1+\int_0^1 \frac{z^{-\lambda}+z^\lambda-2}{(1-z)^2}\,\mathrm dz\,<0. \end{split} \tag{59}\] At \(\lambda=1/2\) the last integral equals one, as follows by the substitution \(z=u^2\). Its integrand is strictly increasing with \(\lambda>0\), so the strict sign follows for \(\lambda<1/2\). The passage from the walk to the integral is justified at small increments by the bounded signed truncated drift and a Taylor remainder. Near the landing singularity and at an exact landing at zero, the point-mass bound \(C\log j/j^2\) gives the required integrable domination. Away from these regions the density equivalents and the tail bound apply directly.

Start at height \(x+w\) and stop on first descent to height at most \(2w\). Before stopping, the heights are large enough for superharmonicity. Nonnegative optional sampling shows that the probability of landing nonnegatively is at most \(C F(x+w)/F(2w)\le C_\lambda(w/x)^\lambda\). A record in the requested depth interval requires such a landing. For several separated target depths, restart at each successive first descent. The starting distance for the next target is comparable to its gap \(d_i'\), because the neighboring widths are small compared with that gap. Multiplication of these conditional bounds proves the second assertion.

For application to both sides, generate two independent infinite signed-move sequences and interleave them using independent fair side choices. Ignoring the segment stopping rule for the necessary record events leaves the two thinned sequences independent. This does not assert independence of later placement histories from the segment; those histories will be handled conditionally. ◻

Proof. Condition on the main word, its seams, the code, and the representative positions, without inspecting filling interiors. The tags determine at most \(O_d(n_0)\) relevant paths, including predecessors needed to place them. These are fixed indices in the prescribed seed schedule: an unavailable start is skipped without renumbering any later seed. A tag requiring that skipped path contributes zero. Group requests in each filling \(J\). At the start of a path, its original-label positions, long-arc availability, and directed depths are known from earlier operations. If those conditions do not hold, the event contributes zero.

On one side of a path there are only \(O(d)\) distinct label prescriptions, regardless of how many scales charge them. At a fixed level \(j\), distinct regular singleton requests are separated in directed depth by \(\gg_{d,R}P(j)\) and are that far from zero. Indeed they have a common probe scale and the long mutual arc separations of Lemma 134, with the arcs still pending in the starting polygon. An origin is charged at most once at that level. The indicated constants need only exceed sufficiently large constants depending on \(d,R\).

For each obstruction depth retain its smallest required width \(w=CP(i)\). Process these depths in increasing width, deleting a candidate if it is within \(M_1\) times its width of an already kept depth, where \(M_1\) is fixed and large compared with \(R\). Then order the retained depths spatially. Their gaps satisfy the width separation required in Lemma 139. At level \(j\), each requested charge is assigned to a retained depth whose width is at most \(P(j+1)\) and whose distance from the requested depth is at most \(M_1P(j+1)\); the omitted levels near a probe ensure this assertion. If its preceding gap is less than \(P(j)\), move to its predecessor and repeat. Every representative encountered still has width at most \(P(j+1)\). The separation between charges and from zero, and the bounded number of prescriptions per side, ensure that distinct arriving charges terminate at distinct retained depths with preceding gap at least \(P(j)\).

Consequently the sum \(\sum_i\log(d_i'/w_i)\) pays at least \(g\) per obstruction charge, summed over all levels, up to harmless rounding constants per path. Lemma 139 gives the corresponding obstruction gain \(\lambda g\). For origin bits or singularities, the one-sided version of the shell estimate and nested integration proved below in Lemmas 144 and 145 gives \(\lambda g-O(1)\) per level. That special case has only a forward window about time zero, an augmented iid path, and no forced moves or smoothing chunks. Its proof is independent of the present auxiliary estimate. The balanced trajectory likelihood costs a bounded factor per path.

