Spectral convergence for critical FK–Ising planar maps

Proof. For a nonloop edge with endpoints u,v, the two incident half-edge contributions to -\ip{f}{L_ng}_{L^2(\mu_n)} are

The two half-edges of a loop contribute zero. Summing proves both operator identities. A function of zero energy agrees across every nonloop edge, and the map is connected, so it is constant.□

Proof. Taking complements across dual edges exchanges A and A^*, and

The dual operation is invertible, including its root convention, and keeps the number of edges. It therefore preserves the normalized finite law. The flag correspondence is (v,e,F)\leftrightarrow(F,e^*,v). It preserves the equilateral triangles and their incidence gluings, including every repeated occurrence. Drawing a primal edge along its vertex-to-midpoint sides and a dual edge along its face-to-midpoint sides proves the crossing assertion in each flag neighborhood. Conditional on the common metric surface, the same three points are independent uniform area samples in both marginal experiments. Genus-zero three-point normalization then gives the last assertion, allowing complex conjugation if the surface orientation is reversed.□

Proof. Bounded convergence shows that F_Y(r)=\EE[\arctan(Y-r)] is continuous with limits \pi/2 and -\pi/2 at the two ends of the real line. It is strictly decreasing because the integrand is strictly decreasing. The zero exists and is unique, and the shift identity follows directly. If Y_j converges in distribution to Y, then F_{Y_j}(r)\to F_Y(r) for every fixed r. For each \varepsilon>0, the two values at \mathfrak m(Y)\pm\varepsilon have opposite strict signs; eventually the same is true for F_{Y_j}. Its zero is between them, proving continuity.

Apply this fact to Y_n=\log(b_n\Delta_n). The continuous mapping theorem applies because \Delta>0. Shift equivariance gives

□

Proof. The canonical limit [30, Theorem 1.1] gives weak convergence of (\phi_n)_*m_n, Hausdorff convergence of the full distance graph with a deterministic scale b_n\to0, and vanishing maximum flag diameter in the same coordinate. The third coordinate of the graph uses the uncompactified distance, so its maximum converges to \Delta=\diam(\Sph,D_h). This diameter is finite and strictly positive. Replacing the graph diameter by \Delta_n=\max\{1,\diam(V_n,d_n)\} does not affect this limit because b_n\to0. Lemma 2.3 gives a_n/b_n\to1, hence the graph convergence in (2.3) and a_n\to0. Couple each flag-area point to its primal vertex. Their images are within \delta_n; therefore (2.2) gives the weak corner-measure convergence. All flags in one incidence star contain the star’s center, which proves the 2\delta_n bound.

The duration-one area tour and its continuity come from [30, Theorem 3.4]. The homeomorphisms of [30, Theorem 4.8] track every flag and its tour representatives uniformly. The uniformization comparison [30, Theorem 8.8] has only the conformal and anticonformal three-point limits. Passing further and retaining the sign gives the uniform tracking in the canonical coordinate; uniform continuity of the limiting tour extends the cut-time statement to (2.4). These arguments apply jointly to a finite number of rooted marginals by tightness and a further extraction. Their pushed area measures are the same limit of the physical flag measures, so each tour has pushforward \mu_h.

Lemma 3.7 of [30] says that an independent quantum-area point has exactly one visit by the tour. Integrating this probability-one statement conditionally on the surface and tour gives the asserted singleton fibers for \mu_h-almost every z; a finite intersection handles all retained rerootings. Conditional uniform rerooting has tail law \mu_n by (2.2). Weak convergence of these measures, tested against products of bounded continuous functions, identifies the joint law by first taking \mu_h and then independently sampling the two roots from it. Quantum area is nonatomic, so two such roots are distinct.

The continuous projections and complete time-preimage protection are [30, Proposition 4.2 and Lemma 4.3]. The reference homeomorphisms and their validity with extra observations are exactly [30, Theorem 4.8]; the canonical comparison is [30, Theorem 8.8]. The protected bilateral sewing and its inverse-image capture are [30, Proposition 4.9]. These statements concern fixed tests with room and supply no rate for a test whose boundary changes with n.

Finally, the sphere field comparison in [31, Lemma 18.1] uses the exploration root and an auxiliary pair of area marks. Use two separate pair marginals. Their four marks are distinct almost surely, and the complements of the two pairs cover the sphere off the root. Strict chart interiors exhaust each complement. Repeating this for two distinct roots covers the entire sphere. Compactness yields a finite subcover. This argument intersects probability-one conclusions from the separate marginals; it does not strengthen either conditional field comparison.□

Proof. We first explain why the reference-word calculation permits the added coordinate. This extends the proof of [30, Lemma 5.8 and Theorem 5.9]; the fixed-chart theorem [31, Theorem 7.10] already permits arbitrary measurable observations of raw words and exact parameters.

Work on one compact stable continuity cell for the chart inputs, with frozen atoms having positive actual mass in that cell. Let \widehat e_m be the frozen microscopic exterior in the companion’s resampling calculation, with macroscopic record e_m, and let \mathsf R_m^{\widehat e_m} be its independent reference law. The actual marked conditional law on the cell is exactly

This is an identity of raw-word masses, cut and search-mark masses, and exact retained shifts, established before any observation is selected. On a sufficiently small such cell, the guards and stable comparison instructions make every prescribed flexible resolution correct for every admitted raw configuration. In the finite bridge, the protected prefix inventories and exact terminal heights also make the prescribed suffix resolutions and empty reduction automatic. Before restriction to these cells the corresponding indicators remain in the exact weight; the omitted actual mass is removed only by the exhaustion below. The remaining cell tests have continuity boundaries in the macroscopic variables. After cancelling factors depending only on the frozen exterior, the weights are bounded on the cell. Write I_m for the scaled chart input and J_m for the scaled auxiliary full heights, noise paths, and whole-contour tests kept in the calculation. Along every convergent sequence of good microscopic exteriors whose limiting frozen value e belongs to the full-measure continuity set for the chosen limiting frozen-data law, their joint reference law satisfies

Here J is the limit of J_m. The limiting weight depends only on these macroscopic inputs. These are the protected-word density and positive-normalizer conclusions of Section 5 of [30], and the resampling and macroscopic-density conclusions used in the proof of Theorem 7.10 of [31].

Adjoining Y_m^\alpha is a measurable push-forward of both sides of (3.4), so the finite identity is unchanged. Let \mathcal P_m be the sigma-field of the full permitted protected package: its raw extended blocks, durations, used-start offsets, exact forest differences, and instruction. Its rescaled intrinsic input is I_m, and the observation (3.1) is \mathcal P_m-measurable. Compactness of E_\alpha permits an extraction whose protected-package limit disintegrates as R_e^I(\dd I)K^\alpha(I,\dd y).

Let D_m be the chosen stable feasible cell in the full reference proposal. In the cell-restricted identity (3.4), globally W_m^{\widehat e_m}=\ind_{D_m}\overline W_m^{\widehat e_m}, with \overline W_m^{\widehat e_m} extended by zero off the cell and uniformly bounded after the exterior cancellation. Thus \ind_{D_m}\mathsf R_m^{\widehat e_m} is an unnormalized subprobability; the package is not conditioned on fitting. The recorded lattice and parity conventions put the retained endpoints on the feasible coset, and the uniform local limit with positive limiting density excludes dummy endpoints on the retained compact positive ranges for all sufficiently large m, by [30, Lemma 5.5] and the proof of [31, Proposition 7.9]. Together with the stable-resolution argument above, this gives an input-only limiting continuity cell D_e.

Disintegrate R_e(\dd I,\dd J)=R_e^I(\dd I)\mathsf J_e(I,\dd J), using the version furnished by the compact increment, fibre-height, and bridge kernels. For every bounded continuous \psi, along the good frozen-exterior sequences above, the source construction implies

This is an averaged conditional estimate; rare exact packages may have large error. To see the implication from the source, condition first on the package, fresh ordinary nuisance pieces, and fibre height. The exact normalized bridge endpoints differ by o_{\mathbb P}(1), under the joint reference law, from those computed from I_m, the scaled fibre height, and the macroscopic nuisance paths. Compact-uniform bridge convergence replaces their bounded continuous conditional tests in L^1; integration over the nuisance kernels and conditional Jensen give (3.6). These are the compact-cell ingredients in [30, Lemma 5.5 and the proof of Lemma 5.7] and the proof of [31, Proposition 7.9]. Here the surrogate bound is used only for protected extensions and genuinely fresh ordinary nuisance blocks. For the hard cell indicator one first uses continuous cutoffs. Approximating from inside and outside, the null limiting boundary makes the remaining reference L^1 gap vanish, first as m\to\infty and then as the cutoff neighborhood shrinks. The resulting \mathsf J_e(\ind_{D_e}\psi) is continuous at R_e^I-almost every input.

Multiplying the conditional error by a bounded continuous f(I_m,Y_m^\alpha), then using the protected-package extraction, proves the localized factorization

Full heights affect the finite observation only through the exact forest differences in \theta_m^\alpha, whose scaled values are retained in I_m; durations remain input variables. Thus no continuity of the numerical observation in the contours was used.

Take the next joint limits under these truncated laws. The corresponding weight marginal, by (3.5) and the same cell cutoff, places the weight at W(e,I,J) almost surely. Since W_m^{\widehat e_m}=\ind_{D_m}W_m^{\widehat e_m} and W=\ind_{D_e}W, the numerator and denominator in (3.4) can be integrated against the truncated laws without change. For a bounded continuous test \Phi of the retained input and output, boundedness of the weights in (3.4) now yields

Thus the weight changes the input law and leaves the conditional law of y equal to K^\alpha(I,\dd y). Including bounded continuous whole-contour tests in J, followed by a monotone class, gives the identity conditional on the complete contours and marks. The conditional-independent attachment of the remaining coordinate data preserves it. The later finite-cell exhaustion removes the cell restriction in the actual law.

For finitely many disjoint buffers, the complete reference packages, including the added outputs, are independent. Their joint weak limit is the product of their joint limits. The resampling weight for their union depends only on the combined macroscopic inputs, so (3.8) preserves the product of output kernels. Cylinder tests give the countable assertion. Exact reconstruction in a common union record makes two descriptions of the same finite observation equal eventually on activation, and this equality passes to the joint limit. For inner and outer descriptions that are only ordered, the corresponding inequalities pass with the fixed open losses instead.

For the canonical feasible-order augmentation, fix a compact protection inside a bounded real buffer, with a fixed larger imaginary buffer. The traversals of the real buffer meeting the protection form an almost surely finite family. Let o range over its feasible orders, with positive conditional weights w_o(I_{\mathrm{IG}}). These weights depend only on the unclocked imaginary input, not on the real field or the time unit. The preceding argument gives the enlarged joint kernel K_o at each order. Keep the actual-order component and sample the other components independently from their K_o’s conditional on the fields. The vector then has conditional law \prod_oK_o, with its joint array of labels intact. Independent auxiliary fills in disjoint raw-buffer packages preserve their product law. This is the feasible-order construction in Section 5 of [30], now applied to the enlarged output.

The compact continuity cells exhaust the actual generic input law. The companion’s reconstruction and guard arguments supply finite covers, positive gaps, and admissible instructions almost surely. Finitely many cells and compact parameter truncations omit at most \varepsilon of that law. Their omitted contribution to a bounded conditional-identity test is at most its bound times \varepsilon. Letting \varepsilon\downarrow0, and then diagonalizing over instructions, finite unions and coordinates, proves the simultaneous identities on activation. Only input and reconstruction events enter this exhaustion, so there is no weight depending on a new numerical success event.

A change of input density r(I,J) multiplies the limiting input–output law by that same density. Bounded approximation and truncation handle every integrable density. For changes of units use [30, Proposition 5.12] and the closure construction in [31, Section 7]. Expressing the same raw map with time unit vm changes the field, time labels, and a labelled frontier length L_\partial to

A raw incidence quantity is unchanged; a normalized output changes only by the specified new deterministic denominator. Raw cuts and incidences remain exact. Rounding a time unit changes its scaled labels by o(1) when that unit diverges. For varying labels, retain the normalized reference input law, its positive duration and shift densities, and all fixed effective units as required by the companion’s indexed construction. Equation (3.9) re-expresses one array; it does not identify differently indexed kernels at an unchanged field.

For the spatial assertion put

where V contains a neighborhood of \overline U, and O is the finite relative order. The reconstruction in [31, Theorem 7.13], and its canonical counterpart in [30, Section 5], reads from \xi the finitely many directed traversals meeting K_*, their clocked cuts, labelled frontier relations, forest differences, separating bands, and used flexible-order instructions. The containments (3.2) and the intrinsic instruction are therefore selected from \xi on activation. The full preimage and star conditions make their finite reconstruction exact eventually.

The global labels locating those blocks in \mathcal C need not be measurable from \xi. What is local is the evaluated kernel. Two realizations of the same intrinsic description use the same reference blocks, durations, shifts, and observation after relabeling. Denote that description by s(\xi), its continuous input by c(\xi), and the map on the joint label/output space that sends time labels to spatial positions, leaving intrinsic numerical values unchanged, by \Phi^\xi_{s(\xi)}. The selected spatial kernel is

which is measurable from \xi. If \alpha_*(\mathcal C) is the globally located atlas member selected by this description, then on its activation event summing the fixed-chart identities over \{\alpha_*=\alpha\} and using the tower property gives

for bounded measurable f. The same computation for products and common union records proves spatial product and restriction compatibility. This selection occurs after the limiting array has been extracted.□

Proof. Let \mathsf P_{T,U} be the law of the actual duration-T sphere real field on U, and fix \mathsf P_{\mathrm{ord},U}:=\operatorname{Law}(G_R|_U) with \overline U\Subset B_R as above. The actual-height comparison [30, Lemma 5.13 and Proposition 5.12], or [31, Lemma 3.3, as used in Lemma 18.1], gives \mathsf P_{T,U}\ll\mathsf P_{\mathrm{ord},U} for T=1. For T\ne1 the field is shifted by \gamma^{-1}\log T; a Cameron–Martin cutoff equal to that constant near \overline U gives the same local absolute continuity.

Let \mathsf P_{\mathrm{IG}} be the law of the entire independent stationary-phase imaginary field and its unparameterized directed space-filling curve, and let \Pi_h be the common conditional law of the local area marks. The full imaginary object retains the order O in (3.10), and all the corresponding finite orders for an array of tests. For such an array, \xi and \mathsf K_\xi denote its joint local input and kernel. The input laws satisfy

The local input \xi is a measurable function of (h,\zeta,\pi). Proposition 3.1 gives the same observation kernel \mathsf K_\xi on activation on both sides; the cemetery convention extends it to the activated array everywhere. If r(h) is the input density, the sphere probability of B equals

The integrand without r vanishes almost everywhere by the reference null assumption, so this integral is zero. A compact patch avoiding the exploration root has full time preimage in a strict interior of (0,T), as required for the finite-word kernel. Countable intersection gives the simultaneous statement. Repetition with fresh area-mark pairs uses each marginal separately, without conditioning on the other pairs.

Convergence in [0,\infty] to a value at most C gives an eventual bound C'>C; convergence to a finite value gives an eventual finite bound. Exact reconstruction on activation supplies the finite-graph interpretation. This deduction is made in the representation extraction and supplies no finite-scale rate or conditional probability.□

Proof. Proposition 7.16 of [31] bounds the compatible subprobability law of the entire raw retained blocks and exact shifts. A measurable push-forward preserves that inequality, proving (3.15). There is no division by the probability of compatibility. Put p_m=\mathsf S_m(B_m\cap G_m) and q_m=\mathsf F_m(B_m\cap G_m). When 0<q_m<1, the binary relative-entropy inequality gives

If q_m=0, finite entropy already gives p_m=0. Since q_m\le C\mathsf R_m(B_m)\to0, the conclusion follows. Conditional entropy integrates to joint entropy when the history marginal is unchanged; a tight entropy bound may first be truncated.

