Proof. The construction and the first three assertions are the construction preceding Lemma “Uniform topological joining”, that lemma, Lemma “Exterior connection and small-loop filling”, Proposition “Degree and robust planar topology”, and Lemma “Separation of entire fibers” in [22]. We explain their applicability to both graphs and to the plane.

The hypotheses are uniform contour convergence, vanishing projected cell mesh, and approximation of the limiting fiber generators by discrete links with small projected connecting paths. For a Brownian excursion above a level, use last and first crossings of nearby levels on the two sides of an interior time. Their discrete analogues approximate the chord. At a triple, apply this on the two excursions on either side of its middle minimum. The fiber description above therefore supplies every generator required for topological joining. Interchanging the coordinates gives the same argument for the dual. Applying planar separation to the two embedded graph paths gives the mixed crossing assertion: a primal path and a dual path can meet only through a primal edge and its dual, with the usual corner subdivision making the intersection literal on the surface.

For a bounded plane region, choose a window containing all its visits and its fixed additional star layers. Both contour coordinates make strictly lower visits on both sides of a sufficiently large window. Lemma “Protected capping of a bounded window” in [22] then supplies a spherical completion preserving all these incidences. Its continuum completion is absolutely continuous with respect to a Brownian excursion, which supplies the sphere quotient needed for the preceding topological results.

For the integer assertion, connect all lifts of each well away from the tested loop, using the first two assertions. A two-chain bounded by that loop has constant multiplicity along each such connected fiber. Projection multiplies the difference of its multiplicities at the two fibers by \(\deg\pi_m=\pm1\). This is exactly the equality of winding differences. Equivalently, one can use relative homology after a simplicial perturbation inside the fixed margins. ◻

Proof. We use the exact-incidence part of the proof of Lemma “Finite-band reconstruction” in [22]. We give the argument because exactness, rather than an approximation of path costs, is essential for energy tests.

Choose \(r>0\) so that the closed \(3r\)-neighborhood of \(K\) lies in \(U\). Properness bounds all its visit times in a finite window. Refine the Poisson cuts until every cell meeting this neighborhood has diameter less than \(r\). Select all cells meeting the closed \(2r\)-neighborhood of \(K\) and merge consecutive selected cells into maximal runs. These runs are disjoint intervals; all visits sufficiently near \(K\) lie in them. Their endpoints stay away from \(K\), since otherwise the neighboring cell would also have been selected. Local traversal data identify the selection: first inspect a slightly larger buffer and require the portions around every visit to the smaller compact to be flanked by marks while they remain in that buffer. Properness, continuity, and refinement make this condition hold eventually. No global index for the retained runs is part of this inspection.

Consider one coordinate and two selected runs \(b<d\) in chronological order. An outgoing point of \(b\) and an incoming point of \(d\) at height \(H\) are linked exactly when \[ H\le\min_{\text{between the two runs}} Z^a. \tag{17}\] The same assertion holds discretely, with the full-strip convention for a matched step. Split the intervening interval into selected runs and omitted gaps. If a limiting link projects near \(K\), any intervening time at its height represents that same continuum point. It must therefore belong to a selected cell. Thus every omitted-gap minimum is strictly above the height of that link. The relevant endpoint pairs form a compact set, and there are only finitely many gaps. Their positive slack is consequently uniform: a sequence of slacks tending to zero would limit to a contact in an omitted gap, contrary to the selection.

An obstruction without slack can only be a minimum of a selected run. At a triple contact, the three directed seams in Figure 1 identify the middle run and recover its offset through (15). Formula (16), evaluated on the discrete string, supplies the cap in (14). At a natural lower seam endpoint, choose the lower rational band endpoint below the record minimum; the exact incoming or outgoing record condition then clips the band at that minimum. This retains links at the minimum as well as links at nearby lattice heights.

For every pair of runs, cover the compact set of relevant link heights by finitely many such bands. Choose each band to be admissible over its entire effective height interval: that interval lies in a genuine shared seam, its omitted-gap comparisons have positive slack, and every allowed non-strict cap is the exact selected minimum just described. The natural record conditions provide the exact clipping at lower seam endpoints. Include the positive seam intervals needed to read the relative heights, and retain a spanning forest of these comparisons in each coordinate. All needed offsets are sums along that forest. Seam directions specify the order of linked runs and distinguish each middle cap; the instructions do not need an order between unrelated runs. Each band retains every discrete link satisfying its rule. Uniform contour convergence, the positive omitted-gap slack, and the exact selected minima show two things: every actual link near \(K\) belongs to one of these bands eventually, and every permitted link in those effective bands is an actual link eventually. In particular the latter assertion applies to any permitted link with one endpoint over a smaller core collar: its two limiting endpoints lie on the same shared-seam point and hence remain inside the captured collar. The assertions hold simultaneously for all links, regardless of their number.

Vertex equality is the same first-coordinate minimum test, so it is captured as well. Start a chain of enabled identifications at a core representative. Its first link is an actual identification by the preceding band validity. All visits to that actual vertex have uniformly close projections, independently of the instruction graph, and lie in the captured collar. Inductively the next enabled link is therefore actual as well and remains at that vertex. Thus a peripheral chain cannot introduce an additional equality at the core. Horizontal tree traversals with the same identified endpoint pair give one tree edge; no two distinct tree edges have that pair. Added edges are instead identified by their matched step-occurrence pair, preserving distinct parallel edges. Hence the result is equality of graphs with incidences, not merely equality of their path relations. Applying the argument to the other coordinate gives the dual conclusion.

All seam, cap, and positive-margin conditions are measurable in the oriented local cells and frontier lengths. Rational bands, integer lower bounds on positive margins, and finite forest choices give a countable selection rule. Taking a further collar before the final restriction gives the stated neighborhood of \(K\). Inside a fixed interval cell, every visit to a compact subset of its interior lies in that interval with a time margin; the same argument then uses only its internal string. ◻

Proof. On the finitely many selected windows, the limiting parameterizations are continuous. Cover the inverse image of \(K\) by finitely many relative-time intervals with closures projecting inside \(O\). Their endpoints may be chosen from the prescribed dense countable relative-time library. Uniform projection accuracy and the complete collar of Lemma 5 give the desired inclusions. Select a vertex when any of its visits is in this union. The same argument applies to each port, while vanishing edge mesh allows an induced-edge rule within the complete prescription.

For value constraints, refine the intervals until the oscillation of \(u\circ\eta\) on each is less than a fixed fraction of \(\delta_+-\delta_-\). Choose a rational value interval between the two allowed tolerances. Impose it at every selected visit in that interval, retaining all simultaneous constraints when a vertex has several visits. The strict tolerance gap and uniform projection error give the two implications. The construction chooses among an already countable collection of tests; it does not add a new exceptional null set for each target \(u\). Compact exhaustion and rational positive tolerances therefore yield common measurable versions for all continuous targets. ◻

Proof. This is the argument of Lemma “Finite reference experiment and conditional transfer” in [22]. We include its measure calculation to specify its application to potentially discontinuous variational outputs.

Write \(\epsilon_m=m^{-1/2}\). For one coordinate, order all selected blocks chronologically as \(1,\ldots,k\). Let \(s_i\) be their starts, \(e_i\) their net increments, and \(g_i>0\) the intervening durations. Complete the union of the package forests to a tree \(T\) on these blocks. Give its extra offsets independent positive densities. If the density on an edge \(e\) is \(\rho_e\), use lattice mass \[r_{m,e}(a)=\int_{a-\epsilon_m/2}^{a+\epsilon_m/2} \rho_e(u)\,\mathrm du, \qquad a\in\epsilon_m\mathbb Z.\] Let \(p_t(j)\) be the transition mass of one coordinate of the unscaled rate-one axial Poisson walk, so its variance rate is \(\sigma^2=1/2\). The exact density of the actual relative starts with respect to these reference offsets is \[ W_m^a= \frac{\displaystyle\prod_{i=1}^{k-1} p_{mg_i}\bigl(\epsilon_m^{-1}(s_{i+1}-s_i-e_i)\bigr)} {\displaystyle\prod_{e\in T}r_{m,e}(a_e)}. \tag{19}\] Indeed, tree differences and consecutive differences are related by an integer linear map with integer inverse, obtained by summing along paths in \(T\). Its determinant is \(\pm1\), so there is no lattice multiplicity. Multiply (19) for the two coordinates and include the density ratio for the durations and cuts to obtain \(W_m\).

On compact input sets with all gaps bounded below by a positive constant, the local central limit theorem gives \[\epsilon_m^{-1}p_{mt}(j) \longrightarrow (2\pi\sigma^2t)^{-1/2} \exp\{-x^2/(2\sigma^2t)\}, \qquad x=\epsilon_m j,\quad \sigma^2=\tfrac12,\] uniformly in bounded \(x\) and in such \(t\). The numerator and denominator of (19) have the same number of lattice factors. They cancel, leaving uniform convergence \(W_m\to W\) to a continuous finite function on these compact sets.

Conditional on the selected inputs, the omitted strings are independent bridges and are independent of the numerical outputs. Their rescaled laws converge uniformly on these input sets to Brownian bridge laws. For completeness, their finite-dimensional densities are ratios of transition masses, so the preceding local limit theorem applies. On the first half of a bridge the density relative to the free walk is bounded by \[\frac{\sup_j p_{m(t-u)}(j)}{p_{mt}(j_0)}\le C, \qquad u\le t/2,\] when \(t\) and \(\epsilon_mj_0\) range in the chosen compact sets. The numerator is at most \(C\epsilon_m\) and the denominator at least \(c\epsilon_m\). Free-walk tightness and the reversed argument on the other half prove uniform bridge tightness. The same applies to a finite list of gaps and to exterior contour portions on a bounded window.

The full independent reference experiments converge jointly to the product of their limits, because their output spaces are compact and their input marginals converge. Let \(a\) denote the completion offsets and \(z\) the omitted bridges. Integrating bounded continuous bridge tests first, the preceding uniform convergence gives the limiting actual law \[ W(x,a)\,\nu(\mathrm dx,\mathrm da)\, \mathcal B(x,a;\mathrm dz)\prod_rK_r(x_r,\mathrm dy_r). \tag{20}\] The weight and the bridge law depend only on contour inputs. The complete contour and cuts recover \((x,a,z)\), so disintegration of (20) gives (18).

To remove the compact restrictions, first keep all finitely many limiting gaps positive, then bound endpoints and offsets, and finally restrict the block paths to compact sets. Under the actual law the excluded probability tends to zero by finite-window contour tightness. If \(P_m\) is the actual law and \(Q_m\) the reference law, the exact identity \[\mathbb E_{Q_m}[W_m\mathbf 1_{A^c}]=P_m(A^c)\] controls the omitted density mass. No global bound on \(W_m\) is required. Exhaustion, followed by a monotone-class argument, gives the assertion for bounded measurable tests and for the entire bilateral contour.

A change of reference density adds another input-only density ratio, continuous in the limit on compact sets, and thus leaves \(K_r\) unchanged. Forest bases are related by unimodular integer maps. The pushed-forward lattice cells and the directly discretized cells have the same volume and vanishing diameter. Their mass ratios converge to one on compact sets; tightness makes the two lattice laws converge in total variation. This comparison remains valid after applying a discontinuous graph-output map. Absolute time labels and the order of unrelated blocks occur only in \(W_m\), and disappear from the conditional output law.

For merged tests one must also check the old output. Begin with the old independent blocks, their offsets, and their old numerical outputs. Insert the formerly omitted positive-duration gaps by the exact density and bridge construction. The resulting string has the new master-block law. Recompute the old graph from its unchanged active windows and bands, and compute the new union graph as a separate output on that same string. Formula (20), applied before conditioning on the new master string, proves the old conditional marginal. It never evaluates a previously chosen kernel on a forced-offset diagonal. Windows with a common endpoint were already within one maximal run, so no positive-density argument is applied to a zero-length gap. Jointly retaining finite vectors proves consistency for all refinements and unions, and then for the countable library.

