Proof. For a nonloop edge with endpoints \(u,v\), the two incident half-edge contributions to \(-\langle f,L_ng\rangle_{L^2(\mu_n)}\) are \[\frac{1}{2n}\bigl(f(u)(g(u)-g(v))+f(v)(g(v)-g(u))\bigr) =\frac{d_e f\,d_e g}{2n}.\] The two half-edges of a loop contribute zero. Summing proves both operator identities. A function of zero energy agrees across every nonloop edge, and the map is connected, so it is constant. ◻

Proof. Taking complements across dual edges exchanges \(A\) and \(A^*\), and \[\ell(M,A)=k(A)+k(A^*)-1=\ell(M^*,A^*).\] The dual operation is invertible, including its root convention, and keeps the number of edges. It therefore preserves the normalized finite law. The flag correspondence is \((v,e,F)\leftrightarrow(F,e^*,v)\). It preserves the equilateral triangles and their incidence gluings, including every repeated occurrence. Drawing a primal edge along its vertex-to-midpoint sides and a dual edge along its face-to-midpoint sides proves the crossing assertion in each flag neighborhood. Conditional on the common metric surface, the same three points are independent uniform area samples in both marginal experiments. Genus-zero three-point normalization then gives the last assertion, allowing complex conjugation if the surface orientation is reversed. ◻

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

Apply this fact to \(Y_n=\log(b_n\Delta_n)\). The continuous mapping theorem applies because \(\Delta>0\). Shift equivariance gives \[\log\frac{b_n}{a_n} =\mathfrak m\bigl(\log(b_n\Delta_n)\bigr)-\mathfrak m(\log\Delta) \longrightarrow0.\] ◻

Proof. The canonical limit (OpenAI 2026a, Theorem 1.1) gives weak convergence of \((\phi_n)_*m_n\), Hausdorff convergence of the full distance graph with a deterministic scale \(b_n\to0\), and vanishing maximum flag diameter in the same coordinate. The third coordinate of the graph uses the uncompactified distance, so its maximum converges to \(\Delta=\mathop{\mathrm{diam}}(\mathbb S^2,D_h)\). This diameter is finite and strictly positive. Replacing the graph diameter by \(\Delta_n=\max\{1,\mathop{\mathrm{diam}}(V_n,d_n)\}\) does not affect this limit because \(b_n\to0\). Lemma 4 gives \(a_n/b_n\to1\), hence the graph convergence in (12) and \(a_n\to0\). Couple each flag-area point to its primal vertex. Their images are within \(\delta_n\); therefore (11) gives the weak corner-measure convergence. All flags in one incidence star contain the star’s center, which proves the \(2\delta_n\) bound.

The duration-one area tour and its continuity come from (OpenAI 2026a, Theorem 3.4). The homeomorphisms of (OpenAI 2026a, Theorem 4.8) track every flag and its tour representatives uniformly. The uniformization comparison (OpenAI 2026a, Theorem 8.8) has only the conformal and anticonformal three-point limits. Passing further and retaining the sign gives the uniform tracking in the canonical coordinate; uniform continuity of the limiting tour extends the cut-time statement to (13). These arguments apply jointly to a finite number of rooted marginals by tightness and a further extraction. Their pushed area measures are the same limit of the physical flag measures, so each tour has pushforward \(\mu_h\).

Lemma 3.7 of (OpenAI 2026a) says that an independent quantum-area point has exactly one visit by the tour. Integrating this probability-one statement conditionally on the surface and tour gives the asserted singleton fibers for \(\mu_h\)-almost every \(z\); a finite intersection handles all retained rerootings. Conditional uniform rerooting has tail law \(\mu_n\) by (11). Weak convergence of these measures, tested against products of bounded continuous functions, identifies the joint law by first taking \(\mu_h\) and then independently sampling the two roots from it. Quantum area is nonatomic, so two such roots are distinct.

The continuous projections and complete time-preimage protection are (OpenAI 2026a, Proposition 4.2 and Lemma 4.3). The reference homeomorphisms and their validity with extra observations are exactly (OpenAI 2026a, Theorem 4.8); the canonical comparison is (OpenAI 2026a, Theorem 8.8). The protected bilateral sewing and its inverse-image capture are (OpenAI 2026a, Proposition 4.9). These statements concern fixed tests with room and supply no rate for a test whose boundary changes with \(n\).

Finally, the sphere field comparison in (OpenAI 2026b, Lemma 18.1) uses the exploration root and an auxiliary pair of area marks. Use two separate pair marginals. Their four marks are distinct almost surely, and the complements of the two pairs cover the sphere off the root. Strict chart interiors exhaust each complement. Repeating this for two distinct roots covers the entire sphere. Compactness yields a finite subcover. This argument intersects probability-one conclusions from the separate marginals; it does not strengthen either conditional field comparison. ◻

Proof. Write \(a_\pi=(N_e(\pi))_{e\in E(\mathsf G)}\), and let \(K\) be the closed convex hull of these nonnegative vectors. For \(t\in K\) and \(g\ge0\), \[\inf_{\pi\in\mathcal F}a_\pi\cdot g \le t\cdot g\le\lVert t\rVert_2\lVert g\rVert_2.\] Consequently \(\mathop{\mathrm{EL}}_{\mathsf G}(\mathcal F)\le\inf_{t\in K}\lVert t\rVert_2^2\). Let \(p\) be the point of \(K\) nearest \(0\). If \(p=0\), this already gives equality. Otherwise \(p\ge0\), and the projection inequality gives \[a_\pi\cdot p\ge\lVert p\rVert_2^2 \quad\hbox{for every }\pi\in\mathcal F.\] Taking the admissible nonzero density \(g=p\) in (34) gives the reverse inequality. Finite convex combinations are precisely the traffic vectors of finite-support laws and are dense in \(K\). A general traffic vector with finite coordinates belongs to \(K\), by approximating its law by finite laws and allowing an arbitrarily small error in the coordinates. This proves (36).

For the port identity, first suppose a connecting walk exists. Let \(u|_A=0\), \(u|_B=1\), and take \(g_e=\lvert d_eu\rvert\). A connecting walk and these port values force \(g\ne0\), so this density is admissible in (34). The telescoping inequality gives \(L_g(\pi)\ge1\) on every connecting walk, so \(\mathop{\mathrm{EL}}\ge1/\mathcal E_{\mathsf G}^0(u)\). Conversely, for a density with \(b=\inf_{\pi\in\mathcal F_{\mathsf G}(A,B)}L_g(\pi)>0\), let \(d_g(A,v)\) be weighted graph distance from \(A\), with value \(\infty\) in a component not meeting \(A\). The function \[u(v)=\min\{1,d_g(A,v)/b\}\] has the required port values and satisfies \(\lvert d_eu\rvert\le g_e/b\) on every edge. Thus the energy infimum in (37) is at most \(\lVert g\rVert_2^2/b^2\). Take the supremum over \(g\). If no connecting walk exists, assign a constant to each component, proving the stated convention. The equality with wired effective resistance is the Dirichlet principle for a unit-conductance network. Loop differences vanish throughout. ◻

Proof. On a successful shell at center \(c\) and radius \(r\), the local normalized EL lower bound and (34) give an edge density \(g_k^{c,r}\), with fixed slack, for which \[ L_{g_k^{c,r}}(\pi)\ge r \quad\hbox{on every complete shell traversal},\qquad \lVert g_k^{c,r}\rVert_2^2\le \frac{Cr^2}{\rho_{j_k}}. \tag{47}\] Its support lies in the shell buffer. The subwalk inclusion (40) makes the charge valid for every traversal. The densities may be chosen measurably: a port family may be restricted to its finitely many simple paths, and a first rational density satisfying a strict near-minimum bound gives a choice from the finite incidence record.

At each radius retain the successful centers whose buffers meet \(U\), and define \[g_{k,N,U}=\frac1N\sum_{(c,r)}g_k^{c,r}.\] For fixed \(N\), the edge mesh is eventually smaller than every shell margin in this finite collection. A whole supported edge can then be assigned to a point in the slightly enlarged buffer that contains it. The bounded spatial overlap at one radius bounds the number of summands on this edge by a deterministic constant; no degree bound is used. Grid counting and (47) give \[\lVert \sum_c g_k^{c,r}\rVert_2^2 \le C\sum_c\lVert g_k^{c,r}\rVert_2^2 \le \frac{C\,\operatorname{Area}(U^{+CR_N})}{\rho_{j_k}}.\] There are at most \(CN\) radii. Cauchy–Schwarz for their sum, followed by division by \(N^2\), proves (45) and the support assertion.