For the final assertion, write the reference input and output law as \mu(\dd c)K_c(\dd y). On a compact stable cell the ordinary completion induces a retained subprobability law

Here \lambda_c is the conditional law of omitted Brownian or first-passage filling pieces, \mathcal T is the strict compatible filling event, and a(c,v)>0 is the translated endpoint-density multiplier. Open fillability gives f(c)>0 on the inputs at issue. The multiplier depends on the input and filling, so Proposition 3.1 leaves the enlarged K_c unchanged. If B is null in this completion, then

forces K_c(B_c)=0 for \mu-almost every openly fillable c. Restricting first to f(c)\ge\varepsilon and then letting \varepsilon\downarrow0 requires no uniform filling probability. The tight absolute continuity obtained from (3.15)–(3.16) transfers this null set to the selected limit. Proposition 7.18 of [31] applies the argument in each jointly extracted limit for scale-dependent reference laws, and hence for mixtures over a viewing level. Its off-center and strict-gap hypotheses remain in force.□

Proof. We give the locality argument of [30, Lemma 8.3] for the enlarged arrays. Let \mathcal H be the sigma field of the entire real and imaginary fields. By [30, Lemma 3.7], at every deterministic planar z the whole-plane exploration has a unique preimage almost surely. For every larger neighborhood V, all visits to a sufficiently small neighborhood of z then lie in one traversal of V: otherwise properness gives a second limiting preimage outside that traversal. Success is readable in a slightly larger local buffer, since the imaginary field there reconstructs the finite directed traversal list. Shrinking the inner neighborhood exhausts a probability-one event. On this exhaustion, the small tests defining a germ need no exterior permutation; their order is the direction of that traversal. Proposition 3.1 therefore implies, for bounded measurable \varphi, that

is measurable from the field restrictions in every neighborhood of z. Here the local Poisson auxiliary marks of the kernel construction are integrated out: their restrictions to disjoint spatial sets are conditionally independent given h. This step does not involve the three canonical area samples used to normalize the sphere. At distinct z,w, disjoint sufficiently small real buffers give product conditional kernels, hence conditional independence of \varphi(Y(z)) and \varphi(Y(w)) given \mathcal H.

For a bounded real \varphi, conditional Fubini and this product identity give, for every rational disk V\Subset D,

The diagonal in V\times V has area zero. A countable disk exhaustion and Lebesgue differentiation imply \varphi(Y(z))=p_\varphi(z) for almost every z, almost surely. Fubini reverses these quantifiers at almost every deterministic point.

The point germ of the two ordinary fields is trivial. For a Dirichlet GFF, the intersection of the Gaussian subspaces generated by restrictions to shrinking disks around z is orthogonal to every smooth function vanishing near z. Logarithmic cutoffs show that a point has zero Dirichlet capacity, so these functions are dense and the intersection is zero. Decreasing Gaussian projections give triviality of the completed sigma-field intersection. The same reasoning applies to two independent fields; reducing the imaginary field modulo its period removes information. Local absolute continuity transfers the conclusion to the ordinary chart conventions, as in the proof of Lemma 8.3 of [30]. Since (3.18) is measurable in every shrinking field restriction, it is deterministic. A countable family of bounded continuous functions separating points of the value space proves the claim for Y.

For \psi(z)=az+b, the field transformation is

Proposition 5.12 of [30] identifies the same intrinsic incidence record in these coordinates and gives local equivalence of the field laws. A probability-zero or probability-one statement about Y therefore transports by its stated action, so its deterministic value obeys that action. Under the additional translation hypothesis, every bounded continuous function of the deterministic version is invariant almost everywhere under rational translations whose local buffers fit. Continuity of translations in local L^1 extends this to all translations, making each such function constant almost everywhere on the connected domain. A countable separating family gives a constant value of the germ. The rotation assertion follows with the stated conjugation action.□

Proof. Write a_\pi=(N_e(\pi))_{e\in E(\mathsf G)}, and let K be the closed convex hull of these nonnegative vectors. For t\in K and g\ge0,

Consequently \EL_{\mathsf G}(\mathcal F)\le\inf_{t\in K}\norm{t}_2^2. Let p be the point of K nearest 0. If p=0, this already gives equality. Otherwise p\ge0, and the projection inequality gives

Taking the admissible nonzero density g=p in (4.1) gives the reverse inequality. Finite convex combinations are precisely the traffic vectors of finite-support laws and are dense in K. A general traffic vector with finite coordinates belongs to K, by approximating its law by finite laws and allowing an arbitrarily small error in the coordinates. This proves (4.3).

For the port identity, first suppose a connecting walk exists. Let u|_A=0, u|_B=1, and take g_e=\abs{d_eu}. A connecting walk and these port values force g\ne0, so this density is admissible in (4.1). The telescoping inequality gives L_g(\pi)\ge1 on every connecting walk, so \EL\ge1/\mathcal E_{\mathsf G}^0(u). Conversely, for a density with b=\inf_{\pi\in\mathcal F_{\mathsf G}(A,B)}L_g(\pi)>0, let d_g(A,v) be weighted graph distance from A, with value \infty in a component not meeting A. The function

has the required port values and satisfies \abs{d_eu}\le g_e/b on every edge. Thus the energy infimum in (4.4) is at most \norm g_2^2/b^2. Take the supremum over g. If no connecting walk exists, assign a constant to each component, proving the stated convention. The equality with wired effective resistance is the Dirichlet principle for a unit-conductance network. Loop differences vanish throughout.□

Proof. On a successful shell at center c and radius r, the local normalized EL lower bound and (4.1) give an edge density g_k^{c,r}, with fixed slack, for which

Its support lies in the shell buffer. The subwalk inclusion (4.7) makes the charge valid for every traversal. The densities may be chosen measurably: a port family may be restricted to its finitely many simple paths, and a first rational density satisfying a strict near-minimum bound gives a choice from the finite incidence record.

At each radius retain the successful centers whose buffers meet U, and define

For fixed N, the edge mesh is eventually smaller than every shell margin in this finite collection. A whole supported edge can then be assigned to a point in the slightly enlarged buffer that contains it. The bounded spatial overlap at one radius bounds the number of summands on this edge by a deterministic constant; no degree bound is used. Grid counting and (4.14) give

There are at most CN radii. Cauchy–Schwarz for their sum, followed by division by N^2, proves (4.12) and the support assertion.

For the charge, choose a coordinate whose oscillation on \Psi_{m_k}(\pi(J)) is at least its diameter divided by \sqrt2, and use a subinterval between a minimum and a maximum in that coordinate. Ignore coordinate levels within CR_N of the two ends. At each remaining level, a visit of the walk lies in at least cN successful central holes. A hole of radius r accounts for a coordinate interval of length at most Cr. For every hole counted in this way the path makes a complete traversal of the corresponding buffered shell. More precisely, strict clearance and the fixed-N mesh bound place a full vertex-to-vertex shell subwalk inside J, even if an endpoint of J is in the interior of an edge. Thus each counted shell contributes its full charge in (4.14). Integrating over the coordinate levels gives

The summed density charges at least the sum on the right before the fixed constant. Divide by N and absorb constants. This proves (4.13). Its error is uniform because all projection and edge-mesh errors are uniform for this fixed configuration. Charges of disjoint parameter intervals add by their fractional edge definition; no endpoint edge is rounded up separately. This is the level-integration argument of [30, Section 8] applied to unit edge intervals.□

Proof. For every edge density, the traffic pairing is

For g=g_{k,N,U}, equations (4.12) and (4.15) bound this expectation by C\sqrt{\operatorname{Area}(U^{+CR_N})} for large k.

Take U large enough to contain K with room. Given d>0, choose a fixed successful N with CR_N<cd/16, and then take k large enough that \epsilon_{k,N,U}<cd/16. Every path interval of projected diameter greater than d/2 pays at least cd/4 in (4.13). By continuity, the endpoints of any finite collection of intervals of diameter at least d, with pairwise disjoint interiors, can be approximated from within by rational endpoints while retaining diameter greater than d/2. The supremum of the crossing counts over such rational collections is measurable, since the collections are countable. Charges add, so the simultaneous bound and (4.17) bound its expectation by C/d; formally, increase over finitely many rational collections and use monotone convergence. This also bounds the maximum count for the original intervals.

For completeness, a measurable clock gives the required tightness. First normalize the original parameter of a continuous projected path to [0,1]. Mark its successive first moves by distance 2^{-\ell} and let J_\ell be their number. Give each such mark mass 2^{-\ell}/(1+J_\ell), for every \ell\ge1, and add Lebesgue mass in the original parameter. The total mass is between 1 and 2. Its inverse clock, with a pause at each atom and normalization to [0,1], represents the same path. On a set where J_\ell\le M_\ell for every \ell, a parameter interval shorter than 2^{-\ell-1}/(1+M_\ell) cannot pass through a complete intervening pause; its image oscillation is at most 2^{2-\ell}. The crossing-count bounds permit M_\ell<\infty to be chosen so that the total exceptional probability is arbitrarily small. For each \ell, the finitely many indices preceding its eventual bound are included by increasing M_\ell. Arzelà–Ascoli and Prokhorov’s theorem give tightness and a weak limit on C([0,1],K). The endpoint separation passes to this limit.

To control variation, fix an integer J and a finite rational box union U. Let V_{J,U}(\gamma) be the supremum of the sums

over disjoint closed intervals with rational endpoints whose images have positive clearance inside U, with the empty sum equal to 0. This is lower semicontinuous on path space: each strict containment is open and each corresponding finite sum is continuous there. The simultaneous charge bound, applied before taking this supremum, and (4.17) yield

First pass to the weak path-law limit k\to\infty, with N,J fixed. Then let N\to\infty through successful configurations, keeping J fixed, and finally let J\to\infty. The finite interval sums exhaust variation in U. Applying this first to a larger confinement proves finite total variation for almost every path, and applying it in each U gives (4.16). Each path has length at least d_0, so \Lambda\ne0. Covering a planar null set by rational boxes of arbitrarily small total area, and increasing over finite subunions, proves \Lambda\ll\dd z. This proves the edge version of the flow argument in [30, Section 8].□

Proof. Continue the record-low assumption (4.9). Choose the joint extraction used above to retain, in addition, the duration-one spheres with n_k=m_k/2, their rooted dual experiments, the matching ordinary reference packages for the two sphere marginals, and the cone packages witnessing the quantile events for each reference marginal. Propositions 3.1 and 3.2 provide these matched packages, and Lemma 2.2 couples the two sphere marginals on the same physical maps with the same edge count. The high-unit and budget arguments above apply to each reference marginal. We carry out the germ argument below in each one, using its own conditional kernel for both its cone transfer and its later sphere transfer; the notation first describes one such matched pair.

Retain also the low-event indicators in (4.8). Their limiting event has positive probability. On that event the discrete indicators equal 1 eventually. Since \rho_{j_k}\to0, the default value Z_{j_k}=1 cannot satisfy the low inequality for large k. Lemma 4.1 then gives near-minimum path laws for the middle family, and trimming between the narrower bands gives squared traffic at most C\rho_{j_k}, endpoint separation at least d_0, and compact confinement. A finite outer cell cover can admit every trimmed path while keeping these two geometric properties, with strict margins. Its finite path family contains the trimmed laws for all large k, so its EL divided by \rho_{j_k} is bounded by a fixed finite constant. Let \mathsf A_{\mathrm{out}}^\alpha be the local activation event that \alpha is such an outer cover with the stated strict margins. Thus the low indicator witnesses the local event

The union is countable and has positive probability in the cone restriction. By countability and compact exhaustion of activation, one instruction \alpha, one finite M, and fixed compact confinement and positive endpoint separation can be selected on a positive-probability subevent. This is an event of the local input and its normalized numerical output. Local equivalence of the cone and ordinary field restrictions, with the common observation kernel, transfers this event with positive probability to its matching ordinary reference marginal. There the convergence Y_0^\alpha<M and exact reconstruction on activation give finite normalized EL at most M+1 for all sufficiently large k. Lemma 4.1 supplies discrete path laws in this one local family with squared traffic at most (M+2)\rho_{j_k}. These laws now live on the same ordinary extraction as the probability-one budgets of Lemma 4.3. Apply Lemma 4.4 on their common event to obtain a law \mu of nonconstant rectifiable paths and a nonzero measure \Lambda\ll\dd z.

We describe the local event witnessed by their tangents. Let v be a unit direction and v^\perp its quarter-turn. For z\in\C and s>0, set

A strict crossing has a compact subroute in Q_v(z,s), from the portion with t<-s to the portion with t>s, with its two vertex ports and all buffered confinement described by intermediate cells. For an instruction \alpha of this kind, write Y^\alpha=Y_0^\alpha for the limiting EL of its path family divided by \rho_{j_k}. Define

The unions use the countable instruction list and integer M. Whether an instruction fits the displayed open tube is a measurable condition on its limiting local input. Thus E_v(z) is jointly measurable. Deleting finitely many radii leaves it unchanged, and all supports shrink to z. Its indicator is therefore a germ in Lemma 3.4. The finite bound M may depend on the radius.

At arclength-almost every interior visit of almost every path of \mu, arclength parameterization is differentiable with a unit tangent. Choose a direction from a fixed finite sufficiently fine net so that it lies within the angular slack of this tangent. Differentiability gives a strict crossing of Q_v(z,s) at every sufficiently small s. Almost every path belongs to the support of \mu. For each one of these fixed strict crossings, choose a smaller compact subroute with additional margin. Choose nested open vertex ports P_\pm^{\mathrm{in}}\Subset P_\pm^{\mathrm{out}} around its two ends and nested confinements W^{\mathrm{in}}\Subset W^{\mathrm{out}}\Subset Q_v(z,s) around the subroute; the outer ports still lie beyond the two longitudinal levels. Cover the full preimages of the inner sets by cells lying in the outer sets. This gives one activated outer instruction. An open neighborhood \mathcal O of the path can require the corresponding subroute to visit the inner ports and remain in the inner confinement. Full-preimage reconstruction and the vanishing mesh then imply that, for all large k, every discrete path with projection in \mathcal O has a trimmed vertex subroute admitted by this fixed instruction. Since the path is in the support, \mu(\mathcal O)>0. Weak convergence gives mass at least p>0 in \mathcal O for all sufficiently large k. Restricting and normalizing that mass, and trimming its paths, bounds traffic coordinatewise by \tau_{\mu_k}/p. Its squared traffic is at most C_0\rho_{j_k}/p^2. Lemma 4.1 proves Y^\alpha<\infty for this radius. This proves E_v(z) at \Lambda-almost every such visit for some direction in the finite net.

Since \Lambda is nonzero and absolutely continuous, on the positive-probability flow event at least one of these finitely many directions satisfies E_v on a set of positive planar area. Take here the library of all finite intrinsic instructions and its common refinements. Translation by a carries the fitting condition for E_v(z) to that for E_v(z+a); rotation by R carries it to that for E_{Rv}(Rz). The same intrinsic instructions describe the transformed cells, and their edge counts and normalized outputs are unchanged. The buffered supports transform with them. These actions are therefore exact on the local arrays, while the corresponding ordinary field laws are locally equivalent as in Lemma 3.4. That lemma therefore makes the indicator for each direction one deterministic constant almost everywhere on an ordinary connected chart. The positive-area event forces the constant for one direction to be 1. Rotational covariance then gives constant 1 for every member of a chosen countable dense direction list. Countable intersection and an ordinary chart exhaustion show that, almost surely,

This assertion has so far been proved for the one network fixed at the start of the power-scale construction.

We use the sphere components retained in this joint extraction to obtain a transverse obstruction. For a rational compact disk K in a buffered ordinary patch and a direction in the countable list, failure is the measurable event

Deleting a common finite number of radii for z\in K computes the tail in (4.19) using only supports inside this patch. The failure event is therefore measurable from the local input and the activated array of normalized EL observations in Proposition 3.2, and it is null by (4.20) in each matching reference marginal, with that marginal’s own conditional kernel. Proposition 3.2 transfers the event to the corresponding sphere marginal, with the same denominator \rho_{j_k}, actual height, and traversal order. Auxiliary mark pairs exhaust the patches off the exploration root.

The two retained sphere experiments have the same law and the same number of letters per unit. The chosen network in their two marginals gives the two opposite physical networks. The transferred conclusions hold for both simultaneously. Their flag surfaces are identified, and the same three fresh flag-area marks give one canonical coordinate after composing one coordinate with conjugation if its orientation convention requires this. Let H_k^+ and H_k^- be the two reference homeomorphisms. The canonical comparison in Proposition 2.4, from [30, Theorem 8.8], permits a further extraction on which their changes to the common canonical coordinate converge uniformly to maps F^+ and F^-, each conformal or anticonformal.