Every step used only compactness of the output coordinates and their measurability from their own package. Thus it applies to the energy and extremal-length tests, with any deterministic normalization. It does not require continuity of those outputs or preservation of conditional independence under arbitrary weak limits. Finally, after conditioning the identities hold for all countably retained templates on one event; contour- and cut-measurable selection from them is therefore permitted. ◻

Proof. Proposition “Background locality” in [22] states that \(\widehat h\) in a buffer determines the oriented curve segments through the testing region. The restriction of \(h\) gives their area parameterizations and their directed frontier lengths. These data determine the locally retained marked cells. For a segment from time \(s\) to time \(t\), its coordinate increment is the outgoing frontier length minus the incoming frontier length of that coordinate, with tagged multiplicities retained: \[ Z^q_t-Z^q_s =\nu_h(\partial_q^{\mathrm{out}}\eta([s,t])) -\nu_h(\partial_q^{\mathrm{in}}\eta([s,t])), \qquad q\in\{L,R\}. \tag{21}\] Formula (21) reads the increments without an absolute contour height. Positive shared seams read the forest offsets, and the three-seam configuration in Lemma 5 reads the offsets needed at a minimum cap. Durations and relative cuts are local area data. Thus all four types of input in Lemma 7 are local.

For finitely many prescriptions in disjoint spatial domains, merge within each domain. Different domains use disjoint cells; two runs from different domains cannot share a cut, since its image would belong to both compactly contained cells. Their intervening chronological gaps are therefore positive even if the runs interleave in time. Lemma 7 gives their product law conditional on the contour and cuts. Its invariance under labels and forest bases lets us enumerate the cells locally without retaining a global cut index.

Adjoin the continuum background by the conditional sampling convention above. The product formula remains valid upon conditioning additionally on the fields. Its factors are now measurable functions of the local fields and spatial marks, by the preceding input reconstruction. Conditional expectation therefore gives the same product formula given the complete fields and spatial marks. This is exactly the argument of Proposition “Spatial product kernels” in [22], applied to the enlarged output vector of Lemma 7.

The time cuts are independent of the continuum background. Area parameterization and the Poisson mapping theorem give their spatial conditional intensity \(\rho\mu_h\) for a layer of time intensity \(\rho\). Write \(\Pi_i\) for the marks in the \(i\)th domain and \(K_i^{\mathrm{marks}}(h,\widehat h,\Pi_i;\cdot)\) for the conditional output kernel already obtained. Given the fields, the marks have independent Poisson laws, and hence the output law after removing them is \[\bigotimes_i\int K_i^{\mathrm{marks}} (h,\widehat h,\pi_i;\cdot)\, \operatorname{Pois}_{\rho\mu_h|_{U_i}}(\mathrm d\pi_i).\] This proves the fields-only assertion. It uses no independence between the marks and the numerical arrays, whose tests may depend on those marks. Alternatively, retain each \(\Pi_i\) together with its array as the local output. Finite joint consistency and the countable prescription library then give the kernel on all tests. Lemmas 5 and 6 identify its buffered exhaustion with the corresponding observations from the actual graph. No additional conditioning on exterior numerical outputs is used. ◻

Proof. This is the finite-experiment identity underlying Proposition “Intrinsic transport and fixed quantum units” in [22]. A conformal change leaves quantum areas, frontier lengths, directed seam roles, and their relative heights unchanged. Hence it leaves the finite instruction graph unchanged and only changes the projection of its labels.

Adding \(c_v\) multiplies areas by \(v\) and boundary lengths by \(\sqrt v\). Thus the block inputs transform as \[ \ell\longmapsto v\ell, \qquad z(t)\longmapsto\sqrt v\,z(t/v), \qquad o\longmapsto\sqrt v\,o. \tag{23}\] Cuts transform with time; band endpoints, their anchors, and selected minima transform with height. A rate-\(m\) Poisson string of duration \(v\ell\) is, after this time change, a rate-\(mv\) string of duration \(\ell\). The height lattices agree because \(\sqrt v\,(mv)^{-1/2}=m^{-1/2}\). Matching strips and minimum comparisons are exactly the same integer tests. Therefore the numerical costs on the resulting graph are identical. Smooth sampling densities may change, but Lemma 7 leaves the conditional kernels unchanged.

The cuts must be transformed too. At field \(g+c_v\), the layer corresponding to a fixed layer at field \(g\) has time intensity divided by \(v\). Its spatial intensity is then unchanged. Choose a countable family of layers closed under these changes, with independent layers and nested refinements as required. Their joint input and output identities give (22). A fixed field translation is locally equivalent to the Gaussian reference law, as specified below; this makes the identity valid under that fixed reference marginal as well. No estimate uniform in unbounded \(v\) is asserted or needed. ◻

Proof. This is Lemma “Ordinary cone patches and exterior Gaussian pins” in [22]. Inside the unit disk the circle-average cone field has the law of the circle-average-zero whole-plane GFF minus \(\gamma\log|z|\). On the patch, this deterministic term has a smooth extension supported away from the marked point and the pinning circle. The Cameron–Martin theorem and Gaussian domain comparison give local equivalence, with the actual constant retained. The same argument applies to a fixed additive shift on the patch. The imaginary field is independent and is interpreted modulo its angular period; its local restrictions have the corresponding Gaussian equivalence and stationary-phase coordinate invariance.

If \(Z=\mathrm d\nu/\mathrm d\mu\) is a field density and \(K_i\) any of the conditional kernels, then for every event \(A\) and every \(R>0\), \[ (\nu\otimes K_i)(A) \le R(\mu\otimes K_i)(A) +\mathbb E_\mu[Z\mathbf 1_{\{Z>R\}}]. \tag{25}\] The last term does not depend on \(i\). This proves the uniform probability comparison; local equivalence gives both directions for probability-one events. A test domain containing the full pinning circle is excluded by the stated buffer condition. ◻

Proof. The discrete assertion follows by exchanging the two independent coordinate strings and their forests. In the continuum, spatial reflection combined with negation of the reflected imaginary field exchanges the two angle trees and the two sides of the curve. The space-filling ordering is preserved: it can be read either from the \(+\pi/2\) tree with right merges or from the \(-\pi/2\) tree with left merges, as in [18]. Both descriptions are exchanged by this operation, so no reversal of area time is involved. The cone field and the independent stationary-phase imaginary field have the required reflected law. Boundary lengths exchange \(L\) and \(R\), and conditional sampling of the background therefore gives the same operation on the joint raw-array limit. The intrinsic kernels and the selector construction then transport the spatial tests. Fixed rotations and translations follow from coordinate covariance and local field equivalence on buffered ordinary patches. A countable group of such transformations can be retained simultaneously. ◻

Proof. All numerical coordinates belong to compact spaces, and the input contours are tight on bounded windows. Include the countable transformed instruction lists and mark layers before the diagonal extraction. The preceding results apply to every finite collection. Their exact density and bridge calculation passes to any further limit by compact truncation; the continuum background passes by the \(L^1\) conditional sampling argument. Countable intersection and the selector exhaustions give the stated simultaneous version. ◻

Proof. The traffic vectors form the convex hull of the path-incidence vectors. Let \(j\) be its point nearest to the origin. The supporting inequality \(\langle j,j_P\rangle\ge\|j\|_2^2\) for every path \(P\) shows that the density \(j\) gives a value at least \(\|j\|_2^2\) in (26). Conversely, averaging path length under any \(\nu\) and applying Cauchy–Schwarz gives \(\mathop{\mathrm{EL}}(\mathcal P)\le\|j_\nu\|_2^2\). This proves (27). In the wired-set case loops may be removed from paths. Absolute potential increments give a charging density, while the distance from the first set in any charging density, clipped at one, gives an admissible potential whose energy is at most the squared density norm. These two constructions prove the reciprocal identity. ◻

Proof. Let \(\mathcal H\) denote the pair of fields. By Proposition 8, \(\mathbb E[Y(z)\mid\mathcal H]\) is measurable with respect to the field restrictions to every neighborhood of \(z\). This field germ is trivial. Indeed, for a zero-boundary Gaussian free field, the intersection of the Gaussian subspaces generated by restrictions to shrinking disks is orthogonal to the Gaussian variables associated with Dirichlet functions vanishing near \(z\). Such functions are dense in the Dirichlet space: logarithmic cutoffs express the zero capacity of a point. The intersection is thus zero, and the Gaussian chaos decomposition gives triviality of the corresponding sigma-field. Apply this to both independent fields. Local equivalence in Lemma 10 transfers the conclusion to the ordinary chart, including the imaginary field modulo its period. Consequently \(\mathbb E[Y(z)\mid\mathcal H]=y(z)\) for a deterministic \(y\).

For distinct \(z,w\), disjoint sufficiently small neighborhoods and the product-kernel assertion give conditional independence of \(Y(z)\) and \(Y(w)\) given \(\mathcal H\). Hence, for every rational disk \(D\) compactly in the chart, \[\mathbb E\left[\left(\int_D(Y(z)-y(z))\,\mathrm dz\right)^2\right]=0.\] The diagonal in \(D\times D\) has Lebesgue measure zero. Countably many disks and differentiation of integrals prove the assertion. Applying the argument to bounded separating coordinates proves the final statement. ◻

Proof. Apply the cited lemma to the kernels of Proposition 8, with the intrinsic transport of Proposition 9. Its hypotheses are the local product rule, the actual-field coordinate rule, and a common fixed-index change of quantum unit; all are supplied by Proposition 12. The real-field parameter \(Q=2/\gamma+\gamma/2\) is greater than two. The cost-index factor in the stated annulus lemma is undone in our numerical tests; the proof of that lemma uses the factor only in the indexed observation rule. Thus it applies equally to the present unchanged network costs. Local Poisson marks are included in the kernels with their corresponding intensities.

For completeness, the probability transfer needed here is a fixed one. Work first with the zero-boundary Gaussian reference patch of that lemma. Local equivalence, retaining the actual additive constant of the real field, gives a single absolute-continuity modulus from our chosen ordinary cone patch to this reference patch. Choose \(\varepsilon\) in (28) sufficiently small that a barrier event of probability \(1-2\varepsilon\), with a strict threshold as above, has reference probability at least \(1-\varepsilon_*\). The annulus lemma gives an exponentially small failure probability at scale \(N\). Borel–Cantelli gives the eventual assertion. A fixed local change of field law transfers this probability-one assertion to any ordinary chart. No likelihood bound uniform over those charts is needed. Finally, success of an annulus is the existence of a retained EL coordinate strictly above a fixed threshold. No density or path is retained as an additional output. All centers, radii, indices, and prescription templates belong to countable libraries. On their simultaneous convergence event, the successful local prescription on each realized annulus gives an eventual discrete EL lower bound. Lemma 13 then constructs a density on that finite graph. For fixed \(N\) only finitely many annuli are used, so their thresholds in \(m\) can be met simultaneously, even though the successful annuli and prescriptions were selected from the fields and arrays. Thus \(N\) is fixed before \(m\) is sent to infinity. ◻

Proof. On a successful shell of radius \(r\), multiply its barrier density by \(r\). Its charge is now \(r\) and its squared norm is at most \(Cr^2/b_m\). Sum these densities over the shells whose central holes meet \(U\), using one density for each geometric shell, and divide by \(N\). At any fixed radius the buffered shells have bounded overlap, because their centers lie on a mesh of comparable spacing. There are at most \(CN\) radii. Cauchy–Schwarz across these radii gives (29); all the involved shells lie within distance \(CR_N\) of \(U\).