For the charge, choose a coordinate whose oscillation on \(\Psi_{m_k}(\pi(J))\) is at least its diameter divided by \(\sqrt2\), and use a subinterval between a minimum and a maximum in that coordinate. Ignore coordinate levels within \(CR_N\) of the two ends. At each remaining level, a visit of the walk lies in at least \(cN\) successful central holes. A hole of radius \(r\) accounts for a coordinate interval of length at most \(Cr\). For every hole counted in this way the path makes a complete traversal of the corresponding buffered shell. More precisely, strict clearance and the fixed-\(N\) mesh bound place a full vertex-to-vertex shell subwalk inside \(J\), even if an endpoint of \(J\) is in the interior of an edge. Thus each counted shell contributes its full charge in (47). Integrating over the coordinate levels gives \[cN\bigl(\operatorname{osc}-CR_N-\epsilon_{k,N,U}\bigr) \le C\sum_{\text{distinct shells traversed}}r .\] The summed density charges at least the sum on the right before the fixed constant. Divide by \(N\) and absorb constants. This proves (46). Its error is uniform because all projection and edge-mesh errors are uniform for this fixed configuration. Charges of disjoint parameter intervals add by their fractional edge definition; no endpoint edge is rounded up separately. This is the level-integration argument of (OpenAI 2026a, sec. 8) applied to unit edge intervals. ◻

Proof. For every edge density, the traffic pairing is \[ \mathbb E_{\mu_k}L_g(\pi) =\sum_e g_e\tau_{\mu_k}(e) \le\lVert g\rVert_2\sqrt{\mathcal T(\mu_k)}. \tag{50}\] For \(g=g_{k,N,U}\), equations (45) and (48) bound this expectation by \(C\sqrt{\operatorname{Area}(U^{+CR_N})}\) for large \(k\).

Take \(U\) large enough to contain \(K\) with room. Given \(d>0\), choose a fixed successful \(N\) with \(CR_N<cd/16\), and then take \(k\) large enough that \(\epsilon_{k,N,U}<cd/16\). Every path interval of projected diameter greater than \(d/2\) pays at least \(cd/4\) in (46). By continuity, the endpoints of any finite collection of intervals of diameter at least \(d\), with pairwise disjoint interiors, can be approximated from within by rational endpoints while retaining diameter greater than \(d/2\). The supremum of the crossing counts over such rational collections is measurable, since the collections are countable. Charges add, so the simultaneous bound and (50) bound its expectation by \(C/d\); formally, increase over finitely many rational collections and use monotone convergence. This also bounds the maximum count for the original intervals.

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

To control variation, fix an integer \(J\) and a finite rational box union \(U\). Let \(V_{J,U}(\gamma)\) be the supremum of the sums \[\sum_{\ell=1}^j\lvert \gamma(t_\ell)-\gamma(s_\ell)\rvert,\qquad 0\le j\le J,\] over disjoint closed intervals with rational endpoints whose images have positive clearance inside \(U\), with the empty sum equal to \(0\). This is lower semicontinuous on path space: each strict containment is open and each corresponding finite sum is continuous there. The simultaneous charge bound, applied before taking this supremum, and (50) yield \[\mathbb E_{\mu_k}V_{J,U} \le C\sqrt{\operatorname{Area}(U^{+CR_N})} +CJR_N+J\epsilon_{k,N,U}.\] First pass to the weak path-law limit \(k\to\infty\), with \(N,J\) fixed. Then let \(N\to\infty\) through successful configurations, keeping \(J\) fixed, and finally let \(J\to\infty\). The finite interval sums exhaust variation in \(U\). Applying this first to a larger confinement proves finite total variation for almost every path, and applying it in each \(U\) gives (49). Each path has length at least \(d_0\), so \(\Lambda\ne0\). Covering a planar null set by rational boxes of arbitrarily small total area, and increasing over finite subunions, proves \(\Lambda\ll\,\mathrm dz\). This proves the edge version of the flow argument in (OpenAI 2026a, sec. 8). ◻

Proof. Continue the record-low assumption (42). Choose the joint extraction used above to retain, in addition, the duration-one spheres with \(n_k=m_k/2\), their rooted dual experiments, the matching ordinary reference packages for the two sphere marginals, and the cone packages witnessing the quantile events for each reference marginal. Propositions 6 and 7 provide these matched packages, and Lemma 3 couples the two sphere marginals on the same physical maps with the same edge count. The high-unit and budget arguments above apply to each reference marginal. We carry out the germ argument below in each one, using its own conditional kernel for both its cone transfer and its later sphere transfer; the notation first describes one such matched pair.

Retain also the low-event indicators in (41). Their limiting event has positive probability. On that event the discrete indicators equal \(1\) eventually. Since \(\rho_{j_k}\to0\), the default value \(Z_{j_k}=1\) cannot satisfy the low inequality for large \(k\). Lemma 10 then gives near-minimum path laws for the middle family, and trimming between the narrower bands gives squared traffic at most \(C\rho_{j_k}\), endpoint separation at least \(d_0\), and compact confinement. A finite outer cell cover can admit every trimmed path while keeping these two geometric properties, with strict margins. Its finite path family contains the trimmed laws for all large \(k\), so its EL divided by \(\rho_{j_k}\) is bounded by a fixed finite constant. Let \(\mathsf A_{\mathrm{out}}^\alpha\) be the local activation event that \(\alpha\) is such an outer cover with the stated strict margins. Thus the low indicator witnesses the local event \[ \mathcal G_{\mathrm{low}} =\bigcup_{\alpha}\ \bigcup_{M\ge1} \bigl(\mathsf A_{\mathrm{out}}^\alpha\cap\{Y_0^\alpha<M\}\bigr). \tag{51}\] The union is countable and has positive probability in the cone restriction. By countability and compact exhaustion of activation, one instruction \(\alpha\), one finite \(M\), and fixed compact confinement and positive endpoint separation can be selected on a positive-probability subevent. This is an event of the local input and its normalized numerical output. Local equivalence of the cone and ordinary field restrictions, with the common observation kernel, transfers this event with positive probability to its matching ordinary reference marginal. There the convergence \(Y_0^\alpha<M\) and exact reconstruction on activation give finite normalized EL at most \(M+1\) for all sufficiently large \(k\). Lemma 10 supplies discrete path laws in this one local family with squared traffic at most \((M+2)\rho_{j_k}\). These laws now live on the same ordinary extraction as the probability-one budgets of Lemma 12. Apply Lemma 13 on their common event to obtain a law \(\mu\) of nonconstant rectifiable paths and a nonzero measure \(\Lambda\ll\,\mathrm dz\).

We describe the local event witnessed by their tangents. Let \(v\) be a unit direction and \(v^\perp\) its quarter-turn. For \(z\in\mathbb C\) and \(s>0\), set \[Q_v(z,s)= \{z+tv+uv^\perp:\lvert t\rvert<3s/2,\ \lvert u\rvert<s/16\}.\] A strict crossing has a compact subroute in \(Q_v(z,s)\), from the portion with \(t<-s\) to the portion with \(t>s\), with its two vertex ports and all buffered confinement described by intermediate cells. For an instruction \(\alpha\) of this kind, write \(Y^\alpha=Y_0^\alpha\) for the limiting EL of its path family divided by \(\rho_{j_k}\). Define \[ E_v(z)= \bigcup_{\ell_0\ge1}\ \bigcap_{\ell\ge\ell_0}\ \bigcup_{\alpha}\ \bigcup_{M\ge1} \left\{ \begin{array}{l} \alpha\text{ is activated as a strict crossing}\\[-2pt] \text{with buffered support in }Q_v(z,2^{-\ell}),\ Y^\alpha<M \end{array}\right\}. \tag{52}\] The unions use the countable instruction list and integer \(M\). Whether an instruction fits the displayed open tube is a measurable condition on its limiting local input. Thus \(E_v(z)\) is jointly measurable. Deleting finitely many radii leaves it unchanged, and all supports shrink to \(z\). Its indicator is therefore a germ in Lemma 9. The finite bound \(M\) may depend on the radius.

At arclength-almost every interior visit of almost every path of \(\mu\), arclength parameterization is differentiable with a unit tangent. Choose a direction from a fixed finite sufficiently fine net so that it lies within the angular slack of this tangent. Differentiability gives a strict crossing of \(Q_v(z,s)\) at every sufficiently small \(s\). Almost every path belongs to the support of \(\mu\). For each one of these fixed strict crossings, choose a smaller compact subroute with additional margin. Choose nested open vertex ports \(P_\pm^{\mathrm{in}}\Subset P_\pm^{\mathrm{out}}\) around its two ends and nested confinements \(W^{\mathrm{in}}\Subset W^{\mathrm{out}}\Subset Q_v(z,s)\) around the subroute; the outer ports still lie beyond the two longitudinal levels. Cover the full preimages of the inner sets by cells lying in the outer sets. This gives one activated outer instruction. An open neighborhood \(\mathcal O\) of the path can require the corresponding subroute to visit the inner ports and remain in the inner confinement. Full-preimage reconstruction and the vanishing mesh then imply that, for all large \(k\), every discrete path with projection in \(\mathcal O\) has a trimmed vertex subroute admitted by this fixed instruction. Since the path is in the support, \(\mu(\mathcal O)>0\). Weak convergence gives mass at least \(p>0\) in \(\mathcal O\) for all sufficiently large \(k\). Restricting and normalizing that mass, and trimming its paths, bounds traffic coordinatewise by \(\tau_{\mu_k}/p\). Its squared traffic is at most \(C_0\rho_{j_k}/p^2\). Lemma 10 proves \(Y^\alpha<\infty\) for this radius. This proves \(E_v(z)\) at \(\Lambda\)-almost every such visit for some direction in the finite net.