For a path bearing both kinds of tests, its joint probability is at most the smaller of their separate upper bounds and hence at most their geometric mean. It therefore pays \(\lambda g/2-O(1)\) per auxiliary charge of either type. The \(O(1)\) per origin-bit shell may be chosen bounded as \(R\) grows. Constants per prescription and per path give the stated \(\exp(O_{d,R}(n_0))\) factor.

Finally integrate the fresh polygon interiors backwards in their prescribed exploration order. Keep the indicators of later reachability and placement constraints until their starting histories are known. Each bound is uniform over the applicable starting histories, and the number and tags of the charges on each grouped path have already been fixed by the main data and representative positions. Thus the conditional estimates multiply without assuming independence of those histories. ◻

Proof. For one coordinate, the stable lattice local limit estimate on a comparable number of signed moves gives the uniform point-mass bound \(C/a_\ell\). Let \(B=B_\ell\) be large and fix a sufficiently small \(\zeta>0\). If a move exceeds \(B^{1-\zeta}a_\ell\), then, conditioning on all other moves and summing over its possible time, its contribution to any specified displacement is at most \[Cs\sup_{j>B^{1-\zeta}a_\ell}\frac{\log j}{j^2}.\] Because \(a_\ell\) is comparable to \(s\log s\), this is bounded by \(C_\beta a_\ell^{-1}B^{-1-\beta}\) after choosing \(\zeta\) small relative to \(1-\beta\). Logarithmic losses are absorbed by the strict power margin.

If there is no such large move, an oscillation of order \(Ba_\ell\) requires an oscillation of that order on at least one half of the chunk. Maximal Bernstein, the truncated second moment, and the truncated drift bound make that event smaller than any fixed inverse power of \(B\). Apply the local limit estimate to the independent other half to retain a factor \(C/a_\ell\). This gives the desired point-mass and bin bound in the complementary case.

Given the allocation counts, or the allocation locations, the two thinned signed sums and their ranges separate. Different chunks also use independent increments, proving the product bound. Atypical allocations are exponentially unlikely in the chunk length. The smallest chunk length is a positive power of the large parameter, so Lemma 136 discards their total contribution before the finer estimates. ◻

Proof. Put \(u_e=\log w_e\) and \(v_\ell=\log a_\ell\). Order rows by increasing \(u_e\) and columns by decreasing \(v_\ell\). Elimination using only earlier rows added to later rows, and earlier columns added to later columns, reduces \(M\) to a partial permutation matrix up to nonzero pivot factors. Explicitly, pivot on the earliest nonzero entry in the first nonzero row, clear its later row and column entries, and continue on the remaining ordered lists. These operations preserve the ranks of all northwest prefix submatrices. A pivot at \((e,\ell)\) contributes one to the integrand exactly on \(u_e\le t\le v_\ell\). Hence \(H_M\) is the sum of \((v_\ell-u_e)_+\) over the pivots.

To pass back to a minor of the original matrix, introduce a parameter \(A\to\infty\), and multiply row \(e\) by \(e^{-Au_e}\) and column \(\ell\) by \(e^{Av_\ell}\). The triangular multipliers in the elimination stay bounded after this conjugation: each earlier-to-later row operation receives a factor \(e^{A(u_{\rm earlier}-u_{\rm later})}\), and each corresponding column operation receives \(e^{A(v_{\rm later}-v_{\rm earlier})}\). Both exponents are nonpositive. Matrix dimensions and coefficients are fixed before \(A\) increases.