For the charge estimate choose a coordinate in which the projected path has oscillation comparable to its diameter. Consider levels between its extreme values, discarding \(CR_N\) near either end. At a visit to each remaining level, the point lies in at least \(cN\) central holes. Since all these shells have radius at most \(R_N\), the path crosses each of them before reaching a suitable extreme. A shell of radius \(r\) can account for at most \(Cr\) of the levels. Integrating over levels therefore bounds the total length in the summed densities from below by \(cN\) times the remaining coordinate oscillation. Divide by \(N\). At fixed \(N\) the vanishing projected mesh permits full-edge crossing portions; the discarded endpoint pieces contribute \(o_m(1)\). Distributing each edge’s density uniformly along that edge permits fractional first and last edges and makes additivity over disjoint intervals explicit. ◻

Proof. Suppose otherwise. Choose approximate record lows \(m\to\infty\) such that \(b_m:=q_m\to0\) and \[q_s\ge c b_m\qquad(1\le s\le m,\ s\text{ in the chosen rate set})\] for a fixed \(c>0\). The bounded-rate lower bounds ensure \(m\to\infty\). For each fixed \(i\), the quantile at \(m2^{-i}\) therefore gives the high-probability barrier event in units \(b_m\). Extract jointly with all these backward offsets. Lemmas 16 and 17 apply in the comparison region.

The first inequality of (28) and Lemma 13, on the other hand, supply with positive probability one graph’s probability flow \(\nu_m\) between the two macroscopically separated ports, confined in a fixed compact set, with \[ \sum_e j_{\nu_m}(e)^2\le Cb_m. \tag{32}\] Retain the event indicator, choose a graph on a positive-probability subevent, and work pathwise in a further coupling. We establish two properties of the projected flows before using locality.

Compactness of the flows. For a path \(P\), let \(J_h(P)\) be the maximum number of disjoint subintervals making a projected advance of at least \(h\). Choose \(N\) large enough that \(CR_N<ch/2\) in (30), and then take \(m\) large. Use a fixed larger box union \(U\) containing the compact confinement. By (29), (32), and Cauchy–Schwarz, \[ \int J_h(P)\,\nu_m(\mathrm dP)\le C/h. \tag{33}\] Thus the paths are tight after reparameterization. One explicit choice of parameters is useful. At every dyadic \(h=2^{-l}\) place clock atoms at successive completed moves of distance \(h\), each with mass \(2^{-l}/(1+J_h(P))\). Add the original traversal time of total mass one, invert this clock with pauses at its atoms, and rescale its total duration to one. On sets where every \(J_{2^{-l}}\) has a fixed bound, these parameterized paths are equicontinuous: an oscillation of several times \(2^{-l}\) must cross a whole atom interval of a fixed positive duration. The bounds (33) and a countable union bound make such sets have arbitrarily large flow probability. The finitely many initial resolutions omitted at an individual scale can be included by enlarging its bound. A subsequence therefore converges to a probability law \(\nu\) on continuous paths. Their endpoints remain separated.

An absolutely continuous measure of arclength. For at most \(J\) disjoint path intervals compactly confined in an open box union \(U\), the same argument gives \[ \int\sup\sum_{k=1}^{J} |\pi_m(P(t_k))-\pi_m(P(s_k))|\,\nu_m(\mathrm dP) \le C\sqrt{\mathop{\mathrm{vol}}(U^{+CR_N})}+CJR_N+o_m(1). \tag{34}\] The supremum may be taken over rational intervals with closure in the prescribed open condition. It is lower semicontinuous under uniform convergence. Pass first to the path limit, next let \(N\to\infty\), and finally let \(J\to\infty\). These interval sums exhaust variation on open sets. With a larger \(U\) containing all paths, they first show that \(\nu\)-almost every path is rectifiable with integrable length. Its average arclength measure \(\Lambda\) consequently satisfies \[ 0<\Lambda(\mathbb C)<\infty, \qquad \Lambda(U)\le C\sqrt{\mathop{\mathrm{vol}}(U)}. \tag{35}\] The positivity follows from the separated endpoints. Open box covers now show \(\Lambda\ll\mathop{\mathrm{vol}}\).

We next convert a tangent of a limiting path into a local event. Choose a sufficiently fine finite grid \(\mathcal W\) of unoriented directions, closed under quarter-turn and reflection in a coordinate axis. For \(z\in\mathbb C\), \(w\in\mathcal W\), and \(s>0\), consider the tube \[ T(z,w,s)=\{z+tw+lw^\perp: |t|<2s,\ |l|<s/16\}. \tag{36}\] A strict crossing test in this tube has ports strictly beyond the two lines \(t=-s\) and \(t=s\), and has confinement and ports with positive clearance from the tube boundary. It is computed on a complete local prescription. Accept all countably many selectors with these properties, and require that the prescription be readable within a fixed-factor enlargement of the tube, with positive margins. Lemma 6 makes existence of such a test independent of the particular selector library after arbitrarily small relaxation of the margins.

At \(\Lambda\)-almost every point there is a direction \(w\in\mathcal W\) for which, at every sufficiently small dyadic scale \(s\), one of these tests on the graph supplying \(\nu_m\) has finite limiting \(\mathop{\mathrm{EL}}/b_m\). Indeed, arclength parameterization gives a unit tangent at almost every parameter of almost every rectifiable path. The fine direction grid puts that tangent inside one of the tubes with strict slack at all sufficiently small scales. Almost every path is in the support of \(\nu\). The existence of the strict crossing subpath is an open path event, so it has positive \(\nu\)-probability at each such fixed scale. Restrict \(\nu_m\) to this event, take the crossing portions, and normalize. Its traffic still has squared norm \(O(b_m)\), with a constant allowed to depend on the scale and the event probability. The complete-selector sandwich gives the asserted finite limiting test value. The statement can be imposed simultaneously at the countably many dyadic scales.

Let \(I^{\mathrm p}_w(z)\), respectively \(I^{\mathrm d}_w(z)\), be the indicator of existence of these finite-\(\mathop{\mathrm{EL}}/b_m\) tests at every sufficiently small dyadic scale, on the primal or dual graph. These are jointly measurable point-germ readouts: the tests and their fixed-factor buffers eventually lie in every neighborhood of \(z\). The preceding paragraph and (35) give a positive-area set of witnesses on our positive-probability event. Lemma 14 and Remark 15 therefore imply that one of these indicators has deterministic value one almost everywhere in ordinary charts. Reflection exchanges the graphs; rotations within the finite direction grid then give the other graph’s indicator in the perpendicular direction. The null sets can be discarded simultaneously, since only finitely many directions and two graphs are involved.

Choose a point with both perpendicular indicators equal to one, and then a sufficiently small common dyadic scale. We obtain primal and dual strict crossing families with \(\mathop{\mathrm{EL}}=O(b_m)\) in perpendicular tubes, as illustrated in Figure 2. Every path of one family meets every path of the other upstairs by the transverse-tube rule in Proposition 3. Such an intersection uses an edge and its dual. Choose traffic minimizers \(j_m,j_m^*\) by Lemma 13. Their unsigned pairing satisfies \[1\le\sum_e j_m(e)j_m^*(e^*) \le\Bigl(\sum_e j_m(e)^2\Bigr)^{1/2} \Bigl(\sum_{e^*}j_m^*(e^*)^2\Bigr)^{1/2} =O(b_m),\] contradicting \(b_m\to0\). This proves (31). ◻

Proof. Proposition 18 and the second quantile bound in (28) give the barrier hypotheses with deterministic units equal to one, at every fixed backward offset. Choose an intermediate closed ring whose two radii lie strictly between \(r\) and \(\tau r\), with all three resulting radial widths comparable to \((\tau-1)r\). Cover it by a finite union \(U\) of open boxes with closure strictly inside the target ring and area \(O_\tau(r^2)\). Apply Lemma 17 to this \(U\). First choose \(N\) so large that \(R_N\) is small compared with the intermediate width and both collars, and then take \(m\) large. The density is supported in the target ring. Every full ring crossing has a subcrossing through the intermediate ring, confined to \(U\), with advance comparable to \((\tau-1)r\). The density therefore charges it by a fixed multiple of this width. After division by the width, the squared density norm is bounded by \(C(\tau)\) times the chart constant. The clipped-distance construction of Lemma 13 gives the cutoff, with exact constants on all inner and outer lifts. One may carry out the construction on a protected capped sphere and extend by constants beyond the chart. Only the ring density changes the function.

Countable buffered comparisons make all the fixed tests available in one extraction. A disk outside the chosen countable library is handled by choosing a nearby library center, an inner library disk containing \(\overline{B(z,r)}\), and a concentric outer library disk contained in \(B(z,\tau r)\). Their radii can be chosen comparable to \(r\), with constants depending only on \(\tau\), and their radial separation at least \(c(\tau-1)r\). The preceding density estimate bounds the cutoff energy by a chart constant times the squared ratio of outer radius to radial separation. It therefore gives a uniform \(C(\tau)\) for these library tests, with the required constants on the original inner and outer sets. This use does not require a resolution threshold uniform over all disk radii. ◻

Proof. Take a dual cutoff \(b_m\) which is one on the inner well and zero on the outer well, with energy at most \(C\). These wells include all relevant lifts, with fixed collars. For \(t\in(0,1)\) away from the finitely many vertex values, color the primal faces according to whether their dual vertices satisfy \(b_m>t\). The oriented boundary of the colored faces splits into simple circuits. A face appearing on both sides of an edge contributes no boundary occurrence; other edge occurrences have their usual multiplicities.

The boundary chain has winding difference one between an inner dual vertex and an outer dual vertex, because their face colors are one and zero. After splitting the chain into simple circuits, at least one circuit separates these two vertices. The inner dual well is connected through dual edges on which the cutoff is one; such a path cannot cross any boundary edge, including at a primal articulation vertex. Hence every vertex of this inner well is on the same side of the chosen circuit. The connected outer well has the analogous property. The collars and vanishing overlay mesh put all primal inner lifts on the inner side as well, and all lifts beyond the larger well on the outer side. This uses the all-lift joining statement of Proposition 3 and remains valid when the full boundary chain touches itself. The selected circuit lies in the change annulus and has variation at most that of the entire boundary. The coarea identity and Cauchy–Schwarz give \[\int_0^1\sum_{e\in\partial\{b_m>t\}}|\mathrm dv_m(e)|\,\mathrm dt =\sum_e |\mathrm dv_m(e)|\,|\mathrm db_m(e^*)| \le \mathsf e_m(v_m;A^+)^{1/2}\mathsf e_m^*(b_m)^{1/2}.\] Choose a level at most this average. This proves (37), with an arbitrarily small slack if necessary. The same argument applies after exchanging the graphs. ◻

Proof. Fix a disk with clearance inside \(U\) and an energy bound \(E\) on a larger disk. Around its center place \(J\) disjoint annuli of one fixed aspect, all inside a prescribed small neighborhood. Choose all radii before taking \(m\) large. Their energy sum is at most \(E\), so Lemma 20 supplies a surrounding circuit on which the oscillation is at most \((CE/J)^{1/2}\). The discrete maximum principle bounds the oscillation everywhere inside by the same quantity. Taking \(J\) large, and then using finitely many such neighborhoods, proves equicontinuity. No assertion is needed at a radius depending on \(m\).