Since \(\Lambda\) is nonzero and absolutely continuous, on the positive-probability flow event at least one of these finitely many directions satisfies \(E_v\) on a set of positive planar area. Take here the library of all finite intrinsic instructions and its common refinements. Translation by \(a\) carries the fitting condition for \(E_v(z)\) to that for \(E_v(z+a)\); rotation by \(R\) carries it to that for \(E_{Rv}(Rz)\). The same intrinsic instructions describe the transformed cells, and their edge counts and normalized outputs are unchanged. The buffered supports transform with them. These actions are therefore exact on the local arrays, while the corresponding ordinary field laws are locally equivalent as in Lemma 9. That lemma therefore makes the indicator for each direction one deterministic constant almost everywhere on an ordinary connected chart. The positive-area event forces the constant for one direction to be \(1\). Rotational covariance then gives constant \(1\) for every member of a chosen countable dense direction list. Countable intersection and an ordinary chart exhaustion show that, almost surely, \[ E_v(z)\text{ holds for planar-area-almost every }z \text{ and every direction }v\text{ in this list}. \tag{53}\] This assertion has so far been proved for the one network fixed at the start of the power-scale construction.

We use the sphere components retained in this joint extraction to obtain a transverse obstruction. For a rational compact disk \(K\) in a buffered ordinary patch and a direction in the countable list, failure is the measurable event \[\left\{\int_K\bigl(1-\mathbf 1_{E_v(z)}\bigr)\,\mathrm dz>0\right\}.\] Deleting a common finite number of radii for \(z\in K\) computes the tail in (52) using only supports inside this patch. The failure event is therefore measurable from the local input and the activated array of normalized EL observations in Proposition 7, and it is null by (53) in each matching reference marginal, with that marginal’s own conditional kernel. Proposition 7 transfers the event to the corresponding sphere marginal, with the same denominator \(\rho_{j_k}\), actual height, and traversal order. Auxiliary mark pairs exhaust the patches off the exploration root.

The two retained sphere experiments have the same law and the same number of letters per unit. The chosen network in their two marginals gives the two opposite physical networks. The transferred conclusions hold for both simultaneously. Their flag surfaces are identified, and the same three fresh flag-area marks give one canonical coordinate after composing one coordinate with conjugation if its orientation convention requires this. Let \(H_k^+\) and \(H_k^-\) be the two reference homeomorphisms. The canonical comparison in Proposition 5, from (OpenAI 2026a, Theorem 8.8), permits a further extraction on which their changes to the common canonical coordinate converge uniformly to maps \(F^+\) and \(F^-\), each conformal or anticonformal.

Choose a point \(w\) in a canonical coordinate plane away from the finitely many omitted marks and poles, at which the two transported versions of (53) hold. Such points have full planar area: the limiting conformal and anticonformal maps are smooth local diffeomorphisms and preserve area-null sets. Let \(z^\pm=(F^\pm)^{-1}(w)\). Choose directions from the two dense lists whose images under \(DF^\pm(z^\pm)\) are respectively horizontal and vertical to a sufficiently small angular error. Write \(b_\pm>0\) for the scale factors of these similarity or anti-similarity derivatives. The two germs allow all sufficiently small dyadic radii. We may choose such radii \(s_\pm\) with \[1\le \frac{\max\{b_+s_+,b_-s_-\}}{\min\{b_+s_+,b_-s_-\}} \le2.\] They may also be chosen small enough that, for a prescribed small \(\varepsilon_{\mathrm{lin}}>0\), each limiting coordinate change satisfies \[\sup_{\lvert \zeta-z^\pm\rvert\le2s_\pm} \lvert F^\pm(\zeta)-F^\pm(z^\pm)-DF^\pm(z^\pm)(\zeta-z^\pm)\rvert \le \varepsilon_{\mathrm{lin}} b_\pm s_\pm .\]

Put \(s_*=\min\{b_+s_+,b_-s_-\}\). If the angular error is \(\delta\), the transverse deviation of either transformed tube is at most \[2s_*\bigl(1/16+(3/2)\sin\delta+\varepsilon_{\mathrm{lin}}\bigr),\] and its endpoint advance in its longitudinal direction is at least \(s_*(\cos\delta-\sin\delta/16-\varepsilon_{\mathrm{lin}})\). Choose \(\delta,\varepsilon_{\mathrm{lin}}\) so that the first bound is less than \(s_*/4\) and the second exceeds \(0.8s_*\). Thus, in coordinates centered at \(w\), every path in the first transformed family stays in \(\{\lvert y\rvert<s_*/4\}\) and reaches both \(x<-0.8s_*\) and \(x>0.8s_*\); every path in the second stays in \(\{\lvert x\rvert<s_*/4\}\) and reaches both \(y<-0.8s_*\) and \(y>0.8s_*\). Any two such continuous paths intersect, by separation in the intervening rectangle. Fix these radii, activated covers, and their finite bounds from (52). Uniform convergence of the coordinate changes and the strict tube margins give the same forced intersection for every pair of discrete paths for all sufficiently large \(k\).

The two discrete EL values are at most \(C_+\rho_{j_k}\) and \(C_-\rho_{j_k}\), with finite constants because the tests have now been fixed. Choose near-minimum path laws by Lemma 10, and write their traffics on the paired physical edges as \(\tau_k(e)\) and \(\tau_k^*(e^*)\). Every pair of sampled paths intersects. By Lemma 3, a primal and a dual edge meet only when they are paired, so for every pair of walks \(\sum_eN_e(\pi)N_{e^*}(\pi^*)\ge1\). Fix the full coupled record, the selected families, and both path laws, including their possible dependence on the continuum data, radii, and orientation sign. Take the product of these two fixed laws. Taking expectations under this pathwise product and applying Cauchy–Schwarz gives \[1\le\sum_e\tau_k(e)\tau_k^*(e^*) \le \left(\sum_e\tau_k(e)^2\right)^{1/2} \left(\sum_e\tau_k^*(e^*)^2\right)^{1/2} \le\sqrt{C_+C_-}\,\rho_{j_k}\longrightarrow0.\] This is impossible. It excludes the record-low extraction for the chosen network, and the same argument applies when the initial choice is exchanged. Since each quantile sequence is deterministic, each positive lower limit is deterministic. ◻

Proof of Theorem 11. Let \(m_k\to\infty\) be any sequence of letters per time unit. Write \[m_k=4^{j_k}v_k,\qquad 1\le v_k<4,\] and pass further so that \(v_k\to v\in[1,4]\), jointly with all fixed backward rows. Proposition 14 makes the high barriers uniformly positive in raw EL units in those power rows. Equation (22) expresses the same incidence record in the \(m_k\) units: the area clock is divided by \(v_k\), frontier heights by \(\sqrt{v_k}\), and the real field is shifted by \(-\gamma^{-1}\log v_k\). A backward row \(4^{j_k-i}\) has relative label \(v_k^{-1}4^{-i}\). Thus the aligned theorem has multiplier \(M=v^{-1}\) in the limit. The normalized field identity is, explicitly, \[\bigl(h-\gamma^{-1}\log v\bigr)(c+ru)+Q\log r -\gamma^{-1}\log(v^{-1}4^{-i}) =h(c+ru)+Q\log r+i\gamma^{-1}\log4.\] The raw EL is unchanged. Strict margins allow the converging ratios and the rounded units to be retained in the indexed arrays. The bounded constant shifts have uniformly bounded cutoff Cameron–Martin norms on the field buffers, as used when choosing \(\varepsilon\) above. Therefore the same high unit-reference probability applies before annular alignment. This is the field and unit change of (OpenAI 2026a, Proposition 5.12); it re-expresses one raw array and does not compare differently indexed kernels at an unchanged field.

Apply the aligned theorem again and repeat Lemma 12, now with a fixed positive raw barrier, so that the denominator in (45) is \(1\). For a fixed annular template, choose slightly inset ports between \(a_2r\) and \(a_3r\), and an outer cell confinement inside \(W\). It catches a subwalk of every member of \(\mathcal A_m(z,r;\mathbf a)\). Cover that confinement by a finite rational box union \(U\) of area at most \(C_{\mathbf a}r^2\). Every caught subwalk has projected diameter at least \(c_{\mathbf a}r\). On the probability-one aligned event choose a successful, fixed \(N\) large enough that \(CR_N<c_{\mathbf a}r/4\) and \(\operatorname{Area}(U^{+CR_N})\le C_{\mathbf a}r^2\). For this fixed configuration, all sufficiently large discrete indices have a density charging every such subwalk by at least \(c'_{\mathbf a}r\), with squared norm at most \(C'_{\mathbf a}r^2\). Equation (34) and the subwalk property give \[\mathop{\mathrm{EL}}\bigl(\mathcal A_m(z,r;\mathbf a)\bigr) \ge\frac{(c'_{\mathbf a}r)^2}{C'_{\mathbf a}r^2} =c''_{\mathbf a}>0.\] The constants are deterministic. The choice of a successful \(N\) may depend on the activated sample and the fixed radius, but it is fixed before the discrete limit. Narrow collars are included by choosing their positive relative gaps first.