Choose a point w in a canonical coordinate plane away from the finitely many omitted marks and poles, at which the two transported versions of (4.20) hold. Such points have full planar area: the limiting conformal and anticonformal maps are smooth local diffeomorphisms and preserve area-null sets. Let z^\pm=(F^\pm)^{-1}(w). Choose directions from the two dense lists whose images under DF^\pm(z^\pm) are respectively horizontal and vertical to a sufficiently small angular error. Write b_\pm>0 for the scale factors of these similarity or anti-similarity derivatives. The two germs allow all sufficiently small dyadic radii. We may choose such radii s_\pm with

They may also be chosen small enough that, for a prescribed small \varepsilon_{\mathrm{lin}}>0, each limiting coordinate change satisfies

Put s_*=\min\{b_+s_+,b_-s_-\}. If the angular error is \delta, the transverse deviation of either transformed tube is at most

and its endpoint advance in its longitudinal direction is at least s_*(\cos\delta-\sin\delta/16-\varepsilon_{\mathrm{lin}}). Choose \delta,\varepsilon_{\mathrm{lin}} so that the first bound is less than s_*/4 and the second exceeds 0.8s_*. Thus, in coordinates centered at w, every path in the first transformed family stays in \{\abs y<s_*/4\} and reaches both x<-0.8s_* and x>0.8s_*; every path in the second stays in \{\abs x<s_*/4\} and reaches both y<-0.8s_* and y>0.8s_*. Any two such continuous paths intersect, by separation in the intervening rectangle. Fix these radii, activated covers, and their finite bounds from (4.19). Uniform convergence of the coordinate changes and the strict tube margins give the same forced intersection for every pair of discrete paths for all sufficiently large k.

The two discrete EL values are at most C_+\rho_{j_k} and C_-\rho_{j_k}, with finite constants because the tests have now been fixed. Choose near-minimum path laws by Lemma 4.1, and write their traffics on the paired physical edges as \tau_k(e) and \tau_k^*(e^*). Every pair of sampled paths intersects. By Lemma 2.2, a primal and a dual edge meet only when they are paired, so for every pair of walks \sum_eN_e(\pi)N_{e^*}(\pi^*)\ge1. Fix the full coupled record, the selected families, and both path laws, including their possible dependence on the continuum data, radii, and orientation sign. Take the product of these two fixed laws. Taking expectations under this pathwise product and applying Cauchy–Schwarz gives

This is impossible. It excludes the record-low extraction for the chosen network, and the same argument applies when the initial choice is exchanged. Since each quantile sequence is deterministic, each positive lower limit is deterministic.□

Proof of Theorem 4.2. Let m_k\to\infty be any sequence of letters per time unit. Write

and pass further so that v_k\to v\in[1,4], jointly with all fixed backward rows. Proposition 4.5 makes the high barriers uniformly positive in raw EL units in those power rows. Equation (3.9) expresses the same incidence record in the m_k units: the area clock is divided by v_k, frontier heights by \sqrt{v_k}, and the real field is shifted by -\gamma^{-1}\log v_k. A backward row 4^{j_k-i} has relative label v_k^{-1}4^{-i}. Thus the aligned theorem has multiplier M=v^{-1} in the limit. The normalized field identity is, explicitly,

The raw EL is unchanged. Strict margins allow the converging ratios and the rounded units to be retained in the indexed arrays. The bounded constant shifts have uniformly bounded cutoff Cameron–Martin norms on the field buffers, as used when choosing \varepsilon above. Therefore the same high unit-reference probability applies before annular alignment. This is the field and unit change of [30, Proposition 5.12]; it re-expresses one raw array and does not compare differently indexed kernels at an unchanged field.

Apply the aligned theorem again and repeat Lemma 4.3, now with a fixed positive raw barrier, so that the denominator in (4.12) is 1. For a fixed annular template, choose slightly inset ports between a_2r and a_3r, and an outer cell confinement inside W. It catches a subwalk of every member of \mathcal A_m(z,r;\mathbf a). Cover that confinement by a finite rational box union U of area at most C_{\mathbf a}r^2. Every caught subwalk has projected diameter at least c_{\mathbf a}r. On the probability-one aligned event choose a successful, fixed N large enough that CR_N<c_{\mathbf a}r/4 and \operatorname{Area}(U^{+CR_N})\le C_{\mathbf a}r^2. For this fixed configuration, all sufficiently large discrete indices have a density charging every such subwalk by at least c'_{\mathbf a}r, with squared norm at most C'_{\mathbf a}r^2. Equation (4.1) and the subwalk property give

The constants are deterministic. The choice of a successful N may depend on the activated sample and the fixed radius, but it is fixed before the discrete limit. Narrow collars are included by choosing their positive relative gaps first.

The passage to v_k\to v preserves the quantifier over representation extractions. Start with any such extraction and retain its existing compactified local annular output limits, together with their activation inputs, in the further extraction above. Their marginal law is preserved. If an activated limit failed a fixed slack version of the deterministic lower bound with positive probability, the same failure would have positive probability in the further extraction, contradicting the bound just proved. Thus every original limiting output has that lower bound on activation. Choosing a strictly smaller deterministic constant and using compactified convergence gives the eventual finite inequality in the original representation. A countable intersection treats the chosen library; the index at which an inequality starts can still depend on its sample and its fixed test.

The local lower tests in this argument are finite incidence observations with the band inclusions (4.7). Proposition 3.2 transfers their probability-one limiting bounds to sphere patches, and convergence of the compactified observations gives the eventual finite inequalities with a fixed slack. A finite family of ordinary patches also treats a collar surrounding an isolated omitted point: choose inner patches with a Lebesgue number comparable to the radius and larger patches still avoiding the point. A through walk has a subinterval of diameter comparable to the radius in one of the larger patches. Summing the finitely many corresponding densities charges all through walks with the same kind of shape bound. The number and relative geometry of these patches can be fixed from the collar ratios. The protected enclosure of Proposition 2.4 keeps the resulting separation in the original surface. This proves part (i), including its stated sphere uses.

We turn this estimate into part (ii). Fix relative radii

In the protected finite graph containing the complete stars over \overline B(z,br) and its disk buffer, set

Part (i) for the template (\beta_0,\ldots,\beta_5) supplies a density g, normalized to charge its annular family by at least 1, with \norm g_2^2\le C. Extend it by zero outside its confined edges. Its support lies compactly in A(z;ar,br). Once the mesh is small, every walk from A_m to B_m contains a subwalk of this annular family and therefore has g-length at least 1. On the buffered graph define

where d_g=\infty on components not meeting A_m. The function is 1 on A_m and 0 on B_m. As in Lemma 4.1, \abs{d_eu_m}\le g_e, so \mathcal E^0(u_m)\le\norm g_2^2. Moreover u_m is constant across every edge on which g_e=0. The vanishing complete-star mesh and the gaps a<\beta_1, \beta_4<b put every vertex whose projected star meets \overline B(z,ar) in A_m, and every vertex whose projected star meets the exterior of B(z,br) in B_m. The support of the differences remains inside the stated compact subannulus. For a protected use, extend the constants across the interior and exterior. Full stars are retained throughout the collar, and any path from one side to the other contains the tested passage; hence no omitted incident edge introduces a new difference in this extension.

For part (iii), take the cutoff from part (ii) in the dual network, with all its nonzero differences in a compact subcollar of A(z;br,cr). Its inner constant region contains every face meeting \overline B(z,ar), and its outer constant region contains every face on the other side of the collar. Choose the subcollar with an additional fixed margin to both boundary circles. A paired primal and dual edge lie in their common flags, whose projected diameters tend to zero by Proposition 2.4; the complete primal and dual star mesh has the same property. Thus each primal edge paired with a nonzero dual difference lies, for all large indices, in a slightly larger compact subcollar of A(z;br,cr). Choose t uniformly from (0,1). For all t except the finitely many vertex values, retain each primal edge whose two incident dual values are on opposite sides of t, and orient it along the boundary of the face on the u_m>t side. Cyclic incidence at a primal vertex balances incoming and outgoing edges, so these edges form closed walks. A transverse arc from the inner region to the outer one has total signed transition count of absolute value 1. The closed walks therefore have nonzero total separating winding. Splitting at repeated vertices gives simple circuits, at least one of which surrounds the inner disk. All its edges remain in the subcollar. This reasoning also includes a primal loop and two parallel edges forming a Jordan curve; a primal bridge has a dual loop and cannot be a transition edge.

Choose one surrounding circuit by any fixed rule on the finite record. An edge can be chosen only when t lies between its two dual values. Consequently

The full face stars and the two constant margins used here are part of the protection; they ensure a closed separating boundary rather than a boundary cut off by the edge of a chart. Exchanging the two networks proves the circuit assertion in either one.

Finally, take the route in part (iv). Its compactness gives positive clearance from W^c, and its endpoints have positive clearance inside the two ports. Choose a sufficiently small common radius r_0. Starting at the first endpoint, choose successive centers at the first points of the route at Euclidean distance r_0 from the preceding center. This terminates after finitely many steps because the polygonal route has finite length. When the remaining route stays within that distance, the last circle can already be chosen wholly inside the terminal port by the endpoint clearance. The first circle is wholly in the initial port, and all circles and their narrow collars lie compactly inside W.

Use part (iii) for a circuit surrounding B(c,3r_0/4) and lying in A(c;7r_0/8,9r_0/8) at each center c. Consecutive centers have distance r_0. The two enclosed disks of radius 3r_0/4 overlap, whereas each has a point outside the other’s disk of radius 9r_0/8. Two disjoint Jordan curves would have disjoint or nested interiors, so every pair of consecutive circuits intersects. In one embedded network an intersection gives a common vertex or edge. The union of the sampled circuits is therefore connected from the first port to the last. Select a simple graph path in this union. If there are J_* circuits and \tau_i(e)=\PP(e\in\Gamma_i), the path traffic is at most \sum_{i=1}^{J_*}\tau_i(e) on every edge. Hence

The number J_* and the collar shape depend only on the fixed route and its clearances and are preserved by a common dilation. This proves part (iv) and completes the theorem.□

Proof. We first describe the numerical outputs. A fixed intrinsic cell instruction selects a finite set of counted edges from the exact incidence record. It also attaches a rational closed interval to each of finitely many cell labels. A vertex carrying several labels must satisfy all their intervals. Take the infimum of the sum of squared differences on the selected edges over all such vertex functions, and compactify its value by s\mapsto s/(1+s), with +\infty\mapsto1. This is a measurable function of a finite incidence record: the objective is a finite quadratic polynomial and the constraints are finitely many closed intervals. The list of intrinsic cell instructions and rational intervals is countable. Proposition 3.1 therefore retains their joint limits, including equality through a common union record on overlaps.

Fix V_0\Subset V_1\Subset U and \varepsilon_0>\varepsilon_1>0. On the continuum activation event, cover the full preimage of \overline V_0 by finitely many used cells whose closures lie over V_1; include the complete stars of their vertices and edges. Subdivide the cells so that the oscillation of f on each image is smaller than a fixed fraction of \varepsilon_0-\varepsilon_1. Rational intervals can then be chosen so that, for every represented vertex of that cell and all large m,

The first inclusion holds for every label at a multiply labelled vertex, so these simultaneous demands are compatible. Full preimages, star mesh, and the strict margins ensure that the selected edge set contains every edge located in V_0 and contains only edges located in V_1. The constrained vertex set contains every vertex located in V_0 and contains only vertices located in V_1.

Let s_m be the resulting infimum. The eventual exact-incidence and activation conclusions of Proposition 3.1, followed by (5.4), give

The observation s_m was extracted before this activation was selected; activation only supplies its interpretation in the spatial chart. Its limit exists in [0,\infty]. Although f,V_0,V_1 were arbitrary, every instruction just used belongs to the one countable list. Uniform continuity of f on the relevant compact sets and the separated tolerances are what allow that list to serve all continuous functions. More precisely, each instruction’s activation is defined from its local continuum cells and margins alone; it does not use f or the value intervals. After f is given we choose a suitable entry among the already activated countable entries. This is a pathwise use of their simultaneous conclusions, not a new conditioning event.

It follows from (5.5) that

For each inner V_0 choose V_1\Subset U, and choose \varepsilon_1<\varepsilon_0. Taking the limits and suprema in (5.3) gives \mathcal R^+(f,U)\le\mathcal R^-(f,U). The opposite inequality is immediate. The same sandwich shows that an alternative edge-location convention or an overlapping protected record gives the same result.

For (i), on each fixed V\Subset U and at each positive tolerance, w_m is eventually admissible in (5.2). Its energy on V is at most its energy on U. Take the liminf, then the tolerance limit and the inner-domain supremum.

For (ii), choose W' with W\Subset W'\Subset U. For each fixed \varepsilon>0, the limsup of q_m(f,W',\varepsilon) is at most \mathcal R(f,U). Choose approximate minimizers and decrease \varepsilon to zero sufficiently slowly with m. They converge uniformly on W', and their energies on W', hence on W, have the asserted limsup. This diagonal choice is made after the extracted sample has been fixed; no new conditional law is asserted.

Finally, for fixed \varepsilon>0 and all sufficiently large j,

because f_j\to f uniformly on W. Taking the limsup in m bounds the right side by \mathcal R(f_j,W). Take the liminf in j, then the tolerance limit and the supremum over V'\Subset V. This is (iii).□

Proof. Here is the gluing estimate, including the localization it needs. Let a compact set be covered by finitely many open recovery regions U_1,\ldots,U_h, with all closures and supports kept inside a larger protected chart. Choose finitely many smaller disks covering the compact set, each with a fixed enlargement in one U_i. The disk cutoffs from Theorem 4.2, followed by finite maxima, give functions \eta_i\in[0,1] supported with room in U_i, with \sum_i\eta_i\ge1 on a neighborhood of the compact set and with bounded energies. Dividing by their sum there gives weights \theta_i\ge0 with \sum_i\theta_i=1. Outside that neighborhood one can divide by \max\{\sum_i\eta_i,1/2\}. On the bounded range of these functions this operation is Lipschitz, so the energies of the \theta_i are bounded by a fixed multiple of \sum_i\mathcal E_m(\eta_i).

Let w_{i,m} approach f uniformly in the regions where \theta_i is used, with common error \delta_m\to0, and set w_m=\sum_i\theta_i w_{i,m} near the compact set. For a counted edge e=(x,y) there, its endpoints lie in this neighborhood for large m. Using \sum_i d_e\theta_i=0, we have

Jensen’s inequality bounds the squared first term by \sum_i\theta_i(y)(d_e w_{i,m})^2. The sum over edges of the square of the second term is at most

The cross term tends to zero by Cauchy–Schwarz whenever the energies of the recoveries are bounded. The support room and vanishing star mesh ensure that every edge with a nonzero i-term belongs to the region where the i-th recovery was estimated. Thus gluing costs at most the sum of the recovery energies plus o(1).

Monotonicity of \mathcal R(f,\cdot) is immediate, and its definition gives inner regularity:

For disjoint opens O_1,O_2, use compactly contained test domains in each. The counted edge sets are disjoint, and the infimum with all the constraints is at least the sum of the separate infima. The liminf and inner-domain limits give \mathcal R(f,O_1\cup O_2)\ge \mathcal R(f,O_1)+\mathcal R(f,O_2).

For subadditivity, fix a compactly contained test domain in O_1\cup O_2. Cover its closure by inset regions of O_1,O_2, recover f there using Lemma 5.1(ii), and apply (5.9). The glued functions converge uniformly on the test domain, so Lemma 5.1(i) bounds its relaxation by \mathcal R(f,O_1)+\mathcal R(f,O_2). Taking the inner supremum proves subadditivity. The construction with finitely many weights proves the corresponding finite-cover statement. A compact subset of a countable open union has a finite subcover, so inner regularity then gives countable subadditivity.

For clarity, these open-set facts give the asserted measure as follows. For A\subset U, take the infimum of \mathcal R(f,O) over open O\supset A. Countable subadditivity makes this an outer measure. If two sets have positive distance, disjoint open neighborhoods and open-set superadditivity give the reverse additive inequality. It is therefore a metric outer measure, whose Borel restriction is a measure. Its values on opens are \mathcal R(f,O) by monotonicity, and inner regularity and local finiteness make it Radon. This is the usual open-set measure construction.

The constraints in an inner domain depend only on f there; this proves locality. Adding a constant to all vertex values proves invariance under constants. Also

which proves homogeneity after relaxation.