The same construction at a fixed, nonvanishing scale bounds oscillation on compact balls. Overlapping chains of balls propagate an additive normalization throughout a connected compact subset. Vertex density and the small-image joining property identify the limits on overlaps. The usual finite-net proof of Arzelà–Ascoli now gives local uniform compactness. A local potential branch is an ordinary harmonic function on each of its branch disks, so the proof also applies to branches. ◻

Proof. Given \(K\Subset K'\Subset U\) and \(0<\delta'<\delta\), insert between the two tests a finite selector test from Lemma 6. Cover the relevant visit labels by finitely many selectors on which the target varies by less than a fixed fraction of \(\delta-\delta'\), and use intermediate rational value constraints. Completeness of the selected graph preserves every edge incidence. Restriction of an admissible function for the larger test is admissible for the intermediate one; restriction from the intermediate test is admissible for the smaller one. The intermediate infimum has a retained limit. Consequently, \[\limsup_m M_m(u,K,\delta) \le \liminf_m M_m(u,K',\delta').\] Taking the two inner suprema proves the assertion about \(\liminf\) and \(\limsup\), including infinite values.

The same comparison proves the lower bound in (i). For (ii), choose almost minimizers on increasing compact sets with decreasing tolerances, and take a diagonal sequence using the \(\limsup\) formula. Tolerance slack also proves lower semicontinuity, directly from (38). Exhausting \(U\) proves inner regularity. The selector library is countable; approximation of a continuous target uses its modulus of continuity on each compact, rather than a new exceptional event. This gives the simultaneous assertion. The locality and measurability claims follow from the same selector sandwich, with its spatial buffers. ◻

Proof. The exact difference identity is \[\mathrm dz_m=\overline b_m\,\mathrm dv_m +(1-\overline b_m)\,\mathrm dw_m +\overline{v_m-w_m}\,\mathrm db_m.\] Apply convexity of the square to the first two terms, followed by \(|a+c|^2\le(1+\varepsilon)|a|^2+ (1+\varepsilon^{-1})|c|^2\). Only the transition region contributes to the last term. For finitely many weights the same identity follows by summing and using \(\sum_i\mathrm db_{i,m}=0\).

To construct weights, choose finitely many disks whose smaller disks cover the compact set and whose larger disks stay in the prescribed members of the cover. Use Proposition 19, sum the resulting cutoffs, and normalize where their sum is at least one. Clipping and division on this fixed range preserve bounded energy. Extra cutoffs with collars extend the construction to a neighborhood of the compact set. All these choices precede the limit in \(m\). ◻

Proof. Recovery sequences for \(u\) and \(v\), the discrete parallelogram identity, and the energy lower bound give \[F(u+v,U)+F(u-v,U)\le2F(u,U)+2F(v,U).\] Applying this inequality to \(u+v\) and \(u-v\) gives the reverse inequality. Homogeneity follows directly from the tolerance definition. Constant addition leaves every edge increment unchanged.

For fixed \(u\), restriction gives monotonicity and superadditivity on disjoint open sets. To prove subadditivity on \(U\cup V\), first fix a compact subset of this union. Recover \(u\) in \(U\) and \(V\), and join the recoveries with Lemma 23; the transition is compactly contained in \(U\cap V\). Let \(m\to\infty\) and then \(\varepsilon\downarrow0\) in [en:gluing-estimate]. Exhausting the compact gives \[F(u,U\cup V)\le F(u,U)+F(u,V).\] Finite subcovers and inner regularity imply countable subadditivity. For completeness, these properties give the required measure criterion as follows. Define \(\nu(A)=\inf\{F(u,O):A\subset O,\ O\text{ open}\}\). This is an outer measure by countable subadditivity. If two sets have positive distance, they have disjoint open neighborhoods; disjoint-open superadditivity shows that \(\nu\) is additive on such sets. Thus it is a metric outer measure and all Borel sets are measurable. On an open set \(O\), monotonicity gives \(\nu(O)=F(u,O)\). Local finiteness and inner regularity make this Borel measure Radon.

It remains to prove (40); this will also establish local finiteness for smooth targets. On a compact set inside \(U\), use a square grid of spacing \(r\) and a bounded-overlap family of disk cutoffs with radii comparable to \(r\). Normalize these cutoffs to weights \(b_{z,m}\) which sum to one near the compact, and set \[v_m=\sum_z b_{z,m}u(z).\] For fixed \(r\) this approximates \(u\) within its modulus of continuity at scale \(Cr\). On each edge the weight increments sum to zero. Subtracting one nearby node value and using bounded overlap bounds the energy by \[C\sum_z r^2\sup_{B(z,C'r)}|\nabla u|^2.\] Indeed each unnormalized cutoff has bounded energy, and normalization has the same bound because the denominator is bounded below and only boundedly many supports meet an edge. First take \(m\) large for this finite collection, and then let \(r\downarrow0\). The displayed sum converges to at most a constant times the gradient integral over the chosen region. Exhaustion inside \(U\) proves (40). The arguments are unchanged on the dual. ◻

Proof. We first identify the energy of smooth functions. Polarize the Radon measures from Proposition 24. The measures of the linear functions \(\ell_1(z)=\Re z\), \(\ell_2(z)=\Im z\) and their sums are absolutely continuous by (40). Define the entries of \(A\) by their densities: \[A_{ii}\,\mathrm dz=F(\ell_i,\mathrm dz),\qquad 2A_{12}\,\mathrm dz= F(\ell_1+\ell_2,\mathrm dz)-F(\ell_1,\mathrm dz)-F(\ell_2,\mathrm dz).\] Quadraticity and the upper bound imply that \(A\) is symmetric, nonnegative, and locally bounded. At a Lebesgue point \(z\) of these densities, compare a smooth \(u\) with its affine first-order Taylor polynomial \(p_z\). The seminorm inequality and (40) give \[\left|F(u,B(z,r))^{1/2}-F(p_z,B(z,r))^{1/2}\right| \le C^{1/2}|B(z,r)|^{1/2} \sup_{B(z,r)}|\nabla u-\nabla u(z)|.\] Divide by \(|B(z,r)|^{1/2}\) and let \(r\downarrow0\). Differentiation of Radon measures yields (41) for smooth \(u\).

We next prove the converse bound, including weak differentiability: \[ \int_U|\nabla u|^2\,\mathrm dz\le C F(u,U). \tag{42}\] Suppose the right side is finite. Put a square grid of fixed small spacing \(r\) in a compactly contained region of \(U\). At every grid node \(x\), apply Lemma 20 to a recovery sequence, using an annulus with inner and outer radii \(0.7r\) and \(0.8r\), and choose a value \(a_{x,m}\) at a vertex on the resulting circuit. Buffers are small fixed fractions of \(r\).

Circuits at neighboring grid nodes share a vertex. Here is the necessary topological verification. A point near the midpoint of the two centers is inside both inner disks. Points on the two opposite sides belong to exactly one inner disk and lie beyond the other outer disk; a distant point lies outside both. The all-lift separation rule therefore gives a common interior lift, a common exterior lift, and lifts with each of the two opposite memberships. Two disjoint simple circuits on the sphere with these memberships could be neither nested nor disjoint in their interiors, a contradiction. Since both circuits belong to the primal graph, their intersection contains a vertex. Consequently, \[|a_{x,m}-a_{y,m}|^2 \le 2\left(\sum_{e\in C_{x,m}}|\mathrm dv_m(e)|\right)^2 +2\left(\sum_{e\in C_{y,m}}|\mathrm dv_m(e)|\right)^2\] for neighbors \(x,y\). Write \(B_r\) for the union of the buffered grid annuli. Bounded overlap and (37) give \[\sum_{x\sim y}|a_{x,m}-a_{y,m}|^2 \le C\mathsf e_m(v_m;B_r) \le C\bigl(F(u,U)+o(1)\bigr).\] For the last inequality, this fixed finite union lies in the recovery set \(K_m\) for all sufficiently large \(m\). The piecewise affine interpolation of the \(a_{x,m}\) on grid triangles has Dirichlet energy bounded by this sum times an absolute constant. Take \(m\) large after fixing the grid, and then let \(r\downarrow0\). Continuity of \(u\) and uniform recovery imply that these interpolations converge locally uniformly to \(u\). Weak Sobolev compactness proves (42) on the compact region; exhaustion proves it on \(U\). Applying it to smooth affine functions on arbitrary disks gives \(A\ge cI\) locally.

Finally, let \(u\in C(U)\cap W^{1,2}_{\mathrm{loc}}(U)\). On smaller regions, smoothing approximates \(u\) both uniformly and in \(W^{1,2}\). Lower semicontinuity and the smooth upper bound first extend that upper bound to such continuous Sobolev functions. Apply the extended bound to the difference between \(u\) and its smooth approximations. Their \(F\)-seminorm difference tends to zero, so the smooth formula passes to \(u\). Exhaustion gives (41), including infinite integrals. If local Sobolev regularity fails, (42) forces infinite energy. All steps apply equally to \(F^*\). ◻

Proof. The derivatives of the energy measures of linear functions are point germs of the arrays: use the tolerance definition on successively smaller disks, with arbitrarily small open buffers. Choose measurable \(\limsup\) versions and a fixed default on the null set where Lebesgue differentiation fails. Lemma 14 applies to these matrix-valued readouts (or to a bounded injective encoding of their entries), so \(A\) and \(A^*\) are deterministic almost everywhere. Translation covariance makes each field a constant matrix on overlapping ordinary charts. Under a rotation, the matrix transforms by \(A\mapsto RAR^{\mathsf T}\). Rotation covariance, in particular a quarter-turn, therefore makes each symmetric matrix scalar. These uses of covariance are precisely those in Remark 15, and compare local laws with their actual field transformations. Primal–dual reflection (Lemma 11) equates the two scalar values. The upper bound and coercivity give \(0<\lambda<\infty\). ◻

Proof. All nonzero increments of the minimizer project into any fixed collar of the closed annulus eventually. Indeed, both endpoints of an edge strictly on one plate have its prescribed value, and the overlay mesh vanishes. Thus the cap contributes no uncontrolled energy term.

For the upper bound, approximate the radial potential by smooth functions which are constant on slightly wider plate collars and whose energies tend to \(C_0\). Recover them uniformly on a neighborhood of the annulus. Bounded-energy weights in the flat collars paste in the exact constants at a vanishing energy cost, by Lemma 23. Lemma 26 gives \(\limsup C_m\le\lambda C_0\).

The minimizer lies in \([0,1]\) and is harmonic off the plates. Its bounded energy and Lemma 21 give compactness in the open annulus. At a plate-boundary point, use the stacks of rings from that lemma, small enough to avoid the other plate. Every surrounding circuit meets a path lying strictly on the plate side: choose a tube there from the inner well to outside the ring and lift it by Proposition 3. Thus the circuit contains a vertex with the prescribed value. A circuit of small variation is uniformly close to that value. Inside it all remaining vertices are harmonic or have that same prescribed value, so the maximum principle proves continuity uniformly up to the plate. Fixed stacks followed by finite covers give the assertion simultaneously along both boundaries.

Any uniform subsequential limit has the condenser boundary values. The variational lower bound and Lemma 26 give energy at least \(\lambda C_0\). This proves (45); equality in the round Dirichlet principle identifies the limit and gives the last assertion. ◻

Proof. Closedness around a dual face is the primal harmonic equation. Harmonicity of a potential branch at a dual vertex is the telescoping sum of \(\mathrm du_m\) around the corresponding primal face. These identities count all incidences and are valid also for loops and repeated edges. The squared dual increments sum to at most \(C_m\) on every interior annular piece.

We justify the existence of branches without assuming regular discrete annular boundaries. In a disk with clearance inside the annulus, arbitrary dual lifts in a smaller disk are connected within the larger one. Loops used to integrate the cochain can be filled inside a further buffer, by Proposition 3; every primal vertex there is harmonic. Cellular closedness therefore makes the integral independent of the chosen path. On connected overlaps the resulting potentials differ by constants.