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

The local lower tests in this argument are finite incidence observations with the band inclusions (40). Proposition 7 transfers their probability-one limiting bounds to sphere patches, and convergence of the compactified observations gives the eventual finite inequalities with a fixed slack. A finite family of ordinary patches also treats a collar surrounding an isolated omitted point: choose inner patches with a Lebesgue number comparable to the radius and larger patches still avoiding the point. A through walk has a subinterval of diameter comparable to the radius in one of the larger patches. Summing the finitely many corresponding densities charges all through walks with the same kind of shape bound. The number and relative geometry of these patches can be fixed from the collar ratios. The protected enclosure of Proposition 5 keeps the resulting separation in the original surface. This proves part (i), including its stated sphere uses.

We turn this estimate into part (ii). Fix relative radii \[a<\beta_0<\beta_1<\cdots<\beta_5<b.\] In the protected finite graph containing the complete stars over \(\overline B(z,br)\) and its disk buffer, set \[A_m=\{v:\Psi_m(v)\in\overline B(z,\beta_1r)\}, \qquad B_m=\{v:\Psi_m(v)\notin B(z,\beta_4r)\}.\] Part (i) for the template \((\beta_0,\ldots,\beta_5)\) supplies a density \(g\), normalized to charge its annular family by at least \(1\), with \(\lVert g\rVert_2^2\le C\). Extend it by zero outside its confined edges. Its support lies compactly in \(A(z;ar,br)\). Once the mesh is small, every walk from \(A_m\) to \(B_m\) contains a subwalk of this annular family and therefore has \(g\)-length at least \(1\). On the buffered graph define \[u_m(v)=1-\min\{1,d_g(A_m,v)\},\] where \(d_g=\infty\) on components not meeting \(A_m\). The function is \(1\) on \(A_m\) and \(0\) on \(B_m\). As in Lemma 10, \(\lvert d_eu_m\rvert\le g_e\), so \(\mathcal E^0(u_m)\le\lVert g\rVert_2^2\). Moreover \(u_m\) is constant across every edge on which \(g_e=0\). The vanishing complete-star mesh and the gaps \(a<\beta_1\), \(\beta_4<b\) put every vertex whose projected star meets \(\overline B(z,ar)\) in \(A_m\), and every vertex whose projected star meets the exterior of \(B(z,br)\) in \(B_m\). The support of the differences remains inside the stated compact subannulus. For a protected use, extend the constants across the interior and exterior. Full stars are retained throughout the collar, and any path from one side to the other contains the tested passage; hence no omitted incident edge introduces a new difference in this extension.

For part (iii), take the cutoff from part (ii) in the dual network, with all its nonzero differences in a compact subcollar of \(A(z;br,cr)\). Its inner constant region contains every face meeting \(\overline B(z,ar)\), and its outer constant region contains every face on the other side of the collar. Choose the subcollar with an additional fixed margin to both boundary circles. A paired primal and dual edge lie in their common flags, whose projected diameters tend to zero by Proposition 5; the complete primal and dual star mesh has the same property. Thus each primal edge paired with a nonzero dual difference lies, for all large indices, in a slightly larger compact subcollar of \(A(z;br,cr)\). Choose \(t\) uniformly from \((0,1)\). For all \(t\) except the finitely many vertex values, retain each primal edge whose two incident dual values are on opposite sides of \(t\), and orient it along the boundary of the face on the \(u_m>t\) side. Cyclic incidence at a primal vertex balances incoming and outgoing edges, so these edges form closed walks. A transverse arc from the inner region to the outer one has total signed transition count of absolute value \(1\). The closed walks therefore have nonzero total separating winding. Splitting at repeated vertices gives simple circuits, at least one of which surrounds the inner disk. All its edges remain in the subcollar. This reasoning also includes a primal loop and two parallel edges forming a Jordan curve; a primal bridge has a dual loop and cannot be a transition edge.

Choose one surrounding circuit by any fixed rule on the finite record. An edge can be chosen only when \(t\) lies between its two dual values. Consequently \[\mathbb P(e\in\Gamma_m)\le\lvert d_{e^*}u_m\rvert, \qquad \sum_e\mathbb P(e\in\Gamma_m)^2 \le\sum_{e^*}(d_{e^*}u_m)^2\le C.\] The full face stars and the two constant margins used here are part of the protection; they ensure a closed separating boundary rather than a boundary cut off by the edge of a chart. Exchanging the two networks proves the circuit assertion in either one.

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

Use part (iii) for a circuit surrounding \(B(c,3r_0/4)\) and lying in \(A(c;7r_0/8,9r_0/8)\) at each center \(c\). Consecutive centers have distance \(r_0\). The two enclosed disks of radius \(3r_0/4\) overlap, whereas each has a point outside the other’s disk of radius \(9r_0/8\). Two disjoint Jordan curves would have disjoint or nested interiors, so every pair of consecutive circuits intersects. In one embedded network an intersection gives a common vertex or edge. The union of the sampled circuits is therefore connected from the first port to the last. Select a simple graph path in this union. If there are \(J_*\) circuits and \(\tau_i(e)=\mathbb P(e\in\Gamma_i)\), the path traffic is at most \(\sum_{i=1}^{J_*}\tau_i(e)\) on every edge. Hence \[\mathcal T(\mu_m) \le\lVert \sum_{i=1}^{J_*}\tau_i\rVert_2^2 \le J_*\sum_{i=1}^{J_*}\lVert \tau_i\rVert_2^2 \le J_*^2 C.\] The number \(J_*\) and the collar shape depend only on the fixed route and its clearances and are preserved by a common dilation. This proves part (iv) and completes the theorem. ◻

Proof. We first describe the numerical outputs. A fixed intrinsic cell instruction selects a finite set of counted edges from the exact incidence record. It also attaches a rational closed interval to each of finitely many cell labels. A vertex carrying several labels must satisfy all their intervals. Take the infimum of the sum of squared differences on the selected edges over all such vertex functions, and compactify its value by \(s\mapsto s/(1+s)\), with \(+\infty\mapsto1\). This is a measurable function of a finite incidence record: the objective is a finite quadratic polynomial and the constraints are finitely many closed intervals. The list of intrinsic cell instructions and rational intervals is countable. Proposition 6 therefore retains their joint limits, including equality through a common union record on overlaps.

Fix \(V_0\Subset V_1\Subset U\) and \(\varepsilon_0>\varepsilon_1>0\). On the continuum activation event, cover the full preimage of \(\overline V_0\) by finitely many used cells whose closures lie over \(V_1\); include the complete stars of their vertices and edges. Subdivide the cells so that the oscillation of \(f\) on each image is smaller than a fixed fraction of \(\varepsilon_0-\varepsilon_1\). Rational intervals can then be chosen so that, for every represented vertex of that cell and all large \(m\), \[ [f(z_m(v))-\varepsilon_1,f(z_m(v))+\varepsilon_1] \ \subset\ J_{\rm cell}\ \subset\ [f(z_m(v))-\varepsilon_0,f(z_m(v))+\varepsilon_0]. \tag{57}\] The first inclusion holds for every label at a multiply labelled vertex, so these simultaneous demands are compatible. Full preimages, star mesh, and the strict margins ensure that the selected edge set contains every edge located in \(V_0\) and contains only edges located in \(V_1\). The constrained vertex set contains every vertex located in \(V_0\) and contains only vertices located in \(V_1\).

Let \(s_m\) be the resulting infimum. The eventual exact-incidence and activation conclusions of Proposition 6, followed by (57), give \[ q_m(f,V_0,\varepsilon_0)\le s_m \le q_m(f,V_1,\varepsilon_1) \qquad\text{for all sufficiently large }m. \tag{58}\] The observation \(s_m\) was extracted before this activation was selected; activation only supplies its interpretation in the spatial chart. Its limit exists in \([0,\infty]\). Although \(f,V_0,V_1\) were arbitrary, every instruction just used belongs to the one countable list. Uniform continuity of \(f\) on the relevant compact sets and the separated tolerances are what allow that list to serve all continuous functions. More precisely, each instruction’s activation is defined from its local continuum cells and margins alone; it does not use \(f\) or the value intervals. After \(f\) is given we choose a suitable entry among the already activated countable entries. This is a pathwise use of their simultaneous conclusions, not a new conditioning event.