The minor of the reduced scaled matrix consisting of the positive-profit pivots has magnitude a nonzero constant times \(e^{AH_M}\). Cauchy–Binet applied to the triangular products bounds it by a constant times the largest scaled nonzero minor of the original matrix. Taking logarithms, dividing by \(A\), and letting \(A\to\infty\) gives the claim. If there are no positive-profit pivots, the empty minor suffices. No integrality of the elimination multipliers is asserted or needed; integrality will be used only for the selected original minor. ◻

Proof. Choose rows and an equal number of columns forming a nonzero minor of the original integer matrix. Fixing unselected columns gives at most \(\prod_{\ell\notin J}CB_\ell a_\ell\) possible integer values in their bins. For selected columns, the row slabs contain at most \(\prod_{e\in I}Cw_e\) integer points. To see this, enlarge each integer point to its unit cube. The original matrix has entries \(0,\pm1\), so each row width grows by at most \(O(n_0)\), which is negligible compared with \(w_e\) in our scale range. Change variables using the selected minor. Its determinant is a nonzero integer, of absolute value at least one. The resulting slab volume is bounded by the product of the enlarged widths.

Multiply these counts by Equation (60). The selected columns and rows contribute the exponential of minus their profit. Each unselected column retains \(B_\ell^{-\beta}\); selected columns retain at least that decay as well. The constant cost is exponential in the number of rows and chunks, hence \(\exp(O(n_0))\).

Apply Lemma 142. For \(t\in[\log P(j+1),\log P(j)]\), all precise rows and all gap columns separating the full time blocks are available. Those columns allow independent shifts in both coordinates of every full-gap block, except possibly the first. In the graph on active nodes, a forest has \(v_j-c_j\) independent connecting rows. Independence holds even when edges use different coordinates: remove a nonroot leaf and use the coordinate appearing in its unique incident row. If a block has no available shift, take it as root. Removing the \(l_j\) imprecise rows loses at most \(l_j\) units of rank. The threshold rank is thus at least \((v_j-c_j-l_j)_+\). The logarithmic width of each scale band satisfies \[\log P(j)-\log P(j+1) =g+\log\frac{\log r(j)}{\log r(j+1)}\ge g.\] The gaps greater than \(8r(j)\) leave a fixed margin for the middle chunks and their integer rounding. Summing these rank lower bounds over the bands proves Equation (62). ◻

Proof. The allowance at the inner endpoint of a shell bit is at most \(D_*+CP(j+1)\). When \(D_*/P(j+1)\gtrsim R\), ordinary maximal \(\theta\)-moments suffice. Indeed the right side of Equation (64) dominates both the constant term and \((D_*/P(j))^\theta\), since \(\theta>2\lambda\ge\lambda k'\).

For the remaining values, fix \(R\) and pass to the symmetric Cauchy limits of the shell increments. Their durations, in units of \(r(j)\), are bounded above and below by positive constants. Distinct coordinates and the two time directions separate in this limit. For a symmetric Cauchy process \(X\) and \(0<u\le C\), the probability of remaining above \(-u\) for a unit time is at most \(C_\lambda u^\lambda\). Here is a direct way to obtain the exponent used. At an independent exponential time of rate one, the classical Wiener–Hopf identity (Bertoin 1996, VI, Theorem 5 and Corollary 10) gives the logarithm of the Laplace transform of the supremum as \[ \int_0^\infty e^{-t} \mathbb E\!\left[(e^{-aX_t}-1)\mathbf 1_{\{X_t>0\}}\right] \frac{\,\mathrm dt}{t}. \tag{65}\] The Cauchy scaling \(X_t\overset{\rm law}=tX_1\) bounds this expression by \(-\lambda\log a+C_\lambda\) for every \(\lambda<1/2\). To see the coefficient, integrate over \(t\) between order \(a^{-1}\) and one: the integrand in the expectation approaches \(-1/2\), while the small- and large-time remainders are controlled after choosing any strict exponent \(\lambda<1/2\). A Laplace bound with \(a=u^{-1}\) gives the exponential-time survival estimate. Survival to deterministic time one is bounded above by a constant times exponential-time survival, by integrating over exponential times in \([0,1]\). Symmetry changes the supremum statement to the stated lower-wall statement.