A dual loop with projected winding one can be built along fixed tubes and closed using small-image joining. The integer-winding statement of Proposition 3 makes its winding upstairs one, up to the fixed orientation sign. Its current integral is the total current entering one plate, since all intervening primal vertices are harmonic. Discrete summation by parts gives total current \(C_m\): the potential difference is one and its energy is \(C_m\). Hence the period is \(\pm C_m\).

Normalize branches on a finite overlapping disk cover of any compact subannulus, propagating constants through chosen overlaps. In particular, fix a cyclic chain of disks around one positively oriented circle. Continue a normalized branch along this chain, matching on each connected overlap; on returning to the first disk, the continued branch differs from the original by \(\pm C_m\). Lemma 21 gives local uniform subsequential limits. Their transition constants converge, so their period is \(\pm\lambda C_0\). Exhausting the annulus yields branches everywhere. For the energy bound, use disjoint boxes lying in branch disks. Their discrete edge energies do not overlap, and their sum is at most \(C_m\). Apply the local lower bound and then exhaust the annulus up to sets of Lebesgue measure zero. This gives the asserted bound over one turn. ◻

Proof. The total masses are bounded. Let \(\nu\) be a subsequential vague limit. To check its local lower bound, take open sets \(W\Subset O\Subset V\) and a compact set \(K\subset O\) containing \(W\) with a collar. The energy tests inside \(W\) are eventually counted by the edge measure \(\nu_m(K)\). Lemma 22 and compact-set Portmanteau give \[F^*(p,W)\le\liminf_m\nu_m(K) \le\limsup_m\nu_m(K)\le\nu(K)\le\nu(O).\] Exhaust \(O\) by such \(W\). Thus \(\nu\) dominates \(\lambda|\nabla p|^2\,\mathrm dz\). Its total mass is at most the limiting total recovery energy, which is the total mass of that same measure. Equality follows. Every subsequential vague limit is consequently the stated measure, proving the claim. ◻

Proof. Take finitely many disks covering a slightly larger closed annulus, each avoiding the origin, with angle branches \(p_i=\theta_i/(2\pi)\). Recover \(p_i\) uniformly on larger branch disks. Lemma 29 applies to these recoveries, denoted \(v_{i,m}\). Take a smooth partition of unity \(b_i\) with supports compactly inside the branch disks. Recover its weights, clip them to be nonnegative, cut them off in fixed support collars, and normalize their sum. This gives \(b_{i,m}\to b_i\) uniformly, bounded energies, supports inside the branch disks, and \(\sum_i b_{i,m}=1\) near the annulus.

There is a useful canonical cochain even though its own energy need not be bounded. On an oriented dual edge, let \[a_m(e^*)=\frac1{2\pi}\int_{\pi_m(e^*)}\mathrm d\arg z.\] Here the integral means the endpoint increment of a continuous argument lift along the projected edge. Every sufficiently small edge image lies in a branch disk, so this is well defined. Vanishing face mesh makes \(a_m\) closed around every cell under consideration, and its period around a loop is that loop’s projected integer winding.

On the support of \(b_{i,m}\) the error \(v_{i,m}-p_i\circ\pi_m\) is a single-valued function tending uniformly to zero. Define the globally single-valued error and cochain by \[ q_m=\sum_i b_{i,m}(v_{i,m}-p_i\circ\pi_m), \qquad \alpha_m=a_m+\mathrm dq_m. \tag{48}\] The support collars make this definition independent of how terms are extended by zero. Since a gradient has zero periods, \(\alpha_m\) is closed and has period exactly one, in the projected orientation. This conclusion is an identity at each finite resolution.

To bound its energy, apply the discrete product identity on each edge. On a branch disk, \(\mathrm d(p_i\circ\pi_m)=a_m\). Since the weights sum to one, the terms containing \(a_m\) cancel to give \[\alpha_m(e^*)= \sum_i\overline b_{i,m}(e^*)\,\mathrm dv_{i,m}(e^*) +\sum_i \overline{v_{i,m}-p_i\circ\pi_m}(e^*)\,\mathrm db_{i,m}(e^*).\] The second sum tends to zero in edge \(\ell^2\), by uniform convergence of the errors and bounded energy of the finitely many weights. Convexity bounds the squared norm of the first sum by the weighted sum of local recovery energies. Lemma 29 and \(b_{i,m}\to b_i\) turn its limit into \[\lambda\sum_i\int_{A_\eta} b_i|\nabla p_i|^2\,\mathrm dz =\lambda\int_{A_\eta}\left|\nabla\frac{\theta}{2\pi}\right|^2\,\mathrm dz =\frac{\lambda}{2\pi}\log\frac{R+\eta}{r-\eta}.\] One may compute first on a slightly larger closed annulus inside the cover; its boundary circles have zero limiting energy measure. The edge averages \(\overline b_{i,m}\) converge uniformly to \(b_i\) at the chosen edge locations, by vanishing mesh. This justifies both the weighted passage to the limit and restriction to the annulus. This proves (47). Finally, \(u_m\) is constant on each plate, so its nonzero increments lie in \(A_\eta\) eventually by vanishing edge mesh. Choosing additional fixed collars gives the assertion about stars. ◻

Proof. Only \(\lambda\ge1\) remains after (46). Pair \(\mathrm du_m\) with the cochain \(\alpha_m\) of Lemma 30, matching each primal edge with its crossing dual edge and using the consistent crossing sign. Then \[ \left|\sum_e\mathrm du_m(e)\alpha_m(e^*)\right|=1. \tag{49}\] To see this at the discrete level, replace \(u_m\) by the indicator of the outer side of an intermediate circle. Their difference is supported strictly inside the region of closedness of \(\alpha_m\). Cellular summation by parts therefore leaves the pairing unchanged. The indicator’s gradient pairs with \(\alpha_m\) along the oriented dual boundary chain separating the two wells. This chain has total winding one up to sign by Proposition 3; the cochain has exact unit period by (48). This proves (49). The argument uses the boundary chain with its multiplicities and does not require a simple discrete plate boundary.

Cauchy–Schwarz, (45), and (47) now give \[1\le C_m\sum_{e^*\subset A_\eta}|\alpha_m(e^*)|^2 \quad\Longrightarrow\quad 1\le\lambda C_0\,\lambda\left(C_0^{-1}+o_\eta(1)\right).\] Let \(\eta\downarrow0\). Thus \(1\le\lambda^2\), and (46) proves \(\lambda=1\). ◻

Proof. Combine Proposition 25, Lemma 26, and Proposition 31. The preceding construction used the same countable buffered tests throughout, so its simultaneous and locality assertions persist. All arguments apply in every fixed quantum-unit offset of the extraction. Their compatibility is the transport statement of Proposition 12. ◻

Proof. The lemma Excursion densities in the spanning-tree sphere section of [22] proves the following statements for the density \(f_N\) of this increment block. If \(w_N\to w\) uniformly, then \(f_N(w_N)\to f_{a,b}(w)\); the limit depends continuously on the two coordinate minima and the net increment. Moreover, \(\sup_N\mathbb E[f_N(W_N)^2]<\infty\) under the free law. These assertions follow there from the entrance-and-exit killed-walk density and its Gaussian bounds.

Realize the assumed joint convergence almost surely. For any bounded continuous test \(\Phi\), first truncate the densities at height \(K\). Sequential convergence and bounded convergence give convergence of the weighted expectations with truncated densities. The error in removing truncation is at most \(K^{-1}\|\Phi\|_\infty\sup_N\mathbb E[f_N(W_N)^2]\), and the analogous limit error has the same bound. Hence \[\mathbb E[f_N(W_N)\Phi(W_N,Z_N)] \longrightarrow\mathbb E[f_{a,b}(W)\Phi(W,Z)].\] Taking \(\Phi=1\) gives total mass one. The finite-resolution change of density depends only on the increment block, so its weighted expectation is precisely that of the additional block data under excursion conditioning. This proves the assertion. ◻

Proof. At each resolution retain the energy minima with rational value constraints and finite unions of relative-time intervals as selectors. Only increments within \(I_d\) enter these internal graphs and tests. Their compactified values are therefore permissible additional block data in Lemma 33. The corresponding free experiment is the bilateral sewing at rate \(m=2n\). In that experiment adjoin these jump-indexed internal tests to the Poisson-instruction arrays before extraction. Each added test is computed from its exact integer increment block and is unchanged when an independent Poisson clock is attached to the word. The earlier local identity applies in this enlarged extraction. To read it from the internal tests, use complete pieces and fixed spatial collars as described below. The vanishing discrepancy of the rescaled clocks only locates the selectors inside these collars; the numerical energy outputs themselves are retained unchanged, without a continuity claim for them under relabeling.

Starting with the given conditioned extraction, take a further subsequence on which the corresponding free block and its internal array converge jointly. This is possible by tightness of the free contour and compactness of the array space, simultaneously for the countably many intervals by diagonal extraction. The weighted-limit lemma then identifies the already specified conditioned block limit with the density tilt of this free joint limit.

Theorem 32 identifies the inner relaxed energies in that bilateral experiment, away from its distinguished point. The distinguished point does not belong to this interval surface: time zero is almost surely a singleton Brownian contour fiber. Indeed, each coordinate takes values strictly below its value at zero arbitrarily close to zero on both sides. Thus no nontrivial identification link in either coordinate can have zero as an endpoint, and neither can the equivalence relation generated by those links identify zero with another time. Since \(I_d\subset(0,1)\), its image excludes the distinguished point.

To express the local energy conclusion in terms of the block alone, reconstruct its abstract decorated interval surface \(H_d\) from the limiting increments using the interval reconstruction in Proposition 2. This reconstruction supplies the surface and its restricted curve modulo conformal coordinates. Evaluate the local identity in a conformal representative, using a countable atlas and compact exhaustions of its interior; conformal invariance makes the event independent of that representative. Both the restricted curve locating the labels and the coordinate domains for the targets belong to this reconstructed interval surface. For a compact \(K\) in one such domain and an open enlargement \(U\) with a collar, Lemma 5 supplies all incidences near \(K\) from the internal word. The selectors of Lemma 6 put every constraint on \(K\) between constraints on internal graphs contained in \(U\). Rational tolerances and compact exhaustion therefore express the inner relaxed energy using only the limiting block and its retained arrays. A change of the conformal representative preserves both the transported energy identity and the Dirichlet integral. The resulting assertion is intrinsic to \(H_d\); the surrounding cone or sphere embedding supplies no further data for its definition.

The simultaneous quantifier on targets can also be retained in this transfer. On a compact coordinate set use a countable dense library of continuous functions and rational tolerances. Uniform closeness of two targets enlarges or shrinks a tolerance constraint by at most that distance. The library therefore describes all the relaxed value tests, jointly measurably in the retained arrays and the target. This is a discretization of constraints, not an approximation of Dirichlet energy merely from values on a uniformly dense library. For varying charts use a countable atlas and compact exhaustions. The usual Dirichlet integral, with its infinite-value convention, is likewise Borel on continuous targets: weak derivatives and their energy can be tested against countably many smooth functions on the exhaustion. Existence of a continuous target violating the identity is consequently an analytic event, obtained by projection from its Polish function space. The event of equality for all targets is universally measurable. Theorem 32 gives precisely this simultaneous event under the free law, so absolute continuity applies to it without a target-dependent exceptional set.