It follows from (58) that \[\limsup_m q_m(f,V_0,\varepsilon_0) \le \liminf_m q_m(f,V_1,\varepsilon_1).\] For each inner \(V_0\) choose \(V_1\Subset U\), and choose \(\varepsilon_1<\varepsilon_0\). Taking the limits and suprema in (56) gives \(\mathcal R^+(f,U)\le\mathcal R^-(f,U)\). The opposite inequality is immediate. The same sandwich shows that an alternative edge-location convention or an overlapping protected record gives the same result.

For (i), on each fixed \(V\Subset U\) and at each positive tolerance, \(w_m\) is eventually admissible in (55). Its energy on \(V\) is at most its energy on \(U\). Take the liminf, then the tolerance limit and the inner-domain supremum.

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

Finally, for fixed \(\varepsilon>0\) and all sufficiently large \(j\), \[q_m(f,V',\varepsilon) \le q_m(f_j,V',\varepsilon/2) \quad\text{whenever }V'\Subset V,\] because \(f_j\to f\) uniformly on \(W\). Taking the limsup in \(m\) bounds the right side by \(\mathcal R(f_j,W)\). Take the liminf in \(j\), then the tolerance limit and the supremum over \(V'\Subset V\). This is (iii). ◻

Proof. Here is the gluing estimate, including the localization it needs. Let a compact set be covered by finitely many open recovery regions \(U_1,\ldots,U_h\), with all closures and supports kept inside a larger protected chart. Choose finitely many smaller disks covering the compact set, each with a fixed enlargement in one \(U_i\). The disk cutoffs from Theorem 11, followed by finite maxima, give functions \(\eta_i\in[0,1]\) supported with room in \(U_i\), with \(\sum_i\eta_i\ge1\) on a neighborhood of the compact set and with bounded energies. Dividing by their sum there gives weights \(\theta_i\ge0\) with \(\sum_i\theta_i=1\). Outside that neighborhood one can divide by \(\max\{\sum_i\eta_i,1/2\}\). On the bounded range of these functions this operation is Lipschitz, so the energies of the \(\theta_i\) are bounded by a fixed multiple of \(\sum_i\mathcal E_m(\eta_i)\).

Let \(w_{i,m}\) approach \(f\) uniformly in the regions where \(\theta_i\) is used, with common error \(\delta_m\to0\), and set \(w_m=\sum_i\theta_i w_{i,m}\) near the compact set. For a counted edge \(e=(x,y)\) there, its endpoints lie in this neighborhood for large \(m\). Using \(\sum_i d_e\theta_i=0\), we have \[ d_e w_m =\sum_i\theta_i(y)d_e w_{i,m} +\sum_i\bigl(w_{i,m}(x)-f(z_m(x))\bigr)d_e\theta_i. \tag{62}\] Jensen’s inequality bounds the squared first term by \(\sum_i\theta_i(y)(d_e w_{i,m})^2\). The sum over edges of the square of the second term is at most \[h\delta_m^2\sum_i\mathcal E_m(\theta_i)=o(1).\] The cross term tends to zero by Cauchy–Schwarz whenever the energies of the recoveries are bounded. The support room and vanishing star mesh ensure that every edge with a nonzero \(i\)-term belongs to the region where the \(i\)-th recovery was estimated. Thus gluing costs at most the sum of the recovery energies plus \(o(1)\).

Monotonicity of \(\mathcal R(f,\cdot)\) is immediate, and its definition gives inner regularity: \[\mathcal R(f,O)=\sup_{V\Subset O}\mathcal R(f,V).\] For disjoint opens \(O_1,O_2\), use compactly contained test domains in each. The counted edge sets are disjoint, and the infimum with all the constraints is at least the sum of the separate infima. The liminf and inner-domain limits give \(\mathcal R(f,O_1\cup O_2)\ge \mathcal R(f,O_1)+\mathcal R(f,O_2)\).

For subadditivity, fix a compactly contained test domain in \(O_1\cup O_2\). Cover its closure by inset regions of \(O_1,O_2\), recover \(f\) there using Lemma 15(ii), and apply (62). The glued functions converge uniformly on the test domain, so Lemma 15(i) bounds its relaxation by \(\mathcal R(f,O_1)+\mathcal R(f,O_2)\). Taking the inner supremum proves subadditivity. The construction with finitely many weights proves the corresponding finite-cover statement. A compact subset of a countable open union has a finite subcover, so inner regularity then gives countable subadditivity.

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

The constraints in an inner domain depend only on \(f\) there; this proves locality. Adding a constant to all vertex values proves invariance under constants. Also \[q_m(af,V,\varepsilon)=a^2q_m(f,V,\varepsilon/\lvert a\rvert) \quad(a\ne0),\] which proves homogeneity after relaxation.

To prove the parallelogram identity, first recover \(f\) and \(g\) on a common smaller domain. Their sums and differences converge uniformly to \(f+g\) and \(f-g\), and the exact edge identity gives \[\mathcal E_m(w_m+\widetilde w_m;V) +\mathcal E_m(w_m-\widetilde w_m;V) =2\mathcal E_m(w_m;V)+2\mathcal E_m(\widetilde w_m;V).\] Apply the two lower bounds on \(V\), the recovery upper bounds on a slightly larger domain, and then exhaust the original open set. This gives \(\Gamma_{f+g}+\Gamma_{f-g}\le2\Gamma_f+2\Gamma_g\) on opens and also proves local finiteness of the sums. Recover \(f+g\) and \(f-g\) instead, and take their half-sum and half-difference. The same displayed identity gives the reverse inequality. Equality on opens is equality of Radon measures. Polarization and homogeneity now give real bilinearity. Finally, positivity of \(\Gamma_{f+t g}(E)=\Gamma_f(E)+2t\Gamma(f,g)(E)+t^2\Gamma_g(E)\) for every real \(t\) gives (61). ◻

Proof. For the upper bound, fix \(V\Subset U\) and a square grid of small fixed spacing \(r\). Cover a neighborhood of \(\overline V\) by disks of radius comparable to \(r\), centered at grid points, with fixed enlargements contained in \(U\). Construct a cutoff partition \((\theta_{i,m})\) as in the proof of Lemma 17, and put \[w_{m,r}(v)=\sum_i f(x_i)\theta_{i,m}(v),\] where \(x_i\) is the center of the \(i\)-th disk. At a vertex in \(V\), all contributing centers are at distance \(O(r)\), so \(w_{m,r}\) differs from \(f\) by at most its modulus of continuity at \(O(r)\).

For an edge in this region, subtract from the coefficients \(f(x_i)\) the value at one contributing center. The sum of the differences of the partition weights is zero. At most a fixed number of weights contribute, and each coefficient difference is at most \(Cr\sup_{B_i'}\lvert \nabla f\rvert\) on a fixed enlargement \(B_i'\). The cutoff energies and their normalization are bounded independently of \(r\). Summing the squared edge differences gives \[ \mathcal E_m(w_{m,r};V) \le C\sum_{i:\,B_i'\cap V\ne\varnothing} r^2\sup_{B_i'}\lvert \nabla f\rvert^2 \qquad\text{for all sufficiently large }m. \tag{64}\] The bounded overlap includes all endpoint margins. For smooth \(f\), first let \(m\) tend to infinity with \(r\) fixed. Then let \(r\) decrease to zero. The right side is bounded by the corresponding Riemann estimate on any fixed intermediate open set between \(V\) and \(U\), and hence by \(C\int_U\lvert \nabla f\rvert^2\). The uniform errors tend to zero. The definition of \(\mathcal R\) and inner exhaustion prove the upper bound.

For the lower bound, let \(w_m\to f\) locally uniformly on \(U\), and pass to a subsequence on which \(\mathcal E_m(w_m;U)\) tends to its finite liminf \(L\). At grid spacing \(r\), use at each grid point \(x_i\) a circular collar \[\alpha r<\lvert x-x_i\rvert<\beta r, \qquad \tfrac12<\alpha<\beta<1,\] with fixed inner and outer margins; for example one can take \(\alpha=0.7\) and \(\beta=0.8\). Restrict to grid cells in an inner part of \(U\), so every buffered collar lies in \(U\). A surrounding circuit contains the inner disk in its Jordan interior and has that interior inside the outer disk. For neighboring grid points the inner disks overlap, while each center is outside the other’s outer disk. Thus two such circuits must intersect: otherwise their overlapping interiors would be nested, contrary to the center exclusion. Circuits in the same embedded network intersect at a graph vertex.