To prove the parallelogram identity, first recover f and g on a common smaller domain. Their sums and differences converge uniformly to f+g and f-g, and the exact edge identity gives

Apply the two lower bounds on V, the recovery upper bounds on a slightly larger domain, and then exhaust the original open set. This gives \Gamma_{f+g}+\Gamma_{f-g}\le2\Gamma_f+2\Gamma_g on opens and also proves local finiteness of the sums. Recover f+g and f-g instead, and take their half-sum and half-difference. The same displayed identity gives the reverse inequality. Equality on opens is equality of Radon measures. Polarization and homogeneity now give real bilinearity. Finally, positivity of \Gamma_{f+t g}(E)=\Gamma_f(E)+2t\Gamma(f,g)(E)+t^2\Gamma_g(E) for every real t gives (5.8).□

Proof. For the upper bound, fix V\Subset U and a square grid of small fixed spacing r. Cover a neighborhood of \overline V by disks of radius comparable to r, centered at grid points, with fixed enlargements contained in U. Construct a cutoff partition (\theta_{i,m}) as in the proof of Lemma 5.3, and put

where x_i is the center of the i-th disk. At a vertex in V, all contributing centers are at distance O(r), so w_{m,r} differs from f by at most its modulus of continuity at O(r).

For an edge in this region, subtract from the coefficients f(x_i) the value at one contributing center. The sum of the differences of the partition weights is zero. At most a fixed number of weights contribute, and each coefficient difference is at most Cr\sup_{B_i'}\abs{\nabla f} on a fixed enlargement B_i'. The cutoff energies and their normalization are bounded independently of r. Summing the squared edge differences gives

The bounded overlap includes all endpoint margins. For smooth f, first let m tend to infinity with r fixed. Then let r decrease to zero. The right side is bounded by the corresponding Riemann estimate on any fixed intermediate open set between V and U, and hence by C\int_U\abs{\nabla f}^2. The uniform errors tend to zero. The definition of \mathcal R and inner exhaustion prove the upper bound.

For the lower bound, let w_m\to f locally uniformly on U, and pass to a subsequence on which \mathcal E_m(w_m;U) tends to its finite liminf L. At grid spacing r, use at each grid point x_i a circular collar

with fixed inner and outer margins; for example one can take \alpha=0.7 and \beta=0.8. Restrict to grid cells in an inner part of U, so every buffered collar lies in U. A surrounding circuit contains the inner disk in its Jordan interior and has that interior inside the outer disk. For neighboring grid points the inner disks overlap, while each center is outside the other’s outer disk. Thus two such circuits must intersect: otherwise their overlapping interiors would be nested, contrary to the center exclusion. Circuits in the same embedded network intersect at a graph vertex.

Let E_{i,m} be the energy of w_m in the buffered i-th collar. If \tau_i is the expected counting traffic of its circuit law, then

Choose one circuit satisfying this bound and assign any one of its vertex values a_{i,m} to x_i. At a common vertex of two neighboring circuits,

Every grid point has at most four neighbors. The enlarged collars have a fixed overlap multiplicity M, including the edge-location margins for all large m at this fixed r. Hence

The piecewise affine interpolation on a triangulation of the grid has Dirichlet integral bounded by a fixed multiple of the left side. Moreover

on each fixed compact part of U, where \omega_f is the local modulus of continuity. With r fixed, take a subsequence of the finitely many grid values and let m tend to infinity in (5.12). Then refine r. The interpolants converge uniformly to f on inner compact sets and have bounded Sobolev energy there. Weak compactness in H^1 and lower semicontinuity yield the weak gradient of f and

One may use a finite union of inner grid regions at each stage, with the same overlap bound. Exhausting U gives the lower inequality for a uniformly convergent sequence.

Apply that inequality to the recoveries in Lemma 5.1(ii), and exhaust their inner domains. It gives the lower bound in (5.10) and the assertion that locally finite relaxed energy supplies a local H^1 gradient.

If f\in C(U)\cap H^1_{\rm loc}(U), mollification on a buffered subdomain gives smooth f_j converging uniformly and in H^1 on any smaller fixed region. After multiplication by a smooth cutoff equal to one on that smaller region, extend f_j by zero to U; thus the inputs in the next step belong to C(U). Use Lemma 5.1(iii) on V\Subset W\Subset U and the smooth upper bound on W. It gives \mathcal R(f,V)\le C\int_W\abs{\nabla f}^2. Choose such W inside U and exhaust U by V. This proves the upper bound for continuous H^1_{\rm loc} functions. All estimates used the same constants on every subopen of the fixed ambient D_0, so Radon regularity gives the corresponding measure inequalities.□

Proof. Work first inside D_0\Subset D. Polarize the energy measures of the coordinate functions x_1,x_2, and define

The densities exist because the upper bound of Lemma 5.4 dominates the measures of x_i+x_j and x_i-x_j by Lebesgue measure. Bilinearity makes A symmetric. For each p\in\mathbb Q^2,

Intersecting the exceptional sets for rational p and using continuity in p gives c_{D_0}I\le A\le C_{D_0}I almost everywhere.

Let f\in C(U)\cap H^1_{\rm loc}(U) and write \Gamma_f=e_f\,\dd x, as permitted by the upper bound. The measure Cauchy–Schwarz inequality has the corresponding density inequality; this follows by applying it on Borel sets and differentiating. For every rational p, outside a null set depending on f,p,

Use one full-measure set for all rational p, and let p approach \nabla f(x). We obtain

The construction on smaller subdomains of this coordinate chart agrees on overlaps by locality, so these matrices form a measurable locally elliptic field.

We next identify its randomness and coordinate dependence. At every Lebesgue point,

and it is enough to use a countable sequence of radii. Each open energy in the numerator is obtained from the cell infima in Lemma 5.1, on inner domains within that ball. Balls with arbitrary centers can themselves be exhausted by a countable base of rational open sets. Thus the density is jointly measurable in z and the retained arrays, and is readable from arbitrarily small strict neighborhoods of z. No outside minimizer enters this reading: only endpoints of counted edges are used, and their full stars lie in the slightly larger protected neighborhood. The affine functions and their sums in (5.15) are among the continuous tests handled by the one countable cell library.

We check the coordinate action required for this germ. Transporting an intrinsic cell instruction by a translation or a rotation only relabels its spatial domains and the scalar value intervals in the same exact finite incidence record. The quadratic polynomial on its edges is unchanged. The inner-domain and tolerance sandwich therefore transports the open energy measures. A translation adds constants to the coordinate functions, while a rotation x\mapsto Rx sends their gradient vectors by R; hence the density matrix is transported by A\mapsto RAR^{\mathsf T}. On overlapping disks the union records of Proposition 3.1 give the common restriction of these measures. Thus the shrinking-ball reading has the translation and rotation actions, with common restrictions, required by Lemma 3.4.

Apply Lemma 3.4 to bounded transforms of the three entries of A. It makes A(z) deterministic for almost every deterministic z, and transports that value under the intrinsic affine coordinate changes. A translation changes each coordinate function by a constant, which leaves its energy measure unchanged. The deterministic matrix is consequently translation invariant almost everywhere, hence constant in the ordinary chart. Equivalently, invariance under the countable rational translations and mollification on smaller domains gives the same conclusion. A rotation conjugates this constant matrix by the rotation matrix. For a quarter-turn R, invariance reads A=RAR^{\mathsf T}. A real symmetric 2-by-2 matrix with this property has equal diagonal entries and zero off-diagonal entries. Thus A=kI, with deterministic k>0.

The preceding assertion is a probability-one statement about activated local outputs and local continuum data. Proposition 3.2 transfers its null failure event from the ordinary input and the extended kernel to each strict-interior sphere patch at its actual field height, without changing the raw energy units. Its countable atlas and separate auxiliary area-pair marginals remove the auxiliary chart marks. This proves (5.13) in the sphere reference charts with the same deterministic constant. The argument is applied separately to the primal and dual outputs. We denote the resulting constants by k_{\rm p} and k_{\rm d}.□

Proof. Contract each prescribed arc to a vertex, denoted t for value 0 and s for value 1, and delete the contracted boundary edges. The two remaining boundary paths run from s to t. Add one s-to-t edge through the exterior, splitting the exterior into two faces adjacent to N_0,N_1. This is a connected map on the sphere.

Let u minimize the primal unit-voltage energy and let I be its total current out of s. It is harmonic at every free vertex. Summation by parts gives

so C_{\rm p}=I>0. Orient the original edges and put their signed currents c_e times the corresponding voltage drops on them. Put the balancing current on the added exterior edge. The resulting flow has zero divergence at every vertex.

A divergence-free flow on a spherical planar map is the gradient of a potential on its faces, with the dual orientation convention. To see this directly, integrate the flow across a dual path from a fixed face. Two choices of dual path differ by cuts enclosing vertices; the integral around each such cut is the sum of the divergences it encloses, hence zero. This defines a face potential \psi, unique up to a constant. Its potential drop between the two exterior faces has magnitude I, because their separating dual edge crosses the added edge.

On the dual of an original edge, multiplication of this face-potential difference by c_e^{-1} gives the corresponding primal voltage difference. These dual currents have zero divergence at an interior face: the primal voltage differences telescope around its boundary, including repeated incidences. At the two exterior faces their total currents have magnitude 1, the voltage difference between s and t along either boundary path. Removing the dual of the added edge thus gives a dual current of magnitude 1 at voltage drop I. Rescaling to unit dual voltage gives C_{\rm d}=1/I.

The contracted map has exactly the dual described in the statement. An interior chord whose endpoints were contracted together is a loop and has zero primal voltage drop; its dual carries zero current in this construction. A primal loop and a dual loop are handled by the same incidence and telescoping identities. Thus no simplicity beyond the boundary cycle was used.□

Proof. Suppose two close vertex sequences have a common limiting position z. Choose a closed disk about z inside O and let M bound the energy in a slightly larger disk. For an integer N, place N disjoint concentric collars of the same relative shape inside that larger disk. Their radii decrease geometrically, and they are all fixed before taking the network limit. In collar i, the circuit-traffic estimate selects a surrounding circuit \gamma_i with total variation at most \sqrt{B E_{i,m}}, where E_{i,m} is the energy paid in that collar. The collars, including their fixed small margins, can be disjoint, so some i has E_{i,m}\le M/N.

Every vertex in the disk inside that circuit has value between the minimum and maximum on the circuit. Indeed all its undetermined vertices are harmonic, and every component inside reaches the circuit: an original graph path from such a component to the rest of the connected map must meet the separating circuit. The finite maximum principle therefore bounds the oscillation inside by \sqrt{BM/N}. For all sufficiently large m, both close vertices lie inside the smallest of the N collars. Let m tend to infinity and then let N increase. This proves their difference tends to zero. For a random z, the collars may be chosen from the countable library by rational perturbations that keep a common inner disk and the disjoint margins; their relative shapes remain in one fixed bounded range.

The sequential assertion gives equicontinuity on every compact by contradiction and compactness of its position set. The vertex positions are dense in each compact region up to an error tending to zero, by the flag mesh and the homeomorphisms. Choose vertices approaching a countable dense set, extract their bounded values diagonally, and use the equicontinuity to extend the limits continuously and uniformly on finite nets. This gives the stated local uniform convergence.□

Proof. Embed a cube graph in the canonical sphere with one facial cycle equal to \partial Q, and pull this embedding back to the reference sphere by T^{-1}. The cubic skeleton lift [30, Lemma 4.7] gives a subdivided primal cube whose graph homeomorphism tracks the pulled-back embedding to any fixed accuracy; its facial cycles and their order are preserved up to one reversal. The reference homeomorphisms in Proposition 2.4 have vanishing projection error, and T_m\to T uniformly transfers that tracking to the canonical sphere. Apply the lift at successively smaller fixed accuracies and choose their thresholds increasingly. This gives a simple primal cycle \gamma_m with an ordered parametrization approaching \partial Q uniformly. In particular, its inside contains every compact subset of Q eventually and lies in every fixed enlargement of \overline Q. Let G_m^Q be the submap consisting of this cycle and everything inside it. Its four arcs follow the four square sides with error tending to zero. Define C_m^{\rm p},C_m^{\rm d} as in Lemma 5.6, using unit conductances.

Write Q^\epsilon=\{x:\dist(x,\overline Q)<\epsilon\}. For an upper bound, choose a smooth function f_\delta on a neighborhood of \overline Q, depending only on the horizontal coordinate, equal to zero in a strip about the left side and one in a strip about the right side. It can be chosen with

For example, flatten the linear function in strips of width \delta, use slope 1/(1-2\delta) between them, and smooth the two bends in smaller strips. Extend it to a fixed small enlargement Q^\epsilon. For fixed \delta,\epsilon, every edge of G_m^Q is eventually located in Q^{\epsilon/2}. Apply the recovery clause of Lemma 5.1 with W=Q^{\epsilon/2}\Subset U=Q^\epsilon, together with (5.13). It recovers f_\delta uniformly on a neighborhood of \overline{Q^{\epsilon/2}}, with limsup energy on Q^{\epsilon/2} at most k_{\rm p}\int_{Q^\epsilon}\abs{\nabla f_\delta}^2\dd x. The constant-recovery modification after Lemma 5.3 makes it exactly zero and one on smaller side strips, with additional energy o(1). The boundary arcs lie in these strips for all large m. Hence

Let \epsilon\downarrow0, and then \delta\downarrow0, to obtain \limsup C_m^{\rm p}\le k_{\rm p}.

For the dual upper bound use the corresponding function of the vertical coordinate, with the same recovery domains W=Q^{\epsilon/2}\Subset U=Q^\epsilon. Thus the full-dual recovery converges uniformly on a neighborhood of \overline{Q^{\epsilon/2}}, and its limsup energy on Q^{\epsilon/2} is bounded by k_{\rm d} times its Dirichlet integral over Q^\epsilon. Apply the constant-recovery modification after Lemma 5.3 to this full-dual recovery on smaller bottom and top side strips whose closures lie in Q^{\epsilon/2} and in the constant regions of the vertical profile. It makes the recovery exactly zero and one there, with additional energy o(1). Give an interior face its modified full-dual value and give each wired vertex its prescribed constant. A dual edge crossing one of the two relevant boundary arcs has its other endpoint at a genuine exterior face of the full map. The complete star of that face has diameter tending to zero. Thus this endpoint is in the smaller constant side strip for all large m, and the squared difference is exactly the squared difference to the wired value. The same face cannot meet the two separated side strips. For this fixed \epsilon, all interior dual edges and these boundary crossing edges eventually lie in Q^{\epsilon/2}, including their whole endpoint stars. Their energy is consequently bounded by the recovered full-dual energy on Q^{\epsilon/2}. The identical sequence of fixed limits, with vertical f_\delta, gives \limsup C_m^{\rm d}\le k_{\rm d}.

For the lower bound take a subsequence attaining the liminf of either conductance and its minimizing voltages, clipped to [0,1]. The upper bound just proved gives bounded total energies. The voltages are harmonic at every undetermined vertex in the submap. On each compact subset of Q, the boundary is absent for large m, and all original primal incidences, or all original dual incidences, are present. Lemma 5.7 therefore gives a locally uniformly convergent further subsequence, with continuous limit f.

We verify its prescribed boundary values away from the four corners. Near the relative interior of a primal prescribed side, extend the voltage to its fixed value on all exterior vertices. Locally this adds no energy: an edge leaving the submap passes through a prescribed boundary vertex. Near a prescribed side for the dual problem, extend the voltage to the wired value on the actual exterior faces. The boundary crossing edges in this neighborhood are precisely the edges to that wired vertex in the complementary problem. Other exterior edges have both endpoints at the same value. Whole-face mesh and the positive distance from the corners ensure that no other side enters this neighborhood. Thus the extended energies in a fixed neighborhood are bounded by the already bounded submap energies.

Fix a point in the relative interior of such a side. Use N disjoint fixed circular collars centered there, as in Lemma 5.7, within the neighborhood just described. Every surrounding circuit visits the exterior side of the approximating boundary. Its Jordan interior contains an inner disk; a ray from a point of that disk on the exterior side to the far exterior must cross the circuit. The crossing has a fixed positive distance from the limiting side, so vanishing mesh supplies an exterior vertex of the circuit. That vertex has the prescribed value. One of the N circuits has variation at most \sqrt{BM/N}. The maximum principle inside it applies to all undetermined vertices; the other vertices there have the prescribed constant value. It bounds their difference from that constant by \sqrt{BM/N}. First take the network limit and then increase N. This proves that f extends continuously to the prescribed value on each relative side interior, uniformly on its compact subsegments.