Absolute continuity of the joint block law now transfers the probability-one identity to the excursion experiment. A countable intersection over \(d\), charts, and exhaustions proves the result. The rest of the sphere contour may be retained in this coupling: the complement of the event already has zero probability in the internal marginal, hence also in the joint law. ◻

Proof. Work in a coordinate disk centered at \(p\). A point has zero two-dimensional Sobolev capacity: radial logarithmic cutoffs equal to one on a shrinking inner disk and zero outside a fixed disk have Dirichlet energies tending to zero. Testing the weak-derivative identity off \(p\) with the complementary cutoffs proves the identity across \(p\). More explicitly, the additional term containing the cutoff gradient tends to zero by Cauchy–Schwarz, since \(u\) is bounded on the support of a fixed smooth test function, while the term containing \(\nabla u\) converges by its local square integrability.

For the approximation choose a smooth cutoff \(\chi_r\) equal to one on \(B(p,r)\), zero outside \(B(p,2r)\), taking values in \([0,1]\), and with \(\int|\nabla\chi_r|^2\,\mathrm dA\le C\) independently of \(r\). Set \[u_r=u(p)+(1-\chi_r)(u-u(p)).\] The uniform error is bounded by the oscillation of \(u\) on \(B(p,2r)\). For the gradient error, \[\int|\nabla(u_r-u)|^2\,\mathrm dA \le 2\int_{B(p,2r)}|\nabla u|^2\,\mathrm dA +2C\sup_{B(p,2r)}|u-u(p)|^2,\] and both terms tend to zero. The ordinary \(L^2\) error follows from uniform convergence. ◻

Proof. Retain the internal tests of Proposition 34 and also the whole-sphere tests defining uniform relaxed energy. The latter are countable compactified arrays; they need no local conditional kernel. The selector sandwich applies to them in the same way.

Suppose first that \(\liminf_n\mathsf e_n(u_n)<\infty\) and pass to a subsequence realizing this liminf. On disjoint compact-interior patches away from \(p\), Proposition 34 bounds their limiting Dirichlet integrals by the discrete energies in those patches. Exhausting the complement of \(p\) gives \[\int_{S^2\setminus\{p\}}|\nabla u|^2\,\mathrm dA \le\liminf_n\mathsf e_n(u_n).\] Lemma 35 supplies the weak derivative at \(p\). This proves the first assertion, including its infinite-value case.

For the upper bound, first take a continuous Sobolev target that is constant near \(p\). Cover the complement of a smaller constant disk by finitely many patches from Proposition 34. Subdivide it into finitely many smaller coordinate regions with piecewise smooth boundaries, each contained in one patch, and thicken these regions slightly. The boundaries have zero round area, so their collars can be chosen with arbitrarily small total integral of \(|\nabla u|^2\). Local recovery on the thickened regions and successive applications of Lemma 23 now give recovery on the compact complement: the main energy is bounded by the sum of the target energies on those regions, and only the chosen small collars are counted more than once. The error terms from gluing tend to zero by uniform agreement on each fixed collar. This proves recovery with arbitrarily small energy slack. Extend by the constant on the smaller disk. To make the extension exact, use a cutoff whose transition lies in a fixed annulus where the target is constant. Apply the finite-cover recovery just proved to a slightly larger annular neighborhood and a smooth cutoff \(b\) that is constant on its two collars. If \(b_n\) is its recovery and \(\delta_n>\|b_n-b\|_\infty\) tends to zero, replace \(b_n\) by \[\left(\frac{b_n-\delta_n}{1-2\delta_n}\right)\vee0\wedge1.\] This is exactly zero and one on the respective flat collars, and its energy is at most \((1-2\delta_n)^{-2}\mathsf e_n(b_n)\). The functions being joined differ by \(o(1)\) on the transition, so the extra term in the gluing estimate is \(o(1)\). The mesh tends to zero, hence every edge crossing either gluing boundary is contained in the fixed collar for all sufficiently large \(n\).

For a general continuous Sobolev \(u\), apply the approximation \(u_r\) of Lemma 35. Recover each fixed \(u_r\) first, then use a diagonal choice of \(r\downarrow0\). Uniform convergence and convergence of \(\mathcal D(u_r)\) give the desired upper bound; the first assertion makes it an equality. The proof for the dual is identical.

Away from \(p\), harmonic compactness follows from Lemma 21. To include \(p\), take any fixed finite stack of disjoint annuli centered at \(p\), each of the same aspect ratio, in a domain of harmonicity. Every transition annulus avoids \(p\). By the preceding local recovery, its dual cutoff can be taken with energy bounded by a constant depending only on the aspect ratio, with exact constant values on its two sides. Fill the inner side by the constant; this uses only a flat collar in the annulus. Lemma 20 gives a primal circuit in each annulus whose oscillation squared is bounded by that constant times the primal energy in the annulus. Among \(k\) disjoint annuli, one therefore has oscillation at most \(C k^{-1/2}\) when the total energy is bounded. The maximum principle transfers this oscillation to all vertices inside its circuit. First choose \(k\), and then take large \(n\); no annulus radius depends on \(n\). Finite covers give equicontinuity, and bounded values give compactness. Exchanging primal and dual proves the remaining assertion. ◻

Proof. The sphere-identification proposition and the final normalization argument for spanning-tree maps in [22] give, jointly with the excursion, the contour correspondence \[\bigl\{(v_{\lfloor2nt\rfloor},\eta(t)):0\le t\le1\bigr\}\] with distortion tending to zero. Here \(v_j\) is the primal vertex at the start of tour step \(j\) and \(\eta\) is the sphere contour. Those results identify the metric of this very peanosphere limit; they do not introduce an independent quantum sphere. Their deterministic diameter normalization is the \(a_n\) in Theorem 1. Choosing one visit for each vertex changes the correspondence by a uniformly vanishing discrepancy, which gives (51) and (52).

There are \(2n\) tour step starts. Their vertex occupation measure is \(\mu_n\): horizontal tree traversals are counted at their starting vertices, and vertical steps count the non-tree half-edges at their incident vertices. Every half-edge is counted once, including both half-edges of a loop. The area parameterization of \(\eta\) therefore gives (53). Mesh convergence is the sphere projection construction preceding Proposition 3. Nonatomicity, full support, and agreement of the metric and conformal topologies are the quantum-area and metric properties in the same repository input.

All these variables may be retained together with the compactified energy arrays before extraction. The metric identification is unchanged upon a further extraction, so it holds in the coupling of Theorem 36. ◻

Proof. The one-vertex case is immediate, so suppose \(m:=|V|\geq2\). Orient the non-loop edges arbitrarily and delete the row of a fixed vertex \(o\) from the incidence matrix. Denote the resulting matrix by \(B\), and write \(\widehat q\) for \(q\) with its \(o\)-coordinate deleted. The reduced Laplacian is \(\widehat H=BB^{\mathsf T}\). If \(\tau\) is the number of spanning trees, the matrix-tree theorem and the determinant lemma give \[ \det(\widehat H+z\widehat q\widehat q^{\mathsf T}) =\tau\bigl(1+z\widehat q^{\mathsf T} \widehat H^{-1}\widehat q\bigr). \tag{57}\] For \(z\geq0\), apply Cauchy–Binet to the matrix obtained by appending \(\sqrt z\widehat q\) to \(B\). A term contributing to the coefficient of \(z\) selects \(m-2\) incidence columns and the source column. The incidence columns are independent exactly when their edges form a spanning forest with two components. Indeed, a cycle gives a linear dependence, whereas an acyclic graph with \(m-2\) edges has exactly two components, counting isolated vertices.

For such a forest \(F\), let \(S_F\) be its component not containing \(o\). Its corresponding determinant has absolute value \(|q(S_F)|\). To see this, successively eliminate leaf rows and their incident forest columns. Eliminating a leaf adds its source entry to its neighbor’s entry. The component containing \(o\) disappears into the deleted row, and the last entry in the other component is \(\sum_{v\in S_F}q(v)\). Consequently, comparison of the coefficients of \(z\) in (57) yields \[ \widehat q^{\mathsf T}\widehat H^{-1}\widehat q =\frac1\tau\sum_F q(S_F)^2. \tag{58}\]

Let \(H^+\) be the inverse of \(H\) on counting-mean-zero vectors, extended by zero on constants. Solving the equation with value zero at \(o\) shows that the left side of (58) equals \(q^{\mathsf T}H^+q\). Polarization therefore gives, for any two vectors \(q,r\) of total sum zero, \[ q^{\mathsf T}H^+r=\frac1\tau\sum_F q(S_F)r(S_F). \tag{59}\] Replacing \(S_F\) by its complement changes the sign of each factor, so the product is independent of this choice.

Every two-component forest \(F\) has exactly \(b(F)\) extensions to a spanning tree: add any one of the graph edges across its cut. Thus, when all pairs \((T,e)\) are counted, \(F=T\setminus e\) occurs exactly \(b(F)\) times. This explains the denominator in (56). Apply (59) with \(r=\delta_x-\mu\). Its left side is \[(H^+q)(x)-\mu(H^+q)=u(x),\] and its right side, regrouped by \((T,e)\), is the asserted expression. Loops enter neither the reduced incidence matrix nor a cut; parallel edges are counted separately throughout. ◻

Proof. Associate each step start with the departing step. The two horizontal traversals of a tree edge depart once from each endpoint, and hence count its two incidences. The two steps of a vertical matched pair count the two incidences of its non-tree edge. When that edge is a loop, both are assigned to the same vertex. Every step belongs to one such pair, proving (60); this is also the tour-measure lemma of [22].

The primal vertex changes only along horizontal steps. It passes between the two components of \(T\setminus e\) exactly at the two traversals of \(e\). Thus each side occupies one cyclic interval, and (60) proves the mass identity.

There is exactly one tree edge across this cut, namely \(e\). A non-tree edge crosses it precisely when exactly one step of its vertical matched pair belongs to \(I_e\). Ignore horizontal holds and follow the resulting dual-tree contour segment. An edge of \(T^*\) is traversed once on this segment exactly when its matched steps lie on opposite sides of \(I_e\); every other dual edge is traversed zero or twice. In a tree the edges traversed an odd number of times by a path are exactly the edges on the simple path between its endpoints. These endpoints are the dual vertices at the two horizontal traversals of \(e\), since horizontal steps do not change the dual vertex. There are therefore \(D_e^*\) non-tree edges across the cut, giving \(b_e=1+D_e^*\). ◻

Proof. Rerooting a plane tree at its \(i\)th contour corner is a bijection on rooted plane trees with \(s/2\) edges. Its inverse reroots by \(s-i\) corners. Thus the rerooted tree is uniform, and \(D\) has the law of the height at time \(j\) of a uniform Dyck excursion of length \(s\).

Let \(p_l(a)\) be the probability that a simple symmetric walk started at zero stays nonnegative for \(l\) steps and finishes at \(a\). Reflection and the binomial estimate give \[\begin{align*} p_l(a) &=2^{-l}\left[ \binom{l}{(l+a)/2}-\binom{l}{(l+a+2)/2}\right], \tag{67}\\ p_l(a)&\leq C(a+1)(l+1)^{-3/2} e^{-ca^2/(l+1)}, \end{align*}\] with inadmissible binomial coefficients interpreted as zero. For even \(s\), \(p_s(0)\) is comparable to \((s+1)^{-3/2}\). The Markov property and time reversal of the second segment give \[\mathbb P(D=a)=\frac{p_j(a)p_{s-j}(a)}{p_s(0)}.\] Inserting (67) proves (64). Summing it after multiplication by \((a+1)^{-2}\) or by \(a^p\) proves (65). In the first case the remaining Gaussian sum is at most \(C\sqrt d\), so the second moment is at most \(C/d\).