Let \(E_{i,m}\) be the energy of \(w_m\) in the buffered \(i\)-th collar. If \(\tau_i\) is the expected counting traffic of its circuit law, then \[\mathbb E\!\left[\sum_{e\in\gamma_i}\lvert d_e w_m\rvert\right] =\sum_e\tau_i(e)\lvert d_e w_m\rvert \le \sqrt{B E_{i,m}}.\] Choose one circuit satisfying this bound and assign any one of its vertex values \(a_{i,m}\) to \(x_i\). At a common vertex of two neighboring circuits, \[\lvert a_{i,m}-a_{j,m}\rvert^2 \le 2B(E_{i,m}+E_{j,m}).\] Every grid point has at most four neighbors. The enlarged collars have a fixed overlap multiplicity \(M\), including the edge-location margins for all large \(m\) at this fixed \(r\). Hence \[ \sum_{i\sim j}\lvert a_{i,m}-a_{j,m}\rvert^2 \le 8B\sum_i E_{i,m} \le 8BM\,\mathcal E_m(w_m;U). \tag{65}\] The piecewise affine interpolation on a triangulation of the grid has Dirichlet integral bounded by a fixed multiple of the left side. Moreover \[\lvert a_{i,m}-f(x_i)\rvert \le \omega_f(Cr)+o_m(1)\] on each fixed compact part of \(U\), where \(\omega_f\) is the local modulus of continuity. With \(r\) fixed, take a subsequence of the finitely many grid values and let \(m\) tend to infinity in (65). Then refine \(r\). The interpolants converge uniformly to \(f\) on inner compact sets and have bounded Sobolev energy there. Weak compactness in \(H^1\) and lower semicontinuity yield the weak gradient of \(f\) and \[\int_V\lvert \nabla f\rvert^2\,\mathrm dx\le C L \qquad(V\Subset U).\] One may use a finite union of inner grid regions at each stage, with the same overlap bound. Exhausting \(U\) gives the lower inequality for a uniformly convergent sequence.

Apply that inequality to the recoveries in Lemma 15(ii), and exhaust their inner domains. It gives the lower bound in (63) and the assertion that locally finite relaxed energy supplies a local \(H^1\) gradient.

If \(f\in C(U)\cap H^1_{\rm loc}(U)\), mollification on a buffered subdomain gives smooth \(f_j\) converging uniformly and in \(H^1\) on any smaller fixed region. After multiplication by a smooth cutoff equal to one on that smaller region, extend \(f_j\) by zero to \(U\); thus the inputs in the next step belong to \(C(U)\). Use Lemma 15(iii) on \(V\Subset W\Subset U\) and the smooth upper bound on \(W\). It gives \(\mathcal R(f,V)\le C\int_W\lvert \nabla f\rvert^2\). Choose such \(W\) inside \(U\) and exhaust \(U\) by \(V\). This proves the upper bound for continuous \(H^1_{\rm loc}\) functions. All estimates used the same constants on every subopen of the fixed ambient \(D_0\), so Radon regularity gives the corresponding measure inequalities. ◻

Proof. Work first inside \(D_0\Subset D\). Polarize the energy measures of the coordinate functions \(x_1,x_2\), and define \[A_{ij}(x)=\frac{\,\mathrm d\Gamma(x_i,x_j)}{\,\mathrm dx}(x).\] The densities exist because the upper bound of Lemma 18 dominates the measures of \(x_i+x_j\) and \(x_i-x_j\) by Lebesgue measure. Bilinearity makes \(A\) symmetric. For each \(p\in\mathbb Q^2\), \[\Gamma_{p\cdot x}=p^{\mathsf T}A p\,\,\mathrm dx,\qquad c_{D_0}\lvert p\rvert^2\le p^{\mathsf T}A(x)p \le C_{D_0}\lvert p\rvert^2 \quad\text{for almost every }x.\] Intersecting the exceptional sets for rational \(p\) and using continuity in \(p\) gives \(c_{D_0}I\le A\le C_{D_0}I\) almost everywhere.

Let \(f\in C(U)\cap H^1_{\rm loc}(U)\) and write \(\Gamma_f=e_f\,\,\mathrm dx\), as permitted by the upper bound. The measure Cauchy–Schwarz inequality has the corresponding density inequality; this follows by applying it on Borel sets and differentiating. For every rational \(p\), outside a null set depending on \(f,p\), \[\begin{split} \lvert e_f-p^{\mathsf T}A p\rvert &=\left|\frac{\,\mathrm d\Gamma(f-p\cdot x,f+p\cdot x)}{\,\mathrm dx}\right|\\ &\le \sqrt{e_{f-p\cdot x}\,e_{f+p\cdot x}} \le C_{D_0}\lvert \nabla f-p\rvert\lvert \nabla f+p\rvert. \end{split}\] Use one full-measure set for all rational \(p\), and let \(p\) approach \(\nabla f(x)\). We obtain \[ \mathcal R(f,U)=\int_U\nabla f^{\mathsf T}A\nabla f\,\mathrm dx. \tag{67}\] The construction on smaller subdomains of this coordinate chart agrees on overlaps by locality, so these matrices form a measurable locally elliptic field.

We next identify its randomness and coordinate dependence. At every Lebesgue point, \[ A_{ij}(z)=\lim_{r\downarrow0} \frac{\Gamma(x_i,x_j)(B(z,r))}{\pi r^2}, \tag{68}\] and it is enough to use a countable sequence of radii. Each open energy in the numerator is obtained from the cell infima in Lemma 15, on inner domains within that ball. Balls with arbitrary centers can themselves be exhausted by a countable base of rational open sets. Thus the density is jointly measurable in \(z\) and the retained arrays, and is readable from arbitrarily small strict neighborhoods of \(z\). No outside minimizer enters this reading: only endpoints of counted edges are used, and their full stars lie in the slightly larger protected neighborhood. The affine functions and their sums in (68) are among the continuous tests handled by the one countable cell library.

We check the coordinate action required for this germ. Transporting an intrinsic cell instruction by a translation or a rotation only relabels its spatial domains and the scalar value intervals in the same exact finite incidence record. The quadratic polynomial on its edges is unchanged. The inner-domain and tolerance sandwich therefore transports the open energy measures. A translation adds constants to the coordinate functions, while a rotation \(x\mapsto Rx\) sends their gradient vectors by \(R\); hence the density matrix is transported by \(A\mapsto RAR^{\mathsf T}\). On overlapping disks the union records of Proposition 6 give the common restriction of these measures. Thus the shrinking-ball reading has the translation and rotation actions, with common restrictions, required by Lemma 9.

Apply Lemma 9 to bounded transforms of the three entries of \(A\). It makes \(A(z)\) deterministic for almost every deterministic \(z\), and transports that value under the intrinsic affine coordinate changes. A translation changes each coordinate function by a constant, which leaves its energy measure unchanged. The deterministic matrix is consequently translation invariant almost everywhere, hence constant in the ordinary chart. Equivalently, invariance under the countable rational translations and mollification on smaller domains gives the same conclusion. A rotation conjugates this constant matrix by the rotation matrix. For a quarter-turn \(R\), invariance reads \(A=RAR^{\mathsf T}\). A real symmetric \(2\)-by-\(2\) matrix with this property has equal diagonal entries and zero off-diagonal entries. Thus \(A=kI\), with deterministic \(k>0\).

The preceding assertion is a probability-one statement about activated local outputs and local continuum data. Proposition 7 transfers its null failure event from the ordinary input and the extended kernel to each strict-interior sphere patch at its actual field height, without changing the raw energy units. Its countable atlas and separate auxiliary area-pair marginals remove the auxiliary chart marks. This proves (66) in the sphere reference charts with the same deterministic constant. The argument is applied separately to the primal and dual outputs. We denote the resulting constants by \(k_{\rm p}\) and \(k_{\rm d}\). ◻

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

Let \(u\) minimize the primal unit-voltage energy and let \(I\) be its total current out of \(s\). It is harmonic at every free vertex. Summation by parts gives \[\sum_e c_e(d_eu)^2 =\sum_v u(v)(Q_cu)(v)=I,\] so \(C_{\rm p}=I>0\). Orient the original edges and put their signed currents \(c_e\) times the corresponding voltage drops on them. Put the balancing current on the added exterior edge. The resulting flow has zero divergence at every vertex.

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

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

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

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

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

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

Proof. Embed a cube graph in the canonical sphere with one facial cycle equal to \(\partial Q\), and pull this embedding back to the reference sphere by \(T^{-1}\). The cubic skeleton lift (OpenAI 2026a, Lemma 4.7) gives a subdivided primal cube whose graph homeomorphism tracks the pulled-back embedding to any fixed accuracy; its facial cycles and their order are preserved up to one reversal. The reference homeomorphisms in Proposition 5 have vanishing projection error, and \(T_m\to T\) uniformly transfers that tracking to the canonical sphere. Apply the lift at successively smaller fixed accuracies and choose their thresholds increasingly. This gives a simple primal cycle \(\gamma_m\) with an ordered parametrization approaching \(\partial Q\) uniformly. In particular, its inside contains every compact subset of \(Q\) eventually and lies in every fixed enlargement of \(\overline Q\). Let \(G_m^Q\) be the submap consisting of this cycle and everything inside it. Its four arcs follow the four square sides with error tending to zero. Define \(C_m^{\rm p},C_m^{\rm d}\) as in Lemma 20, using unit conductances.