We also need its oscillation moment on the same survival event. For \(z\ge1\), the probability of survival together with a unit-time maximal oscillation exceeding \(z\) is at most \(Cu^\lambda/z\). Decompose time into \([2^{-m-1},2^{-m}]\), \(m\ge0\). A large total oscillation requires an oscillation of order \(z2^{-\eta m}\) on some such interval, where \(\eta>0\) is small. The symmetric maximal inequality gives cost \(O(2^{-m(1-\eta)}/z)\) there. Survival before that interval has probability at most \(C(u2^m)^\lambda\) and depends only on earlier increments. Multiplication and summation are allowed because \(1-\eta-\lambda>0\). Other independent bits contribute their own survival factors. Integrating the resulting \(1/z\) tail gives the \(\theta\)-moment bound, since \(\theta<1\). An untested shell coordinate can be integrated independently.

Apply these estimates with \(u=C(1+D_*/P(j+1))/R\). If the inner bound is fixed, each survival process starts at its own shell’s inner endpoint, with the left process reversed when required. The wall measured from the leaf implies survival for these shell increments with the stated allowance. The moment of the shell maxima under all bits is at most a constant times \(u^{\lambda k'}\). The transmitted term \((D_*/P(j))^\theta\) obeys the same desired upper bound. Because \(\theta>\lambda k'\), these observations give Equation (64) for the limiting processes.

For the walks, enlarge positive allowances by a fixed factor before using convergence. Their maximal \(\theta\)-powers are uniformly integrable by the tail, truncated variance, and truncated drift bounds in Equation (24). At fixed \(R\), any sequence of bounded input allowances has a convergent subsequence, so the upper estimate is uniform over those allowances. The unbounded-input case was already handled by ordinary moments. This proves the estimate with constants bounded as \(R\) increases, followed by sequence parameters sufficiently large depending on \(R\). Finally a singularity requires \(D_*\ge cP(j)\); ordinary shell moments then give the same bound with factor \(R^{-\theta}\). ◻

Proof. Use an upper bound for each inner-core oscillation that depends on chunk increments only through their bins. Namely, sum the bounds \(D\) from directly enclosed tested windows, the binned chunk maxima outside those windows, the absolute forced sizes there, and the maximal range of the concatenation of all remaining ordinary inner increments outside these holes. This sum dominates the true oscillation. The last concatenated range has one ordinary bounded normalized moment at scale \(P(j+1)\), because its total step count is at most a constant times the inner-core length.

Apply Equation (64) successively to the nested windows. It passes a child weight \(1+(D_{\rm child}/P(j_{\rm child}))^\theta\) to the next integration, charges a factor \(1+CB_\ell^\theta\) for each chunk at its first enclosing core, and charges \(1+C(|x|/P(j+1))^\theta\) for each forced move at its tightest enclosing core, if there is one. A chunk’s length is bounded by a constant times that core’s length at the first enclosing fit. Since \(\theta<1\), the fractional moment of a sum is bounded by the sum of the fractional moments; passing between this sum and the corresponding nonnegative product costs only constant factors per child, window, and nuisance contribution. The total extra shell constants therefore have logarithm \(O_{\lambda,\theta}(V/g)\).

For clarity, equality smoothing is performed in the same calculation, not on a separately conditioned shell law. First fix codes, forced choices, and bins. Replace the original tests by the necessary enlarged tests just described, which see chunks only through the bins. Conditional on nonchunk increments, extract the integer equality profit and the point-mass/bin weights of Proposition 143. The remaining nonchunk increments still have their original iid integration. Integrate the outer shell inequalities, which leave the displayed child weights, and proceed inwards. Every dyadic bin is then summable, because \(B_\ell^{-\beta}(1+CB_\ell^\theta)\) is summable over upper dyadic bins when \(\theta<\beta\). The forward origin case is exactly the same nested integration with no equality holes or forced moves and hence does not invoke Lemma 140.