For completeness, the diameter estimate follows by dyadic chaining. Reroot at the deterministic first corner of \(K\) and parametrize its steps by the integers in \([0,L]\). At refinement level \(l\), use the times \(\lfloor kL/2^l\rfloor\), \(0\leq k\leq2^l\), and stop at \(l=\lceil\log_2L\rceil\). These grids are nested and the last one contains every integer in \([0,L]\). Include the initial increment from \(0\) to \(L\). For \(l\geq1\), there are \(O(2^l)\) new-to-parent increments at level \(l\), each spanning at most \(C L2^{-l}\) steps. By the fourth-moment estimate and \(\|\max_k Z_k\|_{L^4}^4\leq\sum_k\|Z_k\|_{L^4}^4\), the maximum tree distance across these increments has \(L^4\) norm at most \[C2^{l/4}\sqrt{L2^{-l}}=C\sqrt L\,2^{-l/4}.\] Every contour vertex can be joined to the first vertex through its ancestors in the dyadic grids. The triangle inequality and the summability of \(2^{-l/4}\) bound the maximum distance from that first vertex by \(C\sqrt L\) in \(L^4\). The diameter is at most twice this maximum. For \(L=1\) the bound follows directly from the single edge. ◻

Proof. First condition on a count \(J\in[N/4,3N/4]\). Consider all cyclic intervals containing \(t\) whose lengths lie in \([M,2M)\) and are at most \(N/2\). There are at most \(C(M+1)^2\) such intervals. For each one, a uniform schedule puts at least \(c_0M\) vertical steps in both the interval and its complement, except with probability \(Ce^{-cM}\), for suitable absolute \(c_0,c>0\). Indeed, the vertical density is at least \(1/4\), both sets have at least \(M\) slots, and the hypergeometric lower-tail bound applies. Taking a union bound, let \(\mathcal B_{M,t}\) be the event that this property fails for some interval. Uniformly over the balanced counts, \[ \mathbb P(\mathcal B_{M,t}\mid J) \leq C(M+1)^2e^{-cM}. \tag{69}\]

Fix a schedule outside \(\mathcal B_{M,t}\) and then condition on the horizontal Dyck excursion. This fixes all intervals in \(U_{M,t}\). For each one, its dual endpoints are separated in the vertical contour by the number \(j\) of vertical steps in that interval. Both \(j\) and \((N-J)-j\) are at least \(c_0M\). Lemmas 40 and 41, followed by the triangle inequality in \(L^2\) of the remaining vertical excursion, imply \[ \|U_{M,t}\|_{L^2(R\mid\text{schedule},L)} \leq\frac{C}{\sqrt M}\,K_{M,t}, \qquad K_{M,t}:=\#\{e\in\mathcal E_M:t\in I_e\}. \tag{70}\]

We next bound this number of horizontal cuts. Any two components \(S_e,S_{e'}\) containing \(v_t\) are nested unless their union is all of \(V_n\). This elementary tree property follows by considering the components cut off from \(v_t\): they are either nested or disjoint. The union alternative cannot occur for the chosen smaller sides, since \[\mu_n(S_e\cup S_{e'}) =\mu_n(S_e)+\mu_n(S_{e'})-\mu_n(S_e\cap S_{e'})<1.\] Here each of the first two terms is at most \(1/2\), and the intersection contains \(v_t\), whose mass is positive. Thus all components counted by \(K_{M,t}\) are nested.

If the family is nonempty, let \(S\) be its largest component and choose a contour vertex immediately outside its interval. It has a representative at cyclic distance at most \(2M+2\) from \(t\). The tree path from \(v_t\) to that vertex crosses all \(K_{M,t}\) cut edges. Hence \(K_{M,t}\) is at most the diameter of the horizontal contour image of the deterministic window of tour times within \(2M+2\) of \(t\). After the schedule has been fixed and holds deleted, this is a deterministic contour segment with at most \(4M+4\) horizontal steps. If the window covers the entire tour, the full horizontal contour instead has length \(J\leq N\leq4M+4\). A segment with no horizontal steps has diameter zero; otherwise Lemma 41 applies. Uniformly in the schedule, we obtain \[ \mathbb E\bigl[K_{M,t}^2\mid\text{schedule}\bigr]\leq CM. \tag{71}\] Combining (70) and (71) proves the required bound on schedules outside \(\mathcal B_{M,t}\) with balanced counts.

For the exceptional schedules, use a deterministic bound. Every interval counted by \(K_{M,t}\) has its two delimiting tree traversals within cyclic distance \(2M+2\) of \(t\). Distinct tree edges have distinct traversal occurrences, so \(K_{M,t}\leq C(M+1)\). Since \(b_e\geq1\), the same bound holds for \(U_{M,t}\). By (69), the contribution of these schedules to its second moment is at most \(C(M+1)^4e^{-cM}\), uniformly bounded in \(M\). Finally, \(U_{M,t}\leq N\) always. The unbalanced counts contribute at most \(CN^2(N+1)^3e^{-cN}\) by (63), also uniformly bounded. This proves (68) under the full law. ◻

Proof of Proposition 38. The one-vertex case has \(G_nf=0\), so set \(A_n=0\) there. Otherwise, for each dyadic \(M\leq N/2\), choose a deterministic cyclic grid \(\Gamma_M\). For \(M\geq8\), take its gaps at most \(M/4\), its cardinality at most \(CN/M\), and its distinct points separated by at least \(cM\). For \(M<8\), use all step indices; the same cardinality and separation bounds hold with absolute constants.

An interval of length in \([M,2M)\) containing a fixed time \(t\) contains a grid point within cyclic distance \(2M\) of \(t\). There are only a bounded number of grid points in this neighborhood. Consequently, \[ \sup_t U_{M,t}\leq C\max_{t\in\Gamma_M}U_{M,t}. \tag{72}\] Every visit to a given vertex belongs to the same side of a tree cut. Also, counting the visits in each interval gives \[\sum_{e\in\mathcal E_M}\frac{\mu_n(S_e)}{b_e} =\frac1N\sum_{t=0}^{N-1}U_{M,t}.\] It follows that \[\begin{align*} W_M&:=\sup_{x\in V_n}\sum_{e\in\mathcal E_M} \frac{\mathbf 1_{\{x\in S_e\}}+\mu_n(S_e)}{b_e} \leq C\max_{t\in\Gamma_M}U_{M,t}, \tag{73}\\ \mathbb EW_M&\leq C\left(\sum_{t\in\Gamma_M} \mathbb EU_{M,t}^2\right)^{1/2} \leq C\sqrt{N/M}. \tag{74}\end{align*}\] No independence among the grid variables is needed here.

Fix \(f\) as in the proposition and write \(m_f=\mu_n(|f|)\). For \(e\in\mathcal E_M\), the source mass obeys \[ \left|\sum_{v\in S_e}\mu_n(v)f(v)\right| \leq\min(2M/N,m_f) \leq(2M/N)^{1-\rho}m_f^\rho. \tag{75}\] Define, for the realized pair \((M_n,T)\), \[Y_n(M_n,T)=\sum_{\substack{M\leq N/2\\M\text{ dyadic}}} (2M/N)^{1-\rho}W_M.\] By (74), \[ \mathbb EY_n\leq C_\rho \sum_{\substack{M\leq N/2\\M\text{ dyadic}}} (M/N)^{1/2-\rho} \leq C_\rho. \tag{76}\] The last bound is a geometric sum, using \(\rho<1/2\).

Conditional on \(M_n\), the selected tree is uniform. Define \[ A_n(M_n)=\mathbb E\bigl[Y_n(M_n,T)\mid M_n\bigr]. \tag{77}\] Apply Lemma 39 with \(q=\mu_nf\). Taking absolute values, grouping by \(M\), and using (75) proves \[\|G_nf\|_\infty\leq A_n(M_n)m_f^\rho.\] The variable in (77) is independent of the choice of \(f\), so this bound holds simultaneously for all admissible functions. The tower property and (76) give \(\sup_n\mathbb EA_n<\infty\), and Markov’s inequality gives tightness. ◻

Proof. In coordinate disks, mollification of a continuous Sobolev function converges both uniformly on smaller disks and in Sobolev norm. A smooth partition of unity gives smooth functions \(u_j\) on the sphere with these two convergences globally. Since \(\mu\) is finite, uniform convergence also gives \(L^2(\mu)\) convergence. Thus \((u_j)\) is Cauchy for \(\mathcal E(\cdot,\cdot)+\|\cdot\|_{L^2(\mu)}^2\) and has limit \(u\) in the stipulated closure. The gradient convergence identifies its energy. Polarization gives the same identity for the bilinear form on continuous Sobolev functions. Finally a nonzero continuous difference is bounded away from zero on some open set, which has positive \(\mu\)-mass by full support. ◻

Proof. Put \(u_n=R^n_\alpha(f\circ\pi_n)\). The maximum principle and the resolvent equation give \[ \|u_n\|_\infty\le\frac{\|f\|_\infty}{\alpha}, \qquad H_nu_n=\mu_n g_n, \qquad g_n=2(f\circ\pi_n-\alpha u_n). \tag{82}\] In particular \(g_n\) is uniformly bounded, and multiplying the equation by \(u_n\) shows that \(\mathsf e_n(u_n)\) is uniformly bounded.

We first prove equicontinuity. Fix a point \(x\in S^2\), and choose disjoint open neighborhoods \(U\) of \(x\) and \(V\) of a different point. Take \(\mu\)-null boundaries, with \(\mu(V)>0\). The neighborhood \(U\) may be made arbitrarily small in \(\mu\)-mass because \(\mu\) has no atoms. Write \(U_n=\pi_n^{-1}(U)\) and \(V_n'=\pi_n^{-1}(V)\), and set \[q_n=g_n\mathbf 1_{U_n} -\frac{\mu_n(g_n\mathbf 1_{U_n})}{\mu_n(V_n')}\mathbf 1_{V_n'}.\] For large \(n\), the denominator is bounded below. Hence \(q_n\) has mean zero, bounded supremum, and \(\mu_n(|q_n|)\le C_1\mu_n(U_n)\). Scaling (80) yields \[\|G_nq_n\|_\infty\le C_2\mu_n(U_n)^\rho.\] Also \[\mathsf e_n(G_nq_n)=\mu_n((G_nq_n)q_n) \le\|G_nq_n\|_\infty\mu_n(|q_n|).\] Thus \(u_n-G_nq_n\) has bounded energy and supremum and is harmonic on \(U_n\). Hypothesis (iii) makes this difference equicontinuous on compact subsets of \(U\). Given an oscillation tolerance, first choose \(U\) so that \(G_nq_n\) has smaller supremum, and then use harmonic equicontinuity inside \(U\). A finite cover of \(S^2\) proves equicontinuity of the original sequence. Boundedness and density of \(\pi_n(V_n)\) now give uniformly convergent subsequences.

Let \(u\) be one such limit. Hypothesis (ii) and Lemma 44 show that \(u\in\mathcal F\cap C(S^2)\). The discrete resolvent minimizes \[J_n(v)=\frac12\mathsf e_n(v)+\alpha\mu_n(v^2) -2\mu_n((f\circ\pi_n)v).\] The energy lower bound, uniform convergence, and weak convergence of the measures imply \[\mathcal E(u,u)+\alpha\mu(u^2)-2\mu(fu) \le\liminf_n J_n(u_n).\] For any smooth \(v\), take its uniform recovery sequence \(v_n\). Minimality gives \(J_n(u_n)\le J_n(v_n)\), and the latter converges to \(\mathcal E(v,v)+\alpha\mu(v^2)-2\mu(fv)\). Smooth functions are a core of the closed form by its definition, so \(u\) minimizes this functional over all of \(\mathcal F\). Strict convexity for \(\alpha>0\) identifies \(u=R_\alpha f\) in \(L^2(\mu)\). Full support makes its continuous version unique. Every subsequence has the same limit, which proves (81).