For V\Subset Q, all edges located in V belong to the submap eventually, for either network. Extend the minimizing function arbitrarily outside the submap; no edge in this inner test sees the extension. The lower relaxation bound and (5.13) give

in the primal case, and the corresponding inequality with k_{\rm d} in the dual case. Exhausting Q puts f in H^1(Q) when the liminf is finite. In the primal case its boundary values are zero and one on the relative interiors of the left and right sides. The Sobolev line property and Cauchy–Schwarz on horizontal slices give \int_Q\abs{\nabla f}^2\ge1. More explicitly, on y\in[\eta,1-\eta] the boundary convergence makes f(a,y) uniformly close to zero and f(1-a,y) uniformly close to one for small a>0. For almost every such y,

Integrate, let a\downarrow0, and then \eta\downarrow0. The dual problem has the same calculation on vertical slices. Consequently both liminfs are at least their stated scalars. Together with the upper bounds this proves the lemma.□

Proof of Theorem 5.2. Lemma 5.5 identifies the two reference relaxations with the scalar integrals k_{\rm p} and k_{\rm d}. Equation (5.16) puts these integrals in the same canonical coordinate, and (5.17) equates their positive constants. Apply Lemma 5.6 to every submap in Lemma 5.8. Its exact identity and the two limits give

Thus k_{\rm p}=k_{\rm d}=1.

For a continuous function outside H^1_{\rm loc}, local finiteness is ruled out by Lemma 5.4. For a continuous H^1_{\rm loc} function with infinite integral, inner exhaustion and its lower bound give infinite relaxed energy. This completes (5.6) for every continuous input.

Proposition 2.4(iv) supplies ordinary chart interiors from two rerootings covering the canonical sphere. Each carries the formula just proved, by Proposition 3.2 in its own marked marginal and by (5.16). On overlaps these are restrictions of the same finite-network relaxation, as in Lemma 5.1. The Radon measure property therefore patches their common Dirichlet density across a finite subcover of each compact test. This proves the sphere statement, including at points omitted by one of the rootings. The lower and recovery assertions are Lemma 5.1 with the now identified value. Since the finite product has fixed the value to one on every further extraction, the coefficient is independent of the extraction.□

Proof. For a vertex v, there is a path from v to \mathsf a_n on which the voltage is at least u_n(v). To prove this, let S be the component of v in the induced superlevel graph \{w:u_n(w)\ge u_n(v)\}. If \mathsf a_n\notin S, connectivity supplies a boundary edge, and every boundary difference from S to its complement is strictly positive. Consequently

whereas (6.1) makes the left side nonpositive. This is a contradiction. The sublevel argument with signs reversed gives a path from v to \mathsf b_n with voltage at most u_n(v).

For real \alpha<\beta, let T clip a number to [\alpha,\beta], and put D=\beta-\alpha. Monotonicity and the one-Lipschitz property of T give on every edge

Using (6.2),

Fix concentric inner and outer coordinate disks and a circular collar between them surrounding the inner disk, with the entire closed outer disk contained in O. For large n, both source vertices lie outside the outer disk. Choose any two vertices in the inner disk and let their voltages be \alpha\le\beta. The superlevel path from the higher vertex to \mathsf a_n and the sublevel path from the lower vertex to \mathsf b_n meet every surrounding circuit in the collar. The intersections are graph vertices, because these paths and circuits belong to the same embedded network. At the two intersections the clipped voltage takes the values \beta and \alpha. Its total variation on each such circuit is at least D.

Theorem 4.2 supplies a circuit law with \sum_e\tau(e)^2\le B. The expectation below is only over that finite-network circuit law. By (6.6),

Thus the voltage oscillation in the inner disk is at most B, independently of the total energy of u_n.

The punctured sphere O is connected. Cover a compact K\subset O and finitely many paths joining it to z_0 by finitely many such disks, with consecutive disks overlapping. Vertices occur in each overlap for all large n, by the vanishing mesh in Proposition 2.4. Chaining the oscillation bounds and using u_n(\mathsf d_n)=0, z_n(\mathsf d_n)\to z_0, gives the asserted bound M_K. The chart cover in Proposition 2.4(iv) allows these disks at every point of O, including a point omitted by one rooting. Under the uniform conformal or anticonformal comparison, sufficiently small fixed round collars contain and are contained in reference collars of comparable relative shape. Their fixed-test circuit bounds therefore apply in the canonical coordinate.

Choose a slightly larger compact neighborhood K'\subset O of K. For large n, both endpoints of every edge located in K lie in K'. Clip u_n to an interval containing all its values on K'. This clipping leaves those edge differences unchanged. (6.6) bounds their total energy by the finite length of that interval, giving M_K'.

For each fixed V\Subset W\Subset O, both moving source vertices are eventually outside \overline W. Equation (6.1) then makes u_n harmonic at every vertex with full star in W. Apply Lemma 5.7 on W, using the local bounds just proved, to obtain compactness on V. A countable exhaustion by such buffered pairs and a diagonal extraction give the final subsequential compactness assertion on O.□

Proof. We first justify the lower measure bound with the necessary outer room. Fix a compact K inside a relatively compact open D\subset O. Choose K\subset V\Subset W\Subset D, with V,W open and \xi(\partial W)=0. Such a W can be chosen among distance neighborhoods with sufficiently small radius: the boundaries are disjoint as the radius varies, and a finite measure gives positive mass to at most countably many of them. The local uniform convergence and the lower bound of Lemma 5.1 give

The equality uses vague convergence on the relatively compact continuity set W. These choices around each compact set and Lemma 5.4 put u in H^1_{\rm loc}(O). Theorem 5.2 now identifies \mathcal R(u,V)=\int_V\abs{\nabla u}^2. Thus

Take the supremum over compact K\subset D, and then over relatively compact D. This proves

The continuity set W is essential here: the open-set Portmanteau inequality alone has the opposite direction for the desired comparison.

We next use local minimization to obtain the reverse bound and harmonicity. Work in one Euclidean chart, and let B_r=B(z,r)\Subset O be a disk with \xi(\partial B_r)=0. Let \varphi\in C_c^\infty(B_r). Choose fixed radii

The function u+\varphi is continuous and in H^1 on this buffered disk. The recovery clause of Theorem 5.2 gives w_n\to u+\varphi uniformly on a neighborhood of \overline B_{r_3} and

Choose a cutoff \theta_n equal to one on B_{r_1} and zero outside B_{r_2}, with bounded energy for these fixed radii. Put

Extend the recovery arbitrarily where it is multiplied by zero. All changed edges lie in B_{r_3} for large n, by the open room and star mesh. The transition collar is outside B_r, where \varphi=0, so w_n-u_n\to0 uniformly there. Applying the edge identity (5.9) to these two functions gives

Indeed the principal squared differences are bounded by their convex combination. The coefficient of an old difference can be nonzero only outside \overline B_r, since \theta_n=1 on B_{r_1} and the mesh is small. The remaining term has energy bounded by \sup_{\rm collar}\abs{w_n-u_n}^2\mathcal E_n^0(\theta_n)=o(1); its cross term is o(1) by the bounded old and recovered energies.

Neither source value changes. The minimizing identity (6.3), restricted to the changed edges, gives \mathcal E_n^0(u_n;B_{r_3}) \le\mathcal E_n^0(v_n;B_{r_3}). Subtract the old energy on B_{r_3}\setminus\overline B_r in (6.10). The difference on the left is \xi_n(\overline B_r)\ge\xi_n(B_r), so the result is

Take n to infinity. The left side tends to \xi(B_r) because B_r is a continuity set, and (6.9) bounds the right side. Now decrease r_4 to r, choosing the three intermediate radii afresh between them at each step. Each cutoff is used only for a fixed step before the network limit. The continuum energy has no mass on the circle \partial B_r, so

Set \varphi=0. Together with (6.8), this gives equality of the two measures on every such disk. These disks form a cover basis after excluding at most countably many radii at each center. The nonnegative measure \xi-\abs{\nabla u}^2\dd x vanishes on that basis, hence vanishes on O. This proves (6.7).

For \psi\in C_c^\infty(B_r), use \varphi=t\psi in (6.11) and the measure equality. Expanding for both signs of t and letting t\to0 gives \int_{B_r}\nabla u\cdot\nabla\psi\dd x=0. Every point has such a disk, so u is distributionally harmonic on O. Interior harmonic regularity makes it classically harmonic. Finally, vague convergence to the measure in (6.7) gives convergence on each of its relatively compact continuity sets.□

Proof of Theorem 6.1. Take any subsequence. Lemma 6.2 and a further vague energy extraction give a locally uniform limit u on O. Lemma 6.3 makes u harmonic there and identifies its local energy measure. The normalization u_n(\mathsf d_n)=0 and local uniform convergence imply u(z_0)=0.

Choose a conformal disk about x whose closure avoids y, with x at its Euclidean center. Let \chi be a smooth radial function equal to one on a smaller disk and zero outside a larger disk, with its transition compactly in the punctured disk. Choose a closed-buffered annular domain V\Subset O containing \supp\nabla\chi with clearance from both boundary components. Its two boundary circles have zero \abs{\nabla u}^2\dd x-mass.

Fix t\in\R. Choose annuli V\Subset W\Subset U\Subset O by slightly moving the two boundary radii, and recover u+t\chi on a neighborhood of \overline W. The recovery w_n satisfies

For a vertex of the full network put

Take an annular cutoff \theta_n equal to one on a neighborhood of \overline V and zero outside an intermediate annulus compactly contained in W. Such a cutoff is obtained by combining two disk cutoffs; its energy is bounded for these fixed radii. Define

On both transition collars w_n-\widehat u_n\to0 uniformly. Wherever the coefficient 1-\theta_n of an edge difference can be nonzero, the edge lies outside \overline V with a margin, where \chi is constant on each component. Thus d_e\widehat u_n=d_eu_n on every such edge for all large n. This uses the clearance from \supp\nabla\chi and vanishing star mesh. Outside W, \chi is constant on the inner and outer components and \theta_n=0; hence the changed energy is confined to W. The source vertex is in the inner component and the sink is in the outer one, so

The edge gluing estimate, with the old part supported outside \overline V, gives

The error has energy at most the squared uniform collar discrepancy times the fixed cutoff energy, and its cross term tends to zero by local energy bounds. Minimality of the source functional (6.3) now gives

Subtract the displayed old collar energy. The difference on the left is \xi_n(\overline V)\ge\xi_n(V), so the resulting inequality is

By Lemma 6.3, the left side converges to \int_V\abs{\nabla u}^2, since V is a continuity set. Use (6.12), then shrink U to V through annuli with the same clearance from \supp\nabla\chi. At each stage the annuli and cutoffs are fixed before n tends to infinity. Absolute continuity of the continuum energy and the zero mass of the boundary circles give

Expanding this inequality yields

It holds for every real t, so the coefficient of t is zero:

For a cutoff equal to one near y and zero near x, the replacement changes the sink value by t and the source value by zero. The source term in (6.3) then changes by +2t; the expanded inequality gives flux -1 at y.

We verify that these fluxes determine the only possible singularities. Choose an outer coordinate disk about x whose closure excludes y, and a fixed collar inside it surrounding a smaller disk about x. For every sufficiently large n, take a surrounding network circuit in that collar. Eventually \mathsf a_n is in the smaller disk and \mathsf b_n is outside the outer disk, hence outside the circuit. Lemma 6.2 bounds the voltages on that collar uniformly. Inside the circuit, Q_nu_n=0 except at the positive source, where Q_nu_n=1. The minimum principle therefore bounds every interior voltage from below by the minimum on the circuit. Passing to the limit bounds u from below near x. For y, choose an outer disk whose closure excludes x and a collar inside it surrounding a smaller disk about y, and take a surrounding network circuit there. Eventually \mathsf b_n is inside its circuit and \mathsf a_n is outside the outer disk. The maximum principle then bounds u from above near y.

We recall the planar isolated-singularity argument at the needed strength. If v is a positive harmonic function on a punctured disk, its circular mean M(r) satisfies (rM'(r))'=0. Thus

where positivity forces the sign of a. Harnack’s inequality on the fixed-shape annulus \{r/2<\abs z<2r\} bounds the values on \abs z=r by a constant times M(r), uniformly in small r. This version of Harnack follows directly from the Poisson kernel on a finite chain of disks in the annulus. Hence v=O(1+\log(1/r)). For each nonzero angular Fourier mode, harmonicity makes its radial coefficient a linear combination of r^j and r^{-j}, j\ge1. The logarithmic bound forces every r^{-j} coefficient to vanish. The remaining positive-power series is the harmonic Poisson extension from any fixed smaller circle. Consequently

where h extends harmonically through the center. Applying this to u plus a constant near x, and to -u plus a constant near y, shows that u has only logarithmic or removable singularities.

For a\log(1/\abs z), a radial cutoff equal to one near zero and zero outside has

The harmonic remainder contributes zero. Equation (6.14) therefore sets the coefficient at x to 1/(2\pi), and the sink flux sets the coefficient at y to -1/(2\pi).

For comparison, the round zero-mean Green kernel has, up to its additive constant,

Away from w, a radial calculation on the unit sphere gives -\Delta_z G_{\rm rd}(z,w)=-1/(4\pi); the preceding local flux is one at w. Thus its distributional equation is -\Delta_zG_{\rm rd}(z,w)=\delta_w-(4\pi)^{-1}\dd\mathrm{vol}_{\rm rd}. The difference G_{\rm rd}(\cdot,x)-G_{\rm rd}(\cdot,y) has exactly the two fluxes and singular coefficients just found. Subtract it from u. The difference extends harmonically across x,y, and a harmonic function on the compact sphere is constant. Its value at z_0 gives precisely (6.4).

Every subsequence has therefore a further locally uniform subsequence with the same limit U_{x,y;z_0}. This proves (6.5) along the entire extraction. Evaluating at \mathsf c_n gives the asserted four-point value. The reasoning applied to arbitrary varying pole and evaluation vertices with the stated distinct limits, so it proves the claimed pathwise quantifier.□

Proof. The projected rank and jump inequalities are the spanning-forest and displaced-grid estimates in [31, Section 10]. The fixed-code estimate is [31, Proposition 10.15]; its proof charges a left and right fresh slot for each monochrome rank and gives the coefficient \vartheta=1/2+2/\gamma^2. The decorated address count is [31, Lemma 10.13]: distinct depths give \sum_p m_{k_p}\le T_{\rm occ}, and the terminal base list supplies every base’s coarser address. The regular-slot deletion bound and grid-partition estimate are the corresponding lemmas in the same section. Its sparse-diagonal remark gives the list and jump order of limits; the extra-ancestor estimate in the proof of [31, Lemma 11.6] gives the growing-enclosure condition. In these formulas a generation has log width \ell_R; this accounts for every conversion between sums over generations and log lengths.□

Proof. Choose fixed address parameters within Proposition 7.2 so that

Fix k,M and a log stride L>2\ell_R. Sample an independent translation of a stride-L mesh in \mathcal I_N. At each seed choose a maximum-cardinality H-packing truncated at M, with all access witnesses. Use a fixed measurable ordering of the finite reachable set, and keep the independent mesh phase as side information in the code count. Round a seed y up to generation \lceil y/\ell_R\rceil, discarding the final incomplete generation if needed. The seed depths are distinct because L>2\ell_R. The terminal base-list augmentation adds at most one probe and h_{\mathrm b} fixed entries, so P_N=O(N/L) and K=O(kM+h_{\mathrm b}), with constants allowed to depend on the fixed enclosing depth. Only then sample the chronological address-grid displacement. These lists have exactly the order and the form in Proposition 7.2.

We first bound their occupation for this fixed L. Markov’s inequality in part (i) gives a set of grid displacements of fixed positive conditional probability on which \sum D_k\le C_KN/L+O(1). Intersect it with the regular-slot event, retaining a fixed positive probability. Use e_k\ge m_k-h_{\mathrm b}-D_k in (7.4) and sum the codes by (7.5). All terminal and decoration terms at fixed K,L are O_K(N/L)+O(1). Let \mathsf F_t(\Xi) be the existence of a realized regular code with this jump bound and occupancy t. Uniformly in the displacement,

The constants in the leading term are independent of K. This calculation retains the terminal errors explicitly; the asymptotic occupancy statement with L=L_m\to\infty alone would not justify the present order of limits.