It remains to sum forced moves with the propagated moments. For a forced group with finest and coarsest depths \(i_0,i_1\), logarithmic comparability over the tested range bounds its propagated factor by \[C_{g,\delta}\left(1+\frac{|x|}{P(i_0)}\right)^\theta \left[ \max_{\substack{j>i_0:\\ \text{its time lies in a }Cr(j)\text{ active-box neighborhood}}} \left(\frac{r(i_0)}{r(j)}\vee1\right) \right]^\theta.\] The empty maximum is interpreted as one. The size moment over \(|x|>P(i_1)\) still has an exponential span payment: before the time sum it is bounded by \[r(i_0)^{-1}C e^{-cg(i_0-i_1)}\] for some \(c>0\). This follows from the logarithmic Cauchy tail and \(\theta<1\); the strict moment inequality leaves a positive span exponent.

In the normalized time-position sum over the finest assigned box, the expectation of the second factor is at most \[ C\left( 1+\sum_{j>i_0} R^{-(1-\theta)(j-i_0)}m_j \right). \tag{66}\] Indeed the union of enlarged active-box neighborhoods at level \(j\) has at most \(Cr(j)m_j\) integer positions. This count is uniform in all other exact representative choices. Multiplying its relative size by \((r(i_0)/r(j))^\theta\) gives precisely the geometric term in Equation (66).

The majorants used in the different forced-position sums depend on separate time variables. To see this explicitly, for the fixed preliminary code let \(U_j\) be the union of the \(Cr(j)\)-neighborhoods of its occupied active level-\(j\) boxes. These sets and \(m_j\) are fixed before any exact forced position is summed. Every tested center belongs to one of those boxes, so a forced time \(q\) in its inner core belongs to \(U_j\). The position factor for a group of finest depth \(i_0\) is consequently bounded by \[F_{i_0}(q)=\max\left\{1,\sup_{j>i_0} R^{\theta(j-i_0)}\mathbf 1_{U_j}(q)\right\}.\] This depends only on that group’s time and the fixed code. Dropping distinctness and other admissibility restrictions in the nonnegative upper bound makes the sums of products factor over groups. The bound \(|U_j|\le Cr(j)m_j\) then gives Equation (66) for each factor.

There are \(O_d(n_0)\) groups and bounded multiplicity of their \(i_0\) values at each level. Sum the geometric convolution over these values, then use concavity for the logarithm of its product. Its logarithm is bounded by \(O_d(n_0\log(1+C V/n_0))=o(N+V)\). Use the remaining exponential span payment to sum sharing partitions exactly as in Lemma 136. These integrations use deterministic total exponents whenever Equations (55) and (56), the test coverage, and the stated separations hold. They do not require the exact cluster shapes to be independent of forced positions. This completes the simultaneous estimate. ◻

Proof. Suppose it fails along a large-parameter sequence. As in Section 10, choose an interval with more than \(\exp(\tau(b-1)N)\) bad grid times on an event of probability at least \(e^{-B'N}\). Condition on that event and sample \(d\) bad times with replacement. Equation (48) shows that an event of lift probability bounded below has counting-measure mass at least a fixed positive constant times \[ \exp\{d\tau(b-1)N-B'N\}. \tag{67}\] The constant \(B'\) can absorb the number of intervals; it suffices to obtain a contradiction for arbitrarily large fixed \(B'\).

Impose the high-lift-probability requirements of Lemma 134 and the random grid restriction in Equation (55). Discard the negligible bin and allocation exceptions of Lemmas 136 and 141. We estimate the remaining mass by summing its codes under the original law.

Main shells charge \(\lambda g\) per directed bit and \(\theta g\) per singularity. On nonsingleton nodes these charges cover all nodes except irregular ones and at most \(2l_j\) endpoints of imprecise edges. On a qualifying regular singleton, two bits or a singularity give at least \(2\lambda g\), while a main bit and its auxiliary test give at least \(\frac32\lambda g-O_{\lambda,\theta}(1)\) by Lemma 140. Thus the latter is a common lower gain for every covered singleton.