Write \(Q^\epsilon=\{x:\mathop{\mathrm{dist}}(x,\overline Q)<\epsilon\}\). For an upper bound, choose a smooth function \(f_\delta\) on a neighborhood of \(\overline Q\), depending only on the horizontal coordinate, equal to zero in a strip about the left side and one in a strip about the right side. It can be chosen with \[\int_Q\lvert \nabla f_\delta\rvert^2\,\mathrm dx\longrightarrow1 \quad\text{as }\delta\downarrow0.\] For example, flatten the linear function in strips of width \(\delta\), use slope \(1/(1-2\delta)\) between them, and smooth the two bends in smaller strips. Extend it to a fixed small enlargement \(Q^\epsilon\). For fixed \(\delta,\epsilon\), every edge of \(G_m^Q\) is eventually located in \(Q^{\epsilon/2}\). Apply the recovery clause of Lemma 15 with \(W=Q^{\epsilon/2}\Subset U=Q^\epsilon\), together with (66). It recovers \(f_\delta\) uniformly on a neighborhood of \(\overline{Q^{\epsilon/2}}\), with limsup energy on \(Q^{\epsilon/2}\) at most \(k_{\rm p}\int_{Q^\epsilon}\lvert \nabla f_\delta\rvert^2\,\mathrm dx\). The constant-recovery modification after Lemma 17 makes it exactly zero and one on smaller side strips, with additional energy \(o(1)\). The boundary arcs lie in these strips for all large \(m\). Hence \[\limsup_m C_m^{\rm p} \le k_{\rm p}\int_{Q^\epsilon}\lvert \nabla f_\delta\rvert^2\,\mathrm dx.\] Let \(\epsilon\downarrow0\), and then \(\delta\downarrow0\), to obtain \(\limsup C_m^{\rm p}\le k_{\rm p}\).

For the dual upper bound use the corresponding function of the vertical coordinate, with the same recovery domains \(W=Q^{\epsilon/2}\Subset U=Q^\epsilon\). Thus the full-dual recovery converges uniformly on a neighborhood of \(\overline{Q^{\epsilon/2}}\), and its limsup energy on \(Q^{\epsilon/2}\) is bounded by \(k_{\rm d}\) times its Dirichlet integral over \(Q^\epsilon\). Apply the constant-recovery modification after Lemma 17 to this full-dual recovery on smaller bottom and top side strips whose closures lie in \(Q^{\epsilon/2}\) and in the constant regions of the vertical profile. It makes the recovery exactly zero and one there, with additional energy \(o(1)\). Give an interior face its modified full-dual value and give each wired vertex its prescribed constant. A dual edge crossing one of the two relevant boundary arcs has its other endpoint at a genuine exterior face of the full map. The complete star of that face has diameter tending to zero. Thus this endpoint is in the smaller constant side strip for all large \(m\), and the squared difference is exactly the squared difference to the wired value. The same face cannot meet the two separated side strips. For this fixed \(\epsilon\), all interior dual edges and these boundary crossing edges eventually lie in \(Q^{\epsilon/2}\), including their whole endpoint stars. Their energy is consequently bounded by the recovered full-dual energy on \(Q^{\epsilon/2}\). The identical sequence of fixed limits, with vertical \(f_\delta\), gives \(\limsup C_m^{\rm d}\le k_{\rm d}\).

For the lower bound take a subsequence attaining the liminf of either conductance and its minimizing voltages, clipped to \([0,1]\). The upper bound just proved gives bounded total energies. The voltages are harmonic at every undetermined vertex in the submap. On each compact subset of \(Q\), the boundary is absent for large \(m\), and all original primal incidences, or all original dual incidences, are present. Lemma 21 therefore gives a locally uniformly convergent further subsequence, with continuous limit \(f\).

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

Fix a point in the relative interior of such a side. Use \(N\) disjoint fixed circular collars centered there, as in Lemma 21, within the neighborhood just described. Every surrounding circuit visits the exterior side of the approximating boundary. Its Jordan interior contains an inner disk; a ray from a point of that disk on the exterior side to the far exterior must cross the circuit. The crossing has a fixed positive distance from the limiting side, so vanishing mesh supplies an exterior vertex of the circuit. That vertex has the prescribed value. One of the \(N\) circuits has variation at most \(\sqrt{BM/N}\). The maximum principle inside it applies to all undetermined vertices; the other vertices there have the prescribed constant value. It bounds their difference from that constant by \(\sqrt{BM/N}\). First take the network limit and then increase \(N\). This proves that \(f\) extends continuously to the prescribed value on each relative side interior, uniformly on its compact subsegments.

For \(V\Subset Q\), all edges located in \(V\) belong to the submap eventually, for either network. Extend the minimizing function arbitrarily outside the submap; no edge in this inner test sees the extension. The lower relaxation bound and (66) give \[k_{\rm p}\int_V\lvert \nabla f\rvert^2\,\mathrm dx \le\liminf_m C_m^{\rm p}\] in the primal case, and the corresponding inequality with \(k_{\rm d}\) in the dual case. Exhausting \(Q\) puts \(f\) in \(H^1(Q)\) when the liminf is finite. In the primal case its boundary values are zero and one on the relative interiors of the left and right sides. The Sobolev line property and Cauchy–Schwarz on horizontal slices give \(\int_Q\lvert \nabla f\rvert^2\ge1\). More explicitly, on \(y\in[\eta,1-\eta]\) the boundary convergence makes \(f(a,y)\) uniformly close to zero and \(f(1-a,y)\) uniformly close to one for small \(a>0\). For almost every such \(y\), \[\int_a^{1-a}\lvert \partial_x f(x,y)\rvert^2\,\mathrm dx \ge \frac{\lvert f(1-a,y)-f(a,y)\rvert^2}{1-2a}.\] Integrate, let \(a\downarrow0\), and then \(\eta\downarrow0\). The dual problem has the same calculation on vertical slices. Consequently both liminfs are at least their stated scalars. Together with the upper bounds this proves the lemma. ◻

Proof of Theorem 16. Lemma 19 identifies the two reference relaxations with the scalar integrals \(k_{\rm p}\) and \(k_{\rm d}\). Equation (69) puts these integrals in the same canonical coordinate, and (70) equates their positive constants. Apply Lemma 20 to every submap in Lemma 22. Its exact identity and the two limits give \[k_{\rm p}k_{\rm d}=1.\] Thus \(k_{\rm p}=k_{\rm d}=1\).

For a continuous function outside \(H^1_{\rm loc}\), local finiteness is ruled out by Lemma 18. For a continuous \(H^1_{\rm loc}\) function with infinite integral, inner exhaustion and its lower bound give infinite relaxed energy. This completes (59) for every continuous input.

Proposition 5(iv) supplies ordinary chart interiors from two rerootings covering the canonical sphere. Each carries the formula just proved, by Proposition 7 in its own marked marginal and by (69). On overlaps these are restrictions of the same finite-network relaxation, as in Lemma 15. The Radon measure property therefore patches their common Dirichlet density across a finite subcover of each compact test. This proves the sphere statement, including at points omitted by one of the rootings. The lower and recovery assertions are Lemma 15 with the now identified value. Since the finite product has fixed the value to one on every further extraction, the coefficient is independent of the extraction. ◻

Proof. For a vertex \(v\), there is a path from \(v\) to \(\mathsf a_n\) on which the voltage is at least \(u_n(v)\). To prove this, let \(S\) be the component of \(v\) in the induced superlevel graph \(\{w:u_n(w)\ge u_n(v)\}\). If \(\mathsf a_n\notin S\), connectivity supplies a boundary edge, and every boundary difference from \(S\) to its complement is strictly positive. Consequently \[\sum_{w\in S}(Q_nu_n)(w) =\sum_{\substack{e=\{w,w'\}\\w\in S,\ w'\notin S}} (u_n(w)-u_n(w'))>0,\] whereas (72) makes the left side nonpositive. This is a contradiction. The sublevel argument with signs reversed gives a path from \(v\) to \(\mathsf b_n\) with voltage at most \(u_n(v)\).

For real \(\alpha<\beta\), let \(T\) clip a number to \([\alpha,\beta]\), and put \(D=\beta-\alpha\). Monotonicity and the one-Lipschitz property of \(T\) give on every edge \[(d_e Tu_n)^2\le d_e u_n\,d_e Tu_n.\] Using (73), \[ \mathcal E_n^0(Tu_n) \le \mathcal E_n^0(u_n,Tu_n) =Tu_n(\mathsf a_n)-Tu_n(\mathsf b_n)\le D. \tag{77}\]

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

Theorem 11 supplies a circuit law with \(\sum_e\tau(e)^2\le B\). The expectation below is only over that finite-network circuit law. By (77), \[D\le \mathbb E\!\left[\sum_{e\in\gamma}\lvert d_e Tu_n\rvert\right] =\sum_e\tau(e)\lvert d_e Tu_n\rvert \le \sqrt{B\,\mathcal E_n^0(Tu_n)} \le\sqrt{BD}.\] Thus the voltage oscillation in the inner disk is at most \(B\), independently of the total energy of \(u_n\).