For two displacements, each cell of one grid meets at most two cells of the other, hence T_{\rm occ}(\Xi)\le 2T_{\rm occ}(\Xi'). If the maximum occupancy over displacements is large, it is at least half as large on every displacement, including the fixed positive fraction just used. Integrate (7.7) over the grid and sum over t. Then apply (7.3). It follows that, for a constant C_1 independent of K, the total log occupation is at most

with \mathbb Q_m-probability tending to one. The same bound applies to the occupation restricted to \mathcal I_N.

Here is the conversion to packing, which is the mechanism of [31, Lemma 11.4]. A list chosen at a seed y remains reachable throughout the coarser interval [y-L,y]. Use generation \lceil x/\ell_R\rceil boxes on each continuous log cell, trimming incomplete endpoint cells. They are active throughout [y-L,y] for a seed rounded as above, and their integrated occupation is at most \ell_R T_{\rm occ}. If these boxes at level x have diameter at most c_{\rm box}r_m(x), then its chosen terminals stay in distinct boxes throughout that interval whenever H e^{-L}>2c_{\rm box}. Thus the truncated seed count contributes at least L times that count to integrated occupation, apart from the two end intervals and a fixed-width endpoint strip. Averaging the independent mesh translation turns the seed sum into the uniform integral on \mathcal I_N. On the occupation-exception event, bound the truncated packing by M. Dividing by \abs{\mathcal I_N}, the end error is at most O(M(L+\ell_R)/\abs{\mathcal I_N}). First let m\to\infty for fixed H,L,k,M; then take H\to\infty and L=L(H)\to\infty slowly with He^{-L}>2c_{\rm box}. The term O_K(L^{-1}) vanishes for each fixed k,M, and the limiting bound is independent of them. This proves (7.6).□

Proof. Choose fixed 1<\alpha<\beta<a_0 and apply Lemma 7.3 with one selected base and \mathcal I_N=[\alpha N,\beta N]. For every x\in\mathcal I_N, the proposed jumps divided by r_m(x) are at most C_0e^{-(a_0-\beta)N}, while the proposed separation divided by r_m(x) is at least c_0e^{(\alpha-1)N}. For a fixed enlargement of d, the proposed list is therefore contained in \mathcal R_{m,k}(x) and is an H-packing for every fixed H, eventually uniformly in x. Select the base and only M terminals with their witnesses before the grids. On the asserted event the truncated packing equals M for every fixed H,M and all large m. Taking the limits in (7.6) and then choosing M>C_{\rm pack} is a contradiction.□

Proof. The base is marked at every generation, so b_k=1 and m_k\ge1. At a generation where its component has a rank cycle, that component contributes one rank beyond a union spanning tree. Part (i) of Proposition 7.2 therefore gives

On a regular grid with integrated jump error o(N), the event in the statement consequently implies

Here the first loss counts the generations before aN and the exceptional high generations. Extra components separated by jumps have already been charged to \sum D_k. A permitted growing enclosure shifts the log origin by O(\log\widehat C_m)=o(N); its extra ancestors are included in the same error.

Fix R and the errors so that the \delta in the proof of Lemma 7.3 is at least (\vartheta-1)/2. Combining (7.4), (7.5), and (7.8), with the sparse errors o(N), gives total log weight at most

Choose B with \delta B/2>2\vartheta by a fixed margin. The first negative term makes the sum over T_{\rm occ} geometric, and the remaining exponent is -cN+o(N). Multiplication by e^{\kappa_m}=e^{o(N)} preserves convergence to zero.

This estimate was for regular grids. The cycle event itself depends on the grid, so its intersection with a positive fraction of grids is not automatic. Since M_{\rm term}=o(N), part (iv) provides q_m=o(N) and \alpha_m\to0 such that \PP_\Xi(\Delta(\Xi)>q_m\mid\text{markers})\le\alpha_m. Let E^* be the corresponding cycle event for the grid-independent partition, allowing q_m additional exceptions. If E(\Xi) is the event in the statement, then

For a second independent displacement \Xi', the conditional probability of being regular with jump error o(N) and having \Delta(\Xi')\le q_m is at least 1/2-\alpha_m. On E^*, this implies the cycle event for \Xi' with another q_m=o(N) exceptions. Hence

The preceding exponential estimate applies to the right side. This proves the lemma.□

Proof. The simultaneous saturation and all-seed construction in [31, Section 11] uses precisely (7.6), the sparse cutoffs, and the preselection event. Its order is as follows. First sample the independent log-mesh phase, taking L_m=q_m\ell_R with integer q_m\to\infty as in the source; changing the chosen length by at most \ell_R preserves all displayed asymptotic conditions. Run the truncated recipe at every complete-cell endpoint. Let P_N^{\rm seed}=\Theta(N/L_m) count these cells and let \Lambda_m bound one truncated seed list. Before the chronological grid is drawn, round the seed depths as in the source, leaving them distinct, and apply the terminal base-list augmentation at J_R. Coding then uses P_N\le P_N^{\rm seed}+1 and K_m=O(\Lambda_m+h_{\mathrm b}); the two sparse ratios remain vanishing because h_{\mathrm b} is fixed. The independent mesh phase is held as side information in this count. Next draw the independent chronological grid, form one forest from all the lists, and round and freeze origins inside their coded terminal cells. Then choose uniformly among the original P_N^{\rm seed} complete cells, including fallback cells, and coarsen independently. This law differs from the stated uniform single-seed law by O(L_m/N) in total variation; the fallback probability at that sampled cell vanishes along the sparse diagonal. There is no conditioning that every cell succeeded. Conditional on the independent phases, the chosen seed-generation trial is uniform among P_N^{\rm seed}q_m=\Theta(N/\ell_R) trials, including empty and deleted trials. For each of finitely many fixed residue classes of sufficiently spaced generations, the retained enlarged raw slots are disjoint after deletion. The deletion probability tends to zero by taking the discrete limit first and then the fixed ancestor padding to infinity. The address occupancy estimate gives an expected code length O(N) under (7.3). Entropy allocation in each residue class, followed by their finite sum, gives bounded mean conditional entropy for this trial. Its preliminary selected-slot comparison is only a statement about continuum contour paths and finite height differences. It gives the stationary finite-line system without asserting a law for discontinuous lattice outputs. [31, Lemma 11.12] supplies the simultaneous actual return links and whole-gap margins. [31, Lemma 11.24] supplies the identical original and auxiliary patch, including full stars and full preimages. The inner and outer time-preimage bounds in (ii) are verified from this patch in the proof of [31, Proposition 11.30]. The patch construction first bounds all relevant partner coordinates in a centered range [-M,M] about their enrolled origins. The retained rational cuts are chosen about those origins, and a finite recipe truncation bounds the absolute values of their endpoint coordinates in view units. Finally [31, Lemma 11.27] constructs the countable retained-data recipes and their open fillability. The fixed-duration ordinary completion is the separate construction immediately preceding [31, Proposition 11.30]. The source construction samples the class anchors after its fixed-recipe observation vector. All tight quantities can first be bounded by fixed constants with an arbitrary fixed loss; its finite data and sparse diagonals are then exhausted.□

Proof. Suppose such selected laws exist. Mark and label the alleged unavailable bases before the independent seed mesh, address grid, and view sampling, and fix an arbitrary deterministic label \iota at that stage. Let \mathsf S_m denote the selected sampled-trial law on the common array and its successive extensions by the stated construction kernels for base \iota. Let \mathsf A_m^{\mathrm{acyc}} be the retained indicator from Lemma 7.6, belonging to the common trial marginal, that base \iota’s combined center component is acyclic, and write \mathsf A^{\mathrm{acyc}} for its extracted limit. The auxiliary extensions may depend on \iota; no independence between extensions for different labels is used.

Use Lemma 7.6 and restrict first to finitely many recipe indices and fixed duration, line-count, and margin bounds, losing an arbitrary fixed probability. For each fixed recipe, before its class anchors are drawn, record

in raw unit-edge units, where \mathcal P_{m,s} is its full family of paths between the recorded vertex ports in the recorded confinement graph. Use +\infty for an empty family and compactify these coordinates. This is a function of the same exact incidence record and fixed intrinsic instruction, as required by Proposition 3.1.

For each fixed recipe and fixed truncation, consider the compatible anchored selected subprobabilities constructed below, retaining \mathsf A_m^{\mathrm{acyc}} as an extra coordinate. Let \nu be any weak limit of their pushforwards to normalized view coordinates, after the nuisance trial, unit, and parity labels are projected away at the step specified below. Write c_0 for the limiting intrinsic pre-anchor input, a for its class anchors, and y=(y_s)_{s=1}^q for the extended-real electrical output in [0,\infty]^q, retained through compactification. Let F(c_0,a) be the fixed recipe’s anchored feasibility event in the retained limiting view of Lemma 7.6; its strict margins give the eventual lattice crossing interpretation. Our target is

This is a null statement for an unnormalized subprobability; no acyclic, compatibility, or successful-recipe probability is divided out.

In an ordinary completion of the fixed duration, feasibility has a countable family of rational route witnesses. Work first with one such witness: choose compact endpoint balls inside its open ports and a compact connecting route inside its recorded open confinement. Finitely many ordinary charts off the auxiliary marks cover it. Theorem 4.2 gives a path law with finite squared traffic there. Full preimages and the strict inner/outer margins put its endpoints in the recorded ports and every used edge in the recorded graph for all large indices. The countable intersection covers every feasible witness. Therefore the limit of Y_{m,s} is finite on feasibility in every jointly extracted ordinary completion. This uses only the fixed-test electrical conclusion, for arbitrary diverging units and a fixed duration; neither its threshold nor its route constant has to be uniform over recipes.

We spell out the transfer to the selected view. Let c_{0,m} be the retained input before the class anchors are drawn, including the block paths, forest shifts, durations, fixed instructions, and the trial, unit, and parity parameters \theta_m. Let Y_m be the fixed electrical subvector above. In each residue trial the code and recipe fix the requested cuts, durations, and guard/noise search anchors before fresh letters; auxiliary choices have total conditional weight at most one. These search anchors precede the class anchors considered below. The shifts form the prescribed forests, and fixed truncation supplies positive safety bands and compact fitting ranges. Let G_m^0 be genuine raw-slot compatibility for this pre-anchor instruction. Use the pre-anchor marginal of \mathsf S_m in the following comparison, without conditioning on acyclicity or recipe success.

Write \mathsf R_m^0(dc_0,dy) for the independent raw-block reference law of this pre-anchor input and output, integrated against the same retained trial and parameter marginal as \mathsf S_m, and set \mathsf S_m^0(B)=\mathsf S_m\{G_m^0,(c_{0,m},Y_m)\in B\}. The finite domination part of Lemma 3.3, followed by the bounded all-seed trial entropy, gives

This is tight absolute continuity of the compatible subprobability law; no division by the compatibility mass occurs. The comparison sees the lattice output Y_m, whereas the preliminary comparison in Lemma 7.6 only concerned continuum contours.

After the entire pre-anchor selected sample, including its full fixed observation vector, draw the finite class-anchor vector a_m. Its translation may depend on Y_m. First restrict to finitely many anchor counts and compact height ranges, keeping these restrictions unnormalized. Its conditional density given that entire sample is bounded, as in [31, Lemma 11.25]. Choose a positive smooth reference anchor density, with its scaled feasible-lattice version \varrho_{m,\theta_m}(da). The lattice law is expressed relative to covolume-normalized counting measure; on the compact truncation its density is uniformly bounded below. On those ranges the restriction of the actual conditional anchor law is therefore bounded above by C\varrho_{m,\theta_m}. The same bound given (c_{0,m},Y_m) follows by conditional expectation. The reference anchors are independent of Y_m conditional on the recorded \theta_m.

Here is the sectional argument, applied without normalizing \mathsf S_m^0. Let \mathsf S_m^a be the joint selected subprobability after adjoining a_m, restricted to G_m^0 and the chosen anchor-count and height ranges, and put

For an event B_m with u_m=\mathsf R_m^a(B_m)\to0, let p_m(c_0,y)=\varrho_{m,\theta(c_0)} \{a:(c_0,y,a)\in B_m\}. Markov’s inequality gives \mathsf R_m^0\{p_m>\sqrt{u_m}\}\le\sqrt{u_m}, while the conditional anchor bound gives

by (7.11). This is the proof of [31, Lemma 11.25], including its scaled-lattice form. It allows the actual anchors to depend on the output. Apply it separately to each fixed finite recipe vector; the countable null intersection and the preceding finite recipe truncation handle the later recipe selection.

Before taking the kernel limit, project away the trial index and raw unit/parity labels after expressing the retained paths, shifts, durations, and anchors in normalized view units. Tight absolute continuity survives this pushforward. The fiberwise anchor reference laws are lattice discretizations of one fixed smooth law \varrho; on each normalized compact they converge to \varrho uniformly over the allowed unit and parity parameters as the minimum unit diverges. The normalized reference factorization of [31, Theorem 7.10] and Proposition 3.1 therefore gives the independent-anchor limit law \mu_0(dc_0)K_{c_0}(dy)\varrho(da). The positive ordinary filling argument, applied to augmented input (c_0,a), gives the retained subprobability law

As in (3.17), f integrates the strict omitted filling and its input-dependent density weight, so it has no y argument. The ordinary finiteness proved above yields

Positivity of f makes the anchored feasibility/infinite-cost event null for the independent-anchor reference product. The post-anchor tight absolute continuity just proved transfers this null event to the selected anchored limit on compatibility. Every successful selected recipe has G_m^0 by its construction from the actual compatible raw instructions. The projection of \nu restricted to \{\mathsf A^{\mathrm{acyc}}=1\} is a submeasure bounded by this compatible anchored marginal, so (7.10) follows without a kernel or independence assertion for the indicator. Only the independent-anchor reference and ordinary filling laws use the displayed output kernel K_{c_0}; the selected anchored law is used through tight absolute continuity.

All comparisons through the anchor step are finite lattice comparisons at fixed recipe, guard, duration, shift, anchor-count, and height truncations. Their effective slot lengths diverge because r_m(bN)\to\infty. Apply them in each joint extraction with r_m(X) and its recorded unit parameters; the normalized reference construction and the scaled-lattice anchor argument apply to these varying laws. The ordinary finiteness assertion holds for every such extraction, so the argument also covers mixtures over X, as in [31, Proposition 7.18]. The anchor restrictions are exhaustible under the selected law: on a fixed line-count and centered-oscillation truncation, the completion assigns class heights H_A=(4K_0+10)s(A)+u_A, with 1\le s(A)\le k and \abs{u_A}<1, where K_0 bounds the retained centered oscillations. Bounding k and K_0 bounds these normalized heights; bounded density alone is not used for their tightness. Fix the finite recipe list and the other stated truncations. By the strict certificate limits in [31, Lemma 11.27], (7.10) gives vanishing unnormalized selected mass for the finite-threshold cost failure of a successful retained recipe, first as m\to\infty and then as the cost threshold tends to infinity. Only afterward exhaust the finite recipe and truncation losses. No slot meets the center, and feasibility has no omitted filling as an argument. This is the restricted tightness of \sum_sY_{m,s} used below.

By Lemma 4.1, choose a near-minimizing path law for each \mathcal P_{m,s} with squared traffic at most Y_{m,s}+1. Sample those paths and select a simple surrounding circuit from their union. Every sample works by the recipe’s transverse pattern. If its traffic is \tau and the path traffics are \tau_s, then

On each fixed truncation, its unnormalized upper tail is controlled by the restricted cost estimate. Since e^{\zeta N}\to\infty, the corresponding traffic-failure mass tends to zero.

The inner and outer preimage bounds in Lemma 7.6(ii) give the two disk conditions. The event bN-X\le\sqrt N has probability O(N^{-1/2}). Outside it, \delta_m r_m(X)/r_m(bN)\to\infty after the fixed tightness truncation. Also

Every uninterrupted visit in a closed circuit disk, including a run on its edges, belongs to the full outer preimage, so it has length less than \ell_m. These conclusions apply to every circuit in the constructed law.