Combine these charges with equality smoothing and the joint integration. The negative logarithm of the integrated mass for one code is at least \[ \begin{split} g\sum_j\left[ v_j-c_j+\lambda v_j+\frac32\lambda u_j^{\rm cov} \right] &-C(\varepsilon+R^{-1})(N+V)\\ &-O_{\lambda,\theta}(V/g)-o_d(N+V). \end{split} \tag{68}\] Here irregular-node losses are \(O_R(n_0)\), imprecise-edge losses are controlled by Equation (56), and Equation (55) replaces node counts by box counts at the displayed cost. Constants multiplying \(\varepsilon\) can be absorbed by reducing its choice. The auxiliary separation and precision assumptions restrict only the applicable histories for the specified singleton tests; no further choices are introduced in summing the code.

Subtract Equation (68) from the entropy in Equation (52). At the limiting bookkeeping value \(\lambda=1/2\), the main expression is \[g\sum_j\left[ -\frac{v_j}{2}+c_j +\frac14u_j^{\rm cov}+u_j^{\rm uncov} \right].\] Each component contains a base, so \(c_j+u_j\le d\). Also a nonsingleton component has at least two nodes. Hence \[\frac{v_j}{2}-c_j \ge \frac12(n_j-2d)_+.\] Since \(gJ_0=bN+O(g)\), Equations (55) and (57) yield the following upper bound for the logarithm of the total retained mass at any admissible count \(V\): \[ \begin{split} &CN+d\bigl[(a_+-1)+(b-b_-)\bigr]N +\frac14d(b-1)N\\ &\qquad-\frac12(V-2dbN)_+ +\varepsilon'(N+V)+o_d(N+V). \end{split} \tag{69}\] There are polynomially many possible counts, so their summation adds only a small-order term. For \(\varepsilon'\) small, the negative excess-\(V\) term also controls arbitrarily large \(V\); maximizing over \(V\) costs only the displayed errors at \(V=O_d(N)\).

The error \(\varepsilon'>0\) can be made arbitrarily small by first taking \(\lambda\) close enough to \(1/2\), choosing compatible \(\theta,\beta\), then taking \(R\) large and the bin tolerance small, with a sufficiently slow sparse stride. The absolute \(C\) in the \(CN\) term is independent of \(d\). Choose \(a_+-1\) and \(b-b_-\) small relative to \((\tau-1/4)(b-1)\), then \(d\) large relative to \(B'+C\), and finally all remaining errors sufficiently small. The coefficient of \(N\) in Equation (69) is then strictly smaller than \(d\tau(b-1)-B'\). This contradicts Equation (67) on the requirements having lift probability bounded below, including the independent random shifts. Their probability can be made arbitrarily high by the prescribed choices. The bad-grid estimate follows. ◻

Proof. Proposition 146 supplies the estimate required by Proposition 127. The latter gives the all-endpoint modulus and macroscopic diameter bounds. Its temporary upper perimeter cutoff \(p_*=C_*B_n\) can be replaced in these statements by \(B_n\): for fixed \(C_*\), eventual adjacent-scale bounds from Proposition 116 keep \(\mathcal B(p_*)/\mathcal B(B_n)\) bounded. The cutoff and other truncations can then be exhausted in probability.

On a common profile extraction, ordinary open patches have the same deterministic lower and upper factor by Proposition 116. Endpoint control extends their equality to every vertex pair as in Proposition 127. In particular a bounded-cost sequence of paths cannot acquire an additional limiting distance at an exceptional endpoint. The factor is strictly positive and finite, and is deterministic on the chosen profile extraction.

All estimates first made under restricted central intensities transfer to the exact finite sphere law by the volume local limit domination and counting exhaustion in Proposition 55 and Corollary 56. The common conditioning kernels are retained at fixed sphere volume by Proposition 90. Thus the endpoint comparisons join the area and loop comparisons on the same probability space. Proposition 103 identifies the actual discrete conformal uniformization and gives vanishing flag mesh there.