Finally, \(\|R^n_\alpha a-R^n_\alpha b\|_\infty \le\alpha^{-1}\|a-b\|_\infty\). This contraction bound proves the claim for varying right-hand sides. ◻

Proof. First suppose \(f\) is smooth, and put \(w_\alpha=\alpha R_\alpha f\). The spectral theorem for the nonnegative self-adjoint form operator gives \(w_\alpha\to f\) in the form norm. Indeed the spectral multiplier \(\alpha/(\alpha+\lambda)\) tends to one and is bounded by one, so dominated convergence applies with spectral weight \(1+\lambda\). By Lemma 44, \[\|\nabla(w_\alpha-f)\|_{L^2(\mathrm dA)}\longrightarrow0.\] The maximum principle, or its discrete version followed by Lemma 45, gives \[ \min f\le w_\alpha\le\max f. \tag{84}\]

We also need convergence in round \(L^2\), which does not follow by changing the measure in the form norm. Let \(c_\alpha\) be the round average of \(w_\alpha-f\). Round Poincare gives \[\|w_\alpha-f-c_\alpha\|_{L^2(\mathrm dA)}\longrightarrow0.\] The constants \(c_\alpha\) are bounded by (84). If a subsequence has limit \(c\), then \(w_\alpha\to f+c\) in round \(L^2\) on that subsequence, and therefore along a further subsequence almost everywhere. Consequently \(\min f\le f+c\le\max f\) almost everywhere for round area. Continuity of \(f\) makes its essential extrema equal to its extrema: every neighborhood of a minimizer or maximizer has positive round area. The two inequalities force \(c\ge0\) and \(c\le0\). Thus \(c=0\), and \(w_\alpha-f\to0\) in round \(W^{1,2}\).

Fix a small coordinate disk about a point \(x\), and a fixed annulus inside that disk surrounding a smaller neighborhood of \(x\). Radial averaging of the round \(W^{1,2}\) error supplies, for each large \(\alpha\), a circle in this annulus on which the trace of \(w_\alpha-f\) tends to zero in one-dimensional \(H^1\). The radii stay in a fixed compact subinterval of \((0,\infty)\), so the one-dimensional Sobolev inequality gives uniform convergence on these circles. Almost every radius has the Sobolev trace of the continuous representative, which is the trace we choose.

Let \(D_\alpha\) be the disk inside the chosen circle. The resolvent equation is \[ \mathcal E(w_\alpha,v)+\alpha\int(w_\alpha-f)v\,\mathrm d\mu=0, \qquad v\in\mathcal F. \tag{85}\] Choose a constant \(K\) strictly above the boundary values of \(w_\alpha\) and at least \(\sup_{D_\alpha} f\). The positive part \((w_\alpha-K)^+\) on \(D_\alpha\), extended by zero, vanishes in a collar of the circle and is continuous Sobolev. Lemma 44 makes it an admissible test in (85). The two terms are nonnegative; the second is at least \(\alpha\int ((w_\alpha-K)^+)^2\,\mathrm d\mu\). Thus \(w_\alpha\le K\) on the disk, first \(\mu\)-almost everywhere and then everywhere by continuity and full support. The negative-part test gives the lower bound. Letting the strict slack tend to zero shows that the values inside the circle are bounded by its boundary values and the extrema of \(f\) in the disk.

The oscillation of \(f\) on sufficiently small disks is arbitrarily small. The boundary convergence and this comparison therefore give uniform convergence on a neighborhood of \(x\). A finite cover proves (83) for smooth \(f\). For continuous \(f\), approximate it uniformly by a smooth function and use the contraction \(\|\alpha R_\alpha\|_{\infty\to\infty}\le1\). ◻

Proof. Cover the sphere by finitely many round disks of radius less than \(d/8\). For each choose a smooth function \(f\) in \([0,1]\), zero on that disk, and one outside the concentric disk of radius \(d/2\). If \(\pi_n(v)\) is in the smaller disk and displacement from \(\pi_n(v)\) reaches \(d\), the value of \(f\) at the new location is one.

Fix \(b>0\). Lemmas 46 and 45, first with a sufficiently large fixed \(\alpha\) and then with sufficiently large \(n\), give for each of this finite family a function \[v_n=\alpha R^n_\alpha(f\circ\pi_n),\qquad \|v_n-f\circ\pi_n\|_\infty\le b.\] Its generator satisfies \[\mathcal L_nv_n=\alpha(v_n-f\circ\pi_n), \qquad \|\mathcal L_nv_n\|_\infty\le C_b,\] with a common \(C_b\) for the finite family and all sufficiently large \(n\). Start the chain at \(v\) in the corresponding small disk and stop at \(\sigma=\tau^n_d(v)\wedge s\). If \(p=\mathbb P_v[\tau^n_d(v)\le s]\), then \[\mathbb E_v[v_n(Y^n_\sigma)]\ge p-b, \qquad \mathbb E_v[v_n(Y^n_\sigma)] =v_n(v)+\mathbb E_v\int_0^\sigma \mathcal L_nv_n(Y^n_t)\,\mathrm dt \le b+C_bs.\] Hence \(p\le2b+C_bs\), uniformly over all starting vertices. First let \(s\downarrow0\) and then \(b\downarrow0\) to obtain (86).

Apply the strong Markov property to the chain \(Y^n\) after any stopping time. Since (86) is uniform over its current vertex, it gives the usual stopping-time increment condition for the projected paths. The projection itself need not be Markov. Apply Aldous’s stopping-time criterion [1] to the three coordinate functions of the round sphere in \(\mathbb R^3\); their values are bounded. This gives tightness of each coordinate path law. The largest coordinate jump is bounded by the maximum projected edge length, which tends to zero. Under \(J_1\) convergence any jump of a limit is approximated by jumps of the approximating paths, so every coordinate limit is continuous. Coordinate tightness with continuous limits gives joint tightness with continuous limits in \(\mathbb R^3\): on continuous limits, \(J_1\) convergence agrees with uniform convergence, allowing a common time comparison for the three coordinates. The limits remain in the closed set \(S^2\), proving the assertion. ◻

Completion of the proof of Theorem 43. Take a subsequence of the stationary projected path laws with a weak limit. By Lemma 47, diagonal extraction on the horizons \(1,2,\ldots\) gives a consistent continuous-path law on \([0,\infty)\), denoted by \(Q\). Restrictions are consistent because the limits are continuous at the restriction times.

Let \(f_0,\ldots,f_k\in C(S^2)\) and \(\alpha_1,\ldots,\alpha_k>0\). Integrate the stationary finite-dimensional expectation at times \(0,t_1,t_1+t_2,\ldots,t_1+\cdots+t_k\) against \(\prod_{j=1}^k e^{-\alpha_jt_j}\,\mathrm dt_j\). The Markov property and the resolvent identity give exactly \[ \mu_n\!\left[ (f_0\circ\pi_n)R^n_{\alpha_1} \left((f_1\circ\pi_n)R^n_{\alpha_2} \left(\cdots(f_{k-1}\circ\pi_n) R^n_{\alpha_k}(f_k\circ\pi_n)\cdots\right)\right) \right]. \tag{87}\] Successive applications of Lemma 45, including its varying-data assertion, and weak convergence of the measures show that (87) converges to \[\mu\!\left[f_0R_{\alpha_1} \left(f_1R_{\alpha_2} \left(\cdots f_{k-1}R_{\alpha_k}f_k\cdots\right)\right) \right].\] By the semigroup-resolvent relation for the closed form, this is the same integrated expectation for the stationary diffusion \(B\).

It is also the integrated expectation for \(Q\). On bounded time boxes this follows from weak path convergence, since evaluation at fixed times is continuous at continuous paths. Dominated convergence then integrates over the gaps. The contribution of the complement of a large time box is bounded uniformly by the tails of the product of exponentials, times \(\prod_j\|f_j\|_\infty\).

The product expectations under \(Q\) and under \(B\) are bounded continuous functions of \((t_1,\ldots,t_k)\), including zero gaps. Uniqueness of multivariate Laplace transforms makes these functions equal: one may apply one-dimensional uniqueness successively in each coordinate, and then use continuity to remove almost-everywhere qualifications. Product tests determine probability measures on the compact product \((S^2)^{k+1}\). Thus \(Q\) and the law of \(B\) have the same finite-dimensional distributions, and hence the same law on continuous paths. Every subsequential limit is the same, proving the theorem. ◻

Proof. The generator convention gives \(H_n=-\deg\,L_n\). Since \(\mu_n(v)=\deg(v)/(2n)\), \[-\sum_v\mu_n(v)u(v)nL_nu(v) =\frac12\sum_vu(v)H_nu(v) =\frac12\sum_{\{v,w\}}(u(v)-u(w))^2.\] Loops contribute zero to both sides. Their degree contributions are already included in the first identity, so no holding-time correction is needed. ◻

Proof of Theorem 1. Start with any subsequence. Retain the random variable \(A_n\) of Proposition 38 together with all the contour, spatial, and local and whole-sphere array variables used in Theorem 36 and Proposition 37. Its uniform first-moment bound gives tightness, so these variables admit a joint further extraction. Apply the sphere results to this enlarged extraction and use a Skorokhod representation. On the resulting probability-one event, their conclusions hold and \(A_n\) converges to a finite random variable. In particular \(\sup_n A_n<\infty\) after discarding finitely many initial terms.

Fix a realization in this probability-one event. Its limiting quantum sphere is now a fixed measured conformal sphere. The projections \(\pi_n\), measures, edge mesh, uniform energy bounds and recovery, and harmonic compactness satisfy hypotheses (i)–(iii) of Theorem 43. The realized bounded \(A_n\) gives hypothesis (iv). The Liouville form is exactly (78) with \(\mu=\mu_h\). By Lemma 48, the chain in that deterministic theorem is \(Y^n_t=X^n_{nt}\). Consequently its conditional projected path law converges to the conditional law of \(B^h\), for every fixed finite horizon. All probabilities in this invocation are taken on the individual fixed graphs. The random object being compared is the conditional probability law itself.

It remains to express this comparison in the metric \(\delta_T\). The round topology and the \(D_h\) topology agree on the compact limiting sphere, so the identity map between them is uniformly continuous in both directions. The path-law convergence just proved therefore also holds in \(D([0,T],(S^2,D_h))\).

The correspondences in Proposition 37 have distortions tending to zero. The standard correspondence gluing constructs a common metric space containing isometric copies of all \((V_n,a_nd_n)\) and \((S^2,D_h)\), with \[\sup_{v\in V_n}\mathop{\mathrm{dist}}_Z\bigl(v,\pi_n(v)\bigr)\longrightarrow0.\] For completeness, for each \(n\) attach its copy to the sphere by cross-distances obtained by taking the infimum over corresponding pairs, adding numbers \(\varepsilon_n\downarrow0\) larger than half the respective distortions; use the shortest-path metric on the union, identifying only the common sphere, and then take the completion. This preserves the metrics of every copy. The images become close in Hausdorff distance, and their measures become close in Prokhorov distance.

Apply the embedding to each coordinate of a chain path. Its distance in the common Skorokhod space from its \(\pi_n\)-projected path is bounded by the displayed supremum, using the identity time change. The corresponding probability laws therefore have Prokhorov distance tending to zero. Combining this with the projected law convergence shows that all three terms defining \(\delta_T\) tend to zero in the coupled extraction.

We have obtained almost-sure convergence in a further coupling from every original subsequence, to the prescribed quantum sphere and its conditional diffusion law. The subsequence principle gives convergence in distribution through all positive integers \(n\). Forgetting the root, the spanning tree, and the contour marks leaves precisely the states in Theorem 1. The same choice \(c=1\) works for every time horizon. ◻