They also have the required support reach from the given base. If its combined component has K lines, a simple path of consecutive scalar-class connections from the base line to any retained line uses at most K-1 actual return links. Interleaving jumps on those lines and a final jump to a retained edge occurrence uses at most 2K+O(1) steps, each jump bounded by a tight multiple of r_m(X). The same finite return witnesses work for every occurrence in every retained interval. All path support is in the protected patch, whose complete time preimages are on those intervals. Since r_m(X)/r_m(a_0N)\le e^{-(a-a_0)N}, the jumps are eventually at most \ell_m, and tight K is eventually below the diverging D_m. The master event and its complete-star mesh bound put the entire supported edges in the required confinement. Combining these vanishing disk and support errors with the restricted cost estimate and then exhausting the fixed losses gives

Here unavailable means failure of a law satisfying (a)–(c) and (7.9). Under the selected laws assumed for contradiction, base \iota is unavailable with probability one, hence \mathsf S_m\{\mathsf A_m^{\mathrm{acyc}}=1\}\to0.

It remains to use the address charge. The bases in (i) are \ell_m-separated, whereas high-range cell widths are O(r_m(aN))=o(\ell_m); the same remains true along the permitted sparse enclosure diagonal. The later return links certify colored center-pair bits in the saturated graph. For each such bit the all-seed code retained an earlier chosen actual seed witness and jumps associating its endpoints to the corresponding original pre-grid seed centers. The association jumps cost o(N) split generations. A base has a tight seed offset from its original pre-grid seed center. On a fixed offset truncation, the independent-grid probability that they are separated after coarsening by U is O(\min\{1,e^{-U}\}), whose average for U\sim\operatorname{Unif}[0,L_m] is O(L_m^{-1}). Thus only o(N) further trial generations fail, in probability. Rounding an origin preserves its seed center’s terminal cell and all its ancestors. Outside these exceptions the centers, witness endpoints, and bases have the corresponding projected cells. Thus a rank cycle of a center component is a rank cycle in the projected base component. Adding further connected vertices cannot destroy it, since a union spanning tree gains at least one total monochrome rank per added vertex. Apply this argument separately to each fixed label \iota on the common trial array. Let E_{m,\iota} count generations k\ell_R\in[aN,bN] at which its projected base component lacks a rank cycle. After including the conversion exceptions above, averaging the vanishing acyclic probability over the uniform seed-generation trials and applying Markov’s inequality gives E_{m,\iota}/N\to0 in probability under the common selected array law. Since h_{\mathrm b} is fixed, N^{-1}\sum_{\iota=1}^{h_{\mathrm b}}E_{m,\iota}\to0 in that same probability. The number of generations exceptional for at least one label, divided by N, is bounded by this normalized sum. Outside that union every projected base component has a rank cycle. Choose a deterministic \eta_m\downarrow0 so that the exceptional count is at most \eta_mN with probability tending to one. This is the cost-independent witness conversion in the proof of the circuit-sparsity result in [31, Section 12].

In regime (i) this contradicts [31, Proposition 10.18]. In regime (ii) it contradicts Lemma 7.5. The latter uses only fixed B and r_m(BN)\to\infty; the distance normalization and the narrow length-recursion restriction in [31] were not used. Removing the arbitrary finite truncation losses proves the proposition.□

Proof. Fix an edge e belonging to several W_I. From those blocks retain a sublist whose enlargements are separated by a fixed multiple of T, losing only a fixed factor. For each retained block choose the responsible time in \widehat I; it reaches a step occurrence of e with the chosen depth and jump bound, or is itself such an occurrence for a chronological edge.

By the dictionary of incidence occurrences in [30, Proposition 3.1 and Lemma 3.2], each original edge has the two contour steps of one matched burger/order pair: L-pairs represent primal-tree edges, while the quadrangles of R-pairs carry the complementary primal edges. The convention includes loops, bridges, and repeated incidences. If the up-step ends at u and its matching down-step ends at v, their scalar baselines satisfy

Thus (u-1,v) is an actual scalar link; either step occurrence is within one time step of the corresponding endpoint. One link and at most two unit jumps join its two occurrences, including for loops and parallel edges. The units r_m(a_0N) diverge whenever a circuit is used, so these unit jumps are allowed. Joining through e therefore reaches every responsible time from the first within twice the support depth plus a fixed number of steps. The times are cT-separated. The simultaneous version of Lemma 7.4, with \zeta=1/40, bounds their number by e^{N/40}. This bounds the multiplicity of each edge in the left side.

For chronological-only sets, the enlarged intervals have bounded time overlap, and the primal tree tour traverses a tree edge at most twice. Their edge multiplicity is bounded. Summing the edgewise bounds proves the lemma.□

Proof. A maximal \ell-separated set of bad times has at most h_{\mathrm b}-1 elements. Their \ell-neighborhoods cover the bad times and, after merging, have total length O(h_{\mathrm b}\ell). The complement is a union of at most h_{\mathrm b} good integer runs. In each run choose its endpoints and good times spaced at most d. Their number is at most 2+T/d per total run length, and their open interiors cover the complete interpolated run. Enlarging the deleted intervals by a bounded number of steps handles the endpoints and proves (i).

For (ii), in a good run retain the chosen disks maximal for inclusion. Consecutive original interiors overlap, so the maximal interiors have connected intersection graph. Two overlapping incomparable Jordan disks have intersecting boundaries. Their boundary union is consequently a connected subgraph with total \abs{df}-cost at most the sum of the chosen circuit costs. The run’s first vertex reaches the boundary of a disk containing it within \ell+1 forward steps, and its last vertex does so within \ell+1 backward steps, by the no-run condition. A hit is at a graph vertex, since the walk and circuit use the same embedded graph. Join the two hits in the boundary union. Concatenating these connections across the good runs leaves only the deleted intervals and at most two such exit lengths at each gap. Their number and lengths have the asserted bounds. If there is no good run, use the one unresolved endpoint pair. This is the additive-cost proof of [31, Lemma 12.2]; it applies to the nonnegative edge costs \abs{df}.□

Proof. We first prove the assertion on a fixed compact interior part of an ordinary planar field chart. The moment input is the subcritical GMC theorem for a stationary positive-definite covariance \gamma^2\log_+(R/|x-y|)+g(x-y) in dimension two, with g bounded and continuous, with 0<\gamma<2; here \log_+u=\max\{\log u,0\}. For the exact logarithmic covariance (g=0), its scaling and positive and negative moment statements give

uniformly for the small balls in a fixed buffer, whenever 0<q<4/\gamma^2 or q<0. Here M_\gamma is the limiting Wick chaos with Lebesgue base measure. This is the exact logarithmic specialization of [32, Propositions 3.3 and 3.5–3.7]; the exact logarithmic covariance is available in dimension two. In particular the negative moments in (8.2) are moments of the limiting mass of an open ball.

For \gamma^2=3, take q=7/6. Then q<4/3 and \zeta(q)=49/24. For q=-1, the exponent is \zeta(-1)=-5. Markov’s inequality therefore gives, with a=1/56 and b=8,

At level r=2^{-k}, use a grid of mesh r/16 in the fixed buffer. There are O(r^{-2}) grid points. Apply the upper estimate to the balls of radius 2r and the lower estimate to those of radius r/4; constant factors in the radii only change C in these displays. The failure probabilities summed over the grid are respectively O(r^{1/48}) and O(r), and hence are summable over k. Almost surely, both bounds hold on every sufficiently fine grid. If r/2\le s\le r, any ball of radius s centered in the smaller compact part is contained in a grid ball of radius 2r and contains a grid ball of radius r/4. Borel–Cantelli consequently gives the two polynomial bounds for all centers and all small radii in that compact part, with random constants.

Here is the precise passage from this reference chaos to the sphere. A whole-plane GFF pinned to have circle average zero on a circle of radius R has covariance \log(R/|z-w|) on the concentric disk of radius R/4. The field multiplied by \gamma therefore has on that disk the covariance of the exact stationary field used above. The usual local LQG area agrees with this Wick chaos up to deterministic smooth normalization factors bounded above and below on a smaller buffer. This follows from the common-regularizer identification, including circle averages, in [4, Theorem 1.1], whose base-measure energy condition holds for planar Lebesgue measure on a bounded buffer, and the circle-average normalization in [11, Propositions 1.2 and 3.2]. Such factors preserve the two inequalities with modified constants. Adding any finite constant to the field also preserves them, since it multiplies area by one positive scalar.

In the three-mark embedding of the ordinary unit-area sphere, [30, Proposition 3.5] gives local mutual absolute continuity, modulo additive constants, with a whole-plane GFF on bounded buffers avoiding the three marks. Existence of the local polynomial bounds with some constants is invariant under adding a field constant; the grid formulation makes it a measurable event of the local field modulo constants. It is therefore an almost-sure local property transferred by that proposition. We use it on a countable collection of nested compact and open buffers exhausting the complement of the marks. Resample a second triple of area marks and use the same proposition in its three-mark coordinate. The sphere area is nonatomic, as in the sphere input of Proposition 2.4, so the two triples are disjoint almost surely. Their two mark-free open sets cover the sphere. On each realization, select finitely many of the buffered interior charts whose interiors cover it. The coordinate changes are smooth sphere conformal maps, with upper and lower derivative bounds on these compact patches. A Lebesgue number for the finite cover and comparison of round and chart balls now give (8.1), with common random constants and a common small radius. Thus the estimate for the conditioned sphere follows from these local Gaussian comparisons.□

Proof. Decrease r_* to at most one and set L(x,y)=1+\log_+(r_*/d_0(x,y)). The upper ball bound and layer cake give, for j=1,2,

Indeed the nonconstant part is bounded using

The same bounds hold for \sigma_0. Also \mu is nonatomic by the upper bound. In particular the diagonal has zero mass for all products of measures absolutely continuous with respect to \mu+\sigma_0.

For g\in L^2(\mu), write \mu(g)=\int g\dd\mu and consider the zero-mass signed measure

The bound |G(x,y)|\le C L(x,y) and Schur’s inequality imply

For example the \mu\otimes\mu term has this bound by the j=1 case of (8.11); the mixed terms follow by integrating L against \sigma_0 first and using |\mu(g)|\le\|g\|_2. Thus all signed Green integrals below are absolutely defined.

We record why these integrals are Dirichlet energies, including for the signed measures s_g. Let \Delta_0 be the round Laplace–Beltrami operator with nonpositive spectrum. Thus -\Delta_0 is the operator of \mathcal E^0 in L^2(\mathrm{vol}_0). Let P_t=e^{t\Delta_0} and let p_t be its kernel relative to \mathrm{vol}_0. For t>0, put

The small-time Gaussian estimate for the smooth round heat kernel and its exponential convergence to the constant kernel for large times give

For instance, the singular part is controlled by

Let u_{g,t} be the unique smooth function with \sigma_0(u_{g,t})=0 and

its source is the smoothed measure P_t^*s_g. The heat semigroup identity and integration by parts give the exact cross-energy formula

There is no diagonal mass, so (8.12) and (8.13) justify dominated convergence in this formula. Applying it to the three terms in \mathcal E^0(u_{g,t}-u_{g,s}) shows that these potentials are Cauchy in Dirichlet norm. Poincaré’s inequality and their zero round means make them Cauchy in H^1. Their limit u_g satisfies

The last identity follows by letting t\downarrow0 in the weak equation for u_{g,t}, since P_t\phi\to\phi uniformly. This also identifies the limit as a distributional potential. Poincaré gives its uniqueness among zero-round-mean H^1 functions.

For smooth f, (8.17) and (8.15) show that

Taking the L^2(\mu) dual supremum proves (8.7) on smooth functions. Together with the bound on the round mean it gives \|f\|_{L^2(\mu)}\le C\|f\|_{H^1}. The density of smooth functions in H^1 gives a unique bounded extension R_\mu and proves (8.7) for every f\in H^1. It also extends (8.17) to (8.9).

We next identify this extension with the usual quasi-continuous values. For a compact set F, the H^1 capacity can be defined as the infimum of \|\phi\|_{H^1}^2 over smooth \phi\ge1 on a neighborhood of F, and then extended by the usual outer regularization. For each such test function the restriction inequality gives

Infimizing, and then using inner regularity of \mu, proves that every Borel set of zero H^1 capacity is \mu-null. If smooth functions f_n converge to f in H^1, the capacity Chebyshev inequality applied to a sufficiently rapidly converging subsequence shows that this subsequence converges outside a set of capacity zero to the quasi-continuous representative \widetilde f. A further subsequence converges \mu-almost everywhere to R_\mu f, by its L^2(\mu) convergence. The two limits agree because the exceptional polar set is \mu-null. This proves the asserted identification of the trace.

To see why the trace is injective, work in a fixed smooth planar coordinate patch and take a Euclidean disk B_{3r}(x) contained in that patch. Round and Euclidean balls are uniformly comparable on its fixed buffer, so (8.6) holds there for Euclidean balls after changing constants. Rescale B_r(x) onto B_1 and let \widehat\mu_{x,r} be the pushforward of \mu|_{B_r(x)}/\mu(B_r(x)). Let \mathcal L_{B_1} be normalized Lebesgue measure on B_1. For 0<s\le1 and any center y,

Another layer-cake calculation gives

The Dirichlet Green kernel of B_3 is nonnegative and, on B_1\times B_1, is bounded by C(1+\log_+(1/|y-z|)). Its analogous bound for \mathcal L_{B_1} is uniform. The Dirichlet version of the regularization argument (8.14), applied to \widehat\mu_{x,r}-\mathcal L_{B_1}, therefore yields

Here H^{-1}(B_3;\|\nabla\cdot\|_2) is the dual of H^1_0(B_3) with its Dirichlet norm. One obtains the identity by smoothing inside B_3 and then taking limits, just as above; the supports in B_1 leave a fixed boundary buffer, and (8.18) provides the integrable dominating logarithm.

For smooth f, set F(y)=f(x+ry) and choose a smooth cutoff \chi supported in B_3 and equal to one on B_1. Test (8.19) on \chi(F-|B_3|^{-1}\int_{B_3}F\dd y). Poincaré’s inequality on B_3 bounds the Dirichlet norm of this cutoff by C\|\nabla F\|_{L^2(B_3)}. The signed functional has mass zero and is supported where \chi=1. Scaling back, using the two-dimensional invariance of the Dirichlet integral, gives

This inequality extends to H^1 functions. Indeed smooth approximation of the cutoff converges in H^1_0(B_3), and both the global restriction map and the functional in (8.19) are continuous; their integrals agree on the smooth approximants. If R_\mu f=0, the second average in (8.20) is zero. At almost every Lebesgue point of both f and |\nabla f|^2, its first average tends to f(x) while its right side is O_x(r\sqrt{1+|\log r|}), hence tends to zero. It follows that f=0 almost everywhere in each patch, and therefore on the sphere. Thus R_\mu is injective.

For the norm comparison, write f=c+v with c=\sigma_0(f) and \sigma_0(v)=0. Poincaré and (8.7) give

These imply the left inequality in (8.8); boundedness of restriction gives the right one.

It remains to verify the claimed trace of the potential. The j=2 case of (8.11) makes G\in L^2(\mu\otimes\mu), so T_\mu in (8.10) is well defined and Hilbert–Schmidt. For g,k\in L^2(\mu), use (8.9) with u_k and f=u_g. Since \sigma_0(u_g)=0, the cross-energy identity gives

In the last equality the terms involving \sigma_0 in (8.16) vanish by the centering (8.5). The displayed identity for every k proves (8.10) in L^2(\mu).□

Proof. Apply Lemma 8.2 to \mu_h using Lemma 8.1. Injectivity makes the formula (8.21) well defined. The norm equivalence (8.8) proves that this domain is complete in the form norm. It is exactly the closure of the smooth domain, since smooth functions are dense in H^1. Smooth restrictions are also dense in L^2(\mu): continuous functions are dense for a finite Radon measure on a compact space, and smooth functions are uniformly dense in the continuous ones. Sobolev normal contractions commute with the quasi-continuous restriction and decrease the Dirichlet integral. This proves the Markov property. The same smooth functions form a uniform core in C(\Sph) and a form core, proving regularity. The representation theorem for closed forms gives A_h and B_h; multiplying a form by two multiplies its associated operator by two, so B_h=2A_h.