We spell out the distance topology. Every point of the sphere lies in the image of a flag, and each flag contains an original primal vertex. Vanishing mesh therefore makes the embedded primal vertices spatially dense. For every limiting pair \((z,w)\), choose approaching vertices. The endpoint modulus and the upper and lower comparisons identify their distance as \(cD_h(z,w)\). Conversely, for any approaching sequence of vertex pairs, the same modulus and comparisons rule out an additional limiting distance value. Diameter tightness prevents escape in the third coordinate. These are precisely the two inclusions needed for Hausdorff convergence of the full compact distance graph.

Area convergence is weak convergence in these coordinates. For loops, the full-tour comparisons and count exhaustion give partial bijections matching every loop above each positive spherical diameter cutoff, with multiplicity one and all descendants retained. Their uniform individual-loop errors tend to zero; unmatched loops are discarded only below the corresponding vanishing cutoff. This is the loop topology of Theorem 2. Finally, the three-point normalization in Proposition 103 is the area-marked law of Theorem 1, with the independent nested CLE already identified by the cutting and unbiased comparison. All the assertions therefore hold in one common joint law. ◻

Proof. For each of the first two normalization samples, take the original primal vertex of the flag containing it. Samples on flag edges have probability zero, so this is unambiguous almost surely; any fixed convention handles that null set. Let \(Y_n\) be the original all-edge graph distance between the two selected vertices. Set \[Z=D_h(0,1),\qquad t_*=\mathbb E[\tanh Z].\] The continuum metric is finite and separates distinct points, so \(0<Z<\infty\) almost surely and \(0<t_*<1\).

Along any further extraction of Proposition 147, the two selected vertex images approach \(0\) and \(1\), since each lies in the flag containing its normalization sample. Joint mesh and distance-graph convergence imply \[ Y_n/b_n\ \Longrightarrow\ cZ. \tag{70}\] In particular \(\mathbb P(Y_n=0)\to0\) through the full sequence: otherwise an offending subsequence would admit a further extraction contradicting the positive limit in Equation (70).

For all sufficiently large \(n\), the function \(F_n(a)=\mathbb E[\tanh(aY_n)]\) is continuous and strictly increasing for \(a>0\), starts at zero, and tends as \(a\to\infty\) to \(\mathbb P(Y_n>0)>t_*\). Thus there is a unique deterministic \(a_n>0\) satisfying \[ \mathbb E[\tanh(a_nY_n)]=t_*. \tag{71}\] Assign arbitrary positive values to the finitely many initial exceptions. Coincident selected vertices at some finite indices cause no problem for this argument.

On an extraction with factor \(c\), bounded weak convergence in Equation (70) gives, for each fixed \(u>0\), \[\mathbb E[\tanh(uY_n/b_n)]\longrightarrow\mathbb E[\tanh(ucZ)].\] The limiting function is strictly increasing. Bracket \(1/c\) by any two fixed positive values, one below and one above it, and use Equation (71). It follows that \(a_nb_n\to c^{-1}\) on that extraction. This uses no convergence of unbounded distance moments.

For any fixed permitted \(\sigma>0\), Equation (46) with \(P'=1\) gives \[\mathcal B(P)\ge \frac{\mathcal B(1)}{K_\sigma}P^\sigma.\] Since \(B_n=(2/\pi)\sqrt n\log n\to\infty\), also \(b_n\to\infty\). Every sequence has a further extraction with \(a_nb_n\to c^{-1}\in(0,\infty)\); consequently \(a_n\to0\) through the full sequence. Multiplying the limiting distance graph on that extraction by \(a_nb_n\) removes \(c\). All resulting subsequential joint laws are the same law specified in Theorem 2, so the convergence holds through all positive integers.

The calibration changes only the deterministic graph-distance unit. The conformal structure remains that of the specified equilateral flags, and the metric remains the all-edge distance on original primal vertices. Auxiliary sampled topological positions have served only in the comparisons. The area, complete nested loop collection, and vanishing flag mesh remain in the same joint law throughout. ◻