This also identifies the form with the Liouville time change of standard conformal Brownian motion. Take round Brownian motion with generator \frac12\Delta_0 and base form \frac12\mathcal E^0 in L^2(\mathrm{vol}_0). The finite measure \mu charges no polar set, so it is a smooth measure for this form. The extended finite-energy space of the round form is H^1 on this compact sphere. Indeed, for an energy-Cauchy sequence, subtracting round means makes it H^1-Cauchy by Poincaré; a finite almost-everywhere limit then forces a subsequence of the subtracted constants to converge as well. Injectivity of the quasi-continuous trace now gives full quasi-support: a function in this extended space whose trace vanishes \mu-almost everywhere is zero in H^1 and hence quasi-everywhere. The time-change theorem [14, Theorem 6.2.1] therefore gives exactly (8.21) for the time change whose Revuz measure is \mu. In particular its Brownian clock is the half-form clock.

For later use the raw form is coercive on the centered subspace. If v=R_\mu f and \mu(v)=0, then \sigma_0(f)=-\mu(R_\mu(f-\sigma_0(f))). Hence

The zero-energy functions are precisely the constants, by the round Poincaré inequality and trace identification.

Let \Pi_\mu g=\mu(g)\mathbf1 and P_\mu=\Id-\Pi_\mu. Double centering in (8.22) is the orthogonal projection P_\mu\otimes P_\mu of G in L^2(\mu\otimes\mu). Thus its two marginals vanish and it is square integrable. Its operator is K_h=P_\mu T_\mu P_\mu, which is Hilbert–Schmidt and zero on constants. For centered g, Lemma 8.2 gives

This is exactly the domain criterion for the operator of the raw closed form: K_hg\in\operatorname{Dom}(B_h) and B_hK_hg=g. Uniqueness on the centered subspace follows from (8.25). Consequently K_h=B_h^{-1} there. In particular it is positive and injective on that subspace. The output projection in P_\mu T_\mu P_\mu chooses the representative of the weak solution with \mu-mean zero.

The inverse identity gives

The right side is compact, since \Pi_\mu has rank one and K_h is compact. Scaling also gives compact resolvent for A_h. The ball bounds give full support and no atoms, so L^2(\mu) is infinite-dimensional. The compact-resolvent spectral theorem and the one-dimensional kernel now give the stated eigenvalue enumeration. On the centered space the inverse eigenvalues are \nu_j^{-1}. The Hilbert–Schmidt identity yields the first assertion in (8.24). For every \delta>0,

The Weierstrass test therefore proves the heat assertion and its local uniform convergence, with the constant mode counted once.

For completeness, choose a countable rational linear space of smooth functions dense in H^1, for example rational combinations of round spherical harmonics. In the min–max formula for \nu_j one may take the infimum only over (j+1)-dimensional spans from this space: the form-norm density just proved approximates any finite-dimensional test space together with its mass and energy matrices. For each fixed tuple of smooth functions, the entries of the mass matrix are \int\phi_k\phi_l\dd\mu, which are Borel, indeed continuous, in the weak topology on probability measures; the energy matrix is deterministic. The supremum of its Rayleigh quotient can be taken over nonzero rational vectors, assigning the value +\infty when the mass denominator vanishes. This is a countable supremum of Borel functions. The countable infimum over tuples is therefore Borel and equals \nu_j on the measures under consideration. Finite heat sums are Borel maps into C_{\mathrm{loc}}((0,\infty)), and their locally uniform limit is Borel as well.

Finally, a conformal or anticonformal sphere map transports \mu and acts unitarily on the corresponding L^2 spaces. It preserves the two-dimensional Dirichlet integral. Being a smooth diffeomorphism of the compact sphere, it preserves H^1 and capacity-zero sets, so it also transports the quasi-continuous trace domain. The resulting closed forms are unitarily equivalent. This proves the asserted coordinate invariance and completes the continuum construction.□

Proof. By Proposition 2.4, for \mu-almost every z the fiber \eta_2^{-1}(\{z\}) has exactly one element among all times in [0,1]. The pushforward identity for \eta_1 therefore implies that, for almost every t, \eta_1(t) has such a unique second time; call it \tau(t). Fix one of these t, also avoiding the countable union of step endpoints. Every subsequence of \tau_n(t) has a further convergent subsequence, with some limit s. The two matched corners have the same projected vertex. The uniform tracking in (2.4) and the continuity of \eta_2 give

along this further subsequence. Singleton fibers force s=\tau(t). Thus the entire sequence converges. The all-times formulation of the singleton property is what also determines limits that could lie in a null set of second-tour times.

Extend \tau arbitrarily on the exceptional null set. It is measurable as an almost-everywhere pointwise limit of the measurable \tau_n. For every continuous g on [0,1], bounded convergence yields

Hence \tau preserves Lebesgue measure. Away from the two limiting endpoints, membership in J_n eventually agrees with membership in J. For the composed indicators the only additional exceptional times are those for which \tau(t) is an endpoint; these have measure zero by the pushforward just proved. The indicators converge almost everywhere and are bounded by one, so they converge in L^2. The assertion for finite products and differences follows in the same way.

The area pushforward makes U_r an isometry. To see surjectivity, let E_r be the set of points whose fiber under \eta_r is a singleton. For every k, the set of points having two preimages at distance at least 1/k is the continuous image of a compact subset of [0,1]^2, hence is closed. Subtracting the union of these sets from the compact image of \eta_r shows that E_r is Borel. On E_r the inverse of \eta_r is continuous in the relative topology. Indeed, convergence of image points and compactness of the time interval show that any limit of their inverse times belongs to the limiting fiber, which is a singleton. Choose a Borel representative of any element of H and compose it with this inverse on E_r, extending by zero outside E_r. This constructs a measurable function of the image whose pullback agrees with the given time function on \eta_r^{-1}(E_r). Since \mu(E_r)=1, the isometry is onto.□

Proof. Put \tau_n^{(1)}=\Id and \tau_n^{(2)}=\tau_n. Assign a time to the first interval that covers it by setting

For large n, these functions take values in \{0,1\} and sum to one almost everywhere. For a nonempty original block, let

Define the linear map on F=U_n^{(1)}f by

Thus the coefficient is always the average on the full original block; the priority rule only restricts the indicator on which this coefficient is used. There are at most L2^j summands with fixed indicator functions, proving the rank bound with C_0=L.

Equation (9.1) and the disjoint priority assignment give

where C_1=\sum_\ell C_\ell. The last inequality uses that every F_{n,\ell} is a subset of the original edge set, including its multiplicities. The map Q_{n,j} is allowed to take values outside \mathcal H_n; only its rank and its approximation error are used.

Fix j and any sequence of indices and norm-bounded functions as in the statement. Relabel its indices by n, and first extract a subsequence on which the finitely many absolute interval endpoints converge. By Cauchy–Schwarz and the isometry of U_n^{(r)},

Equation (9.2) bounds these coefficients uniformly for a norm-bounded sequence. Lemma 9.1 shows that each of the finitely many indicator factors in (9.6), and hence their product, converges strongly in H. After a subsequence the finitely many scalar coefficients also converge. This proves the fixed-level compactness. A priority piece can have limiting length zero; the bound still holds because the coefficient was taken on the original block in (9.7).

For the asserted energy compactness, start with any sequence of indices and functions with both norms and energies bounded, again relabelling its indices by n. For each successive integer j\ge J, extract a subsequence on which Q_{n,j}F_n converges, and use a diagonal subsequence. Given \varepsilon>0, choose j so large that the square root of the right side of (9.4) is less than \varepsilon/3 uniformly along this sequence. For two sufficiently late terms, the difference of their level-j approximants is also less than \varepsilon/3. The triangle inequality makes the diagonal sequence Cauchy in H, proving strong convergence.

Finally let i<\dim \mathcal H_n be sufficiently large and set

Choose i_0 so that j(i)\ge J for i\ge i_0. The span of the first i+1 orthonormal eigenvectors of B_n, including its constant mode, has dimension greater than \rank Q_{n,j(i)}\le i/2. It contains a unit vector F killed by Q_{n,j(i)}. Its energy is at most \nu_{n,i}, so (9.4) gives

This is (9.5) with c_*=C_1^{-1}(4C_0)^{-p}. Padded eigenvalues are infinite and satisfy the same bound.□

Proof. The nonzero eigenvalues of K_n, in decreasing order and with zeros appended, are \kappa_{n,i}=\nu_{n,i}^{-1}, i\ge1, where (+\infty)^{-1}=0. Its kernel is doubly centered because K_n is self-adjoint and kills constants. For N\ge i_0, assumption (ii) gives the uniform Hilbert–Schmidt tail bound

where r_N\to0 because 2p>1.

Suppose first that M_n=\norm{K_n}_{\mathrm{op}} is unbounded. On a subsequence it tends to infinity, so we may assume M_n\ge1. Choose orthonormal eigenvectors e_{n,i} for the positive eigenvalues and put \alpha_{n,i}=\kappa_{n,i}/M_n. Every \alpha_{n,i} belongs to [0,1]. By a diagonal extraction, assume these numbers converge to \alpha_i for every fixed i. If \alpha_i>0, the vector e_{n,i} exists for large n and

Assumption (i), and a further diagonal extraction over these indices, give strong limits e_{n,i}\to e_i. If \alpha_i=0, its rank-one term has Hilbert–Schmidt norm \alpha_{n,i}\to0 and needs no eigenvector limit. It follows that each fixed finite truncation

converges in Hilbert–Schmidt norm. Here e\otimes e denotes the rank-one operator F\mapsto\ip{F}{e}_H e, and a term with zero eigenvalue is omitted. For the positive limiting coefficients this uses \norm{e_{n,i}\otimes e_{n,i}-e_i\otimes e_i}_{\HS} \le2\norm{e_{n,i}-e_i}_H.

After division by M_n\ge1, the squared tail is still at most r_N^2 by (9.9). The convergent finite truncations and this uniform tail show that k_n/M_n has an L^2 limit k^*. Hilbert–Schmidt convergence implies operator-norm convergence, so the operator with kernel k^* has norm one. On the other hand, assumption (iii) implies \mathfrak D(k_n/M_n)\to0 almost everywhere, since its unnormalized numerator has a finite almost-everywhere limit. Boundedness of \mathfrak D also gives convergence in L^2 to \mathfrak Dk^*. Both imply convergence in measure, whose limit is unique. Thus \mathfrak Dk^*=0. Centering passes to an L^2 limit, so (9.8) gives k^*=0, contradicting its operator norm one. This proves boundedness of M_n.

Take now any subsequence without dividing by M_n. The bounded inverse norms allow a numerical diagonal extraction \kappa_{n,i}\to\kappa_i for every i. A positive \kappa_i gives \mathfrak e_n(e_{n,i})=1/\kappa_{n,i} bounded, so assumption (i) applies to its eigenvectors. A further diagonal extraction over every index with \kappa_i>0 gives simultaneous strong limits e_{n,i}\to e_i. A zero limiting coefficient again makes its rank-one term vanish. The same truncation argument using (9.9) supplies a further L^2 limit k^* of the kernels. It is doubly centered. Assumption (iii), boundedness of \mathfrak D, and uniqueness of limits in measure now give \mathfrak Dk^*=\mathfrak Dk. Equation (9.8) implies k^*=k. Every subsequence has a further subsequence with this same L^2 limit, proving the full claimed convergence.□

Proof of Theorem 1.1. Start with any sequence of sizes tending to infinity and pass to a common representation extraction as above. On its probability-one event, (9.11) holds. Let \kappa_i, i\ge1, be the positive eigenvalues of K in decreasing order, with multiplicity. Proposition 8.3 says that K_h is injective on the centered subspace of L^2(\mu), which is infinite-dimensional. Thus every \kappa_i is positive and

For a nonnegative compact self-adjoint operator T on H, let \kappa_i(T) denote its positive eigenvalues in decreasing order, with multiplicity and with zeros appended after a finite list. The variational formula is

where F ranges over linear subspaces. When \kappa_i(T)>0, the spectral theorem gives the upper bound by choosing the span of the first i-1 eigenvectors, and the lower bound because F^\perp meets the span of the first i eigenvectors nontrivially for every such F. When \kappa_i(T)=0, positivity gives the lower bound and a trial subspace containing the positive range gives the upper bound. For two such operators T,S, their quadratic forms on any unit vector differ by at most \norm{T-S}_{\mathrm{op}}. Comparing the two suprema for each F, and then the two infima, therefore gives

Applying this to K_n,K yields

The characterization applies with multiplicities and with zeros appended to the finite-rank spectra. Since \kappa_i>0, for each fixed i the number \kappa_{n,i} is positive for all sufficiently large n, and reciprocation gives \nu_{n,i}\to\nu_i. The constant eigenvalue is zero on both sides. With the conventions of Lemma 2.1 and Proposition 8.3,

Zero padding for the inverses is exactly the prescribed infinite padding for the finite generator spectra. This proves coordinatewise convergence, including every multiplicity, with acceleration exactly n.

For the heat traces, use e^{-t\infty}=0 to write

Fix 0<\delta<T<\infty. For N\ge i_0, the high-index bound (9.5) gives, for all sufficiently large n,

The summability follows, for example, by comparison with the integral of e^{-\delta c_*x^p/2}. Passing to the limit in each fixed index in (9.5) gives the same bound for \nu_i and the same estimate for the continuum tail. For a fixed finite set of indices, (9.12) gives uniform convergence of the corresponding exponential terms on [\delta,T], since for finite x,y\ge0,

Combining the finite sums and (9.13) proves H_n\to H_h uniformly on [\delta,T], hence in C_{\mathrm{loc}}((0,\infty)). Finiteness and measurability of the limiting heat trace were established in Proposition 8.3.

It remains to express the metric and measure convergence in the requested intrinsic topology. This uses the same coordinate sphere as the inverse kernel above. Write \Gamma_n and \Gamma for the two graphs in (2.3), and let \varepsilon_n\to0 bound their Hausdorff distance in the maximum product metric of round distance and ordinary distance on [0,\infty). By uniform continuity of D_h on \Sph\times\Sph, its modulus

tends to zero as r\downarrow0. Approximating each point of \Gamma_n by a point of \Gamma gives

Approximating (x,x,0)\in\Gamma in the opposite direction also shows that z_n(V_n) is \varepsilon_n-dense in the round sphere.

The pushed measures \bar\mu_n=(z_n)_*\mu_n converge weakly to \mu by Proposition 2.4. On the compact round sphere this is convergence in Prokhorov distance. Choose \rho_n\downarrow0 larger than that distance and than \varepsilon_n. The coupling characterization of Prokhorov distance gives a coupling of \bar\mu_n and \mu assigning mass at least 1-\rho_n to pairs at round distance at most \rho_n. Lift it to a coupling of \mu_n and \mu by the finite conditional distribution of a vertex given its projection z_n(v). The relation

is a correspondence, by round density, and the lifted coupling gives it mass at least 1-\rho_n. Its distortion for a_nd_n and D_h is at most

by (9.14). This tends to zero. For completeness, if a correspondence \mathcal R between spaces X,Y has distortion less than 2\xi, the cross-distance

together with the original distances gives a metric on their disjoint union. The distortion inequality verifies the triangle inequalities that cross between the spaces. Related pairs have distance at most \xi, so the Hausdorff distance is at most \xi. A coupling giving the relation mass at least 1-\rho makes the Prokhorov distance at most \max\{\xi,\rho\}. Apply this with \xi=\tfrac12\operatorname{dis}(\mathcal R_n)+1/n and \rho=\rho_n. We obtain

The normalization a_n here is exactly the deterministic diameter normalization in Theorem 1.1; its identification with the scale in the companion distance graph was proved in Lemma 2.3.

All three convergences have been proved on this one representation with the same \mu_h, D_h, and conformal sphere. The conformal or anticonformal sign retained by Proposition 2.4 preserves the round Dirichlet integral, so it also preserves the spectrum defined from this sphere. The continuum spectral readouts are measurable functions of its area in that coordinate by Proposition 8.3. Thus the joint limiting law on this extraction is the one in Theorem 1.1, with its stated clock.

Every original subsequence permits a further common extraction with these almost-sure conclusions. The Gromov–Hausdorff–Prokhorov space of compact metric probability spaces, the coordinatewise product [0,\infty]^{\mathbb N_0}, and C_{\mathrm{loc}}((0,\infty)) are Polish. The subsequence criterion for convergence in distribution therefore gives the asserted joint convergence through all positive integers n.□