The punctured sphere \(O\) is connected. Cover a compact \(K\subset O\) and finitely many paths joining it to \(z_0\) by finitely many such disks, with consecutive disks overlapping. Vertices occur in each overlap for all large \(n\), by the vanishing mesh in Proposition 5. Chaining the oscillation bounds and using \(u_n(\mathsf d_n)=0\), \(z_n(\mathsf d_n)\to z_0\), gives the asserted bound \(M_K\). The chart cover in Proposition 5(iv) allows these disks at every point of \(O\), including a point omitted by one rooting. Under the uniform conformal or anticonformal comparison, sufficiently small fixed round collars contain and are contained in reference collars of comparable relative shape. Their fixed-test circuit bounds therefore apply in the canonical coordinate.

Choose a slightly larger compact neighborhood \(K'\subset O\) of \(K\). For large \(n\), both endpoints of every edge located in \(K\) lie in \(K'\). Clip \(u_n\) to an interval containing all its values on \(K'\). This clipping leaves those edge differences unchanged. (77) bounds their total energy by the finite length of that interval, giving \(M_K'\).

For each fixed \(V\Subset W\Subset O\), both moving source vertices are eventually outside \(\overline W\). Equation (72) then makes \(u_n\) harmonic at every vertex with full star in \(W\). Apply Lemma 21 on \(W\), using the local bounds just proved, to obtain compactness on \(V\). A countable exhaustion by such buffered pairs and a diagonal extraction give the final subsequential compactness assertion on \(O\). ◻

Proof. We first justify the lower measure bound with the necessary outer room. Fix a compact \(K\) inside a relatively compact open \(D\subset O\). Choose \(K\subset V\Subset W\Subset D\), with \(V,W\) open and \(\xi(\partial W)=0\). Such a \(W\) can be chosen among distance neighborhoods with sufficiently small radius: the boundaries are disjoint as the radius varies, and a finite measure gives positive mass to at most countably many of them. The local uniform convergence and the lower bound of Lemma 15 give \[\mathcal R(u,V) \le\liminf_n\mathcal E_n^0(u_n;W)=\xi(W)<\infty.\] The equality uses vague convergence on the relatively compact continuity set \(W\). These choices around each compact set and Lemma 18 put \(u\) in \(H^1_{\rm loc}(O)\). Theorem 16 now identifies \(\mathcal R(u,V)=\int_V\lvert \nabla u\rvert^2\). Thus \[\int_K\lvert \nabla u\rvert^2\,\mathrm dx\le\xi(D).\] Take the supremum over compact \(K\subset D\), and then over relatively compact \(D\). This proves \[ \lvert \nabla u\rvert^2\,\mathrm dx\le\xi. \tag{79}\] The continuity set \(W\) is essential here: the open-set Portmanteau inequality alone has the opposite direction for the desired comparison.

We next use local minimization to obtain the reverse bound and harmonicity. Work in one Euclidean chart, and let \(B_r=B(z,r)\Subset O\) be a disk with \(\xi(\partial B_r)=0\). Let \(\varphi\in C_c^\infty(B_r)\). Choose fixed radii \[r<r_1<r_2<r_3<r_4,\qquad B_{r_4}\Subset O.\] The function \(u+\varphi\) is continuous and in \(H^1\) on this buffered disk. The recovery clause of Theorem 16 gives \(w_n\to u+\varphi\) uniformly on a neighborhood of \(\overline B_{r_3}\) and \[ \limsup_n\mathcal E_n^0(w_n;B_{r_3}) \le\int_{B_{r_4}}\lvert \nabla(u+\varphi)\rvert^2\,\mathrm dx. \tag{80}\] Choose a cutoff \(\theta_n\) equal to one on \(B_{r_1}\) and zero outside \(B_{r_2}\), with bounded energy for these fixed radii. Put \[v_n=\theta_n w_n+(1-\theta_n)u_n.\] Extend the recovery arbitrarily where it is multiplied by zero. All changed edges lie in \(B_{r_3}\) for large \(n\), by the open room and star mesh. The transition collar is outside \(B_r\), where \(\varphi=0\), so \(w_n-u_n\to0\) uniformly there. Applying the edge identity (62) to these two functions gives \[ \mathcal E_n^0(v_n;B_{r_3}) \le \mathcal E_n^0(w_n;B_{r_3}) +\mathcal E_n^0(u_n;B_{r_3}\setminus\overline B_r)+o(1). \tag{81}\] Indeed the principal squared differences are bounded by their convex combination. The coefficient of an old difference can be nonzero only outside \(\overline B_r\), since \(\theta_n=1\) on \(B_{r_1}\) and the mesh is small. The remaining term has energy bounded by \(\sup_{\rm collar}\lvert w_n-u_n\rvert^2\mathcal E_n^0(\theta_n)=o(1)\); its cross term is \(o(1)\) by the bounded old and recovered energies.

Neither source value changes. The minimizing identity (74), restricted to the changed edges, gives \(\mathcal E_n^0(u_n;B_{r_3}) \le\mathcal E_n^0(v_n;B_{r_3})\). Subtract the old energy on \(B_{r_3}\setminus\overline B_r\) in (81). The difference on the left is \(\xi_n(\overline B_r)\ge\xi_n(B_r)\), so the result is \[\xi_n(B_r) \le\mathcal E_n^0(w_n;B_{r_3})+o(1).\] Take \(n\) to infinity. The left side tends to \(\xi(B_r)\) because \(B_r\) is a continuity set, and (80) bounds the right side. Now decrease \(r_4\) to \(r\), choosing the three intermediate radii afresh between them at each step. Each cutoff is used only for a fixed step before the network limit. The continuum energy has no mass on the circle \(\partial B_r\), so \[ \xi(B_r)\le \int_{B_r}\lvert \nabla(u+\varphi)\rvert^2\,\mathrm dx. \tag{82}\]

Set \(\varphi=0\). Together with (79), this gives equality of the two measures on every such disk. These disks form a cover basis after excluding at most countably many radii at each center. The nonnegative measure \(\xi-\lvert \nabla u\rvert^2\,\mathrm dx\) vanishes on that basis, hence vanishes on \(O\). This proves (78).

For \(\psi\in C_c^\infty(B_r)\), use \(\varphi=t\psi\) in (82) and the measure equality. Expanding for both signs of \(t\) and letting \(t\to0\) gives \(\int_{B_r}\nabla u\cdot\nabla\psi\,\mathrm dx=0\). Every point has such a disk, so \(u\) is distributionally harmonic on \(O\). Interior harmonic regularity makes it classically harmonic. Finally, vague convergence to the measure in (78) gives convergence on each of its relatively compact continuity sets. ◻

Proof of Theorem 23. Take any subsequence. Lemma 24 and a further vague energy extraction give a locally uniform limit \(u\) on \(O\). Lemma 25 makes \(u\) harmonic there and identifies its local energy measure. The normalization \(u_n(\mathsf d_n)=0\) and local uniform convergence imply \(u(z_0)=0\).

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

Fix \(t\in\mathbb R\). Choose annuli \(V\Subset W\Subset U\Subset O\) by slightly moving the two boundary radii, and recover \(u+t\chi\) on a neighborhood of \(\overline W\). The recovery \(w_n\) satisfies \[ \limsup_n\mathcal E_n^0(w_n;W) \le\int_U\lvert \nabla(u+t\chi)\rvert^2\,\mathrm dx. \tag{83}\] For a vertex of the full network put \[\widehat u_n(v)=u_n(v)+t\chi(z_n(v)).\] Take an annular cutoff \(\theta_n\) equal to one on a neighborhood of \(\overline V\) and zero outside an intermediate annulus compactly contained in \(W\). Such a cutoff is obtained by combining two disk cutoffs; its energy is bounded for these fixed radii. Define \[v_n=\theta_n w_n+(1-\theta_n)\widehat u_n.\] On both transition collars \(w_n-\widehat u_n\to0\) uniformly. Wherever the coefficient \(1-\theta_n\) of an edge difference can be nonzero, the edge lies outside \(\overline V\) with a margin, where \(\chi\) is constant on each component. Thus \(d_e\widehat u_n=d_eu_n\) on every such edge for all large \(n\). This uses the clearance from \(\mathop{\mathrm{supp}}\nabla\chi\) and vanishing star mesh. Outside \(W\), \(\chi\) is constant on the inner and outer components and \(\theta_n=0\); hence the changed energy is confined to \(W\). The source vertex is in the inner component and the sink is in the outer one, so \[v_n(\mathsf a_n)=u_n(\mathsf a_n)+t,\qquad v_n(\mathsf b_n)=u_n(\mathsf b_n).\]

The edge gluing estimate, with the old part supported outside \(\overline V\), gives \[\mathcal E_n^0(v_n;W) \le \mathcal E_n^0(w_n;W) +\mathcal E_n^0(u_n;W\setminus\overline V)+o(1).\] The error has energy at most the squared uniform collar discrepancy times the fixed cutoff energy, and its cross term tends to zero by local energy bounds. Minimality of the source functional (74) now gives \[\mathcal E_n^0(u_n;W) \le\mathcal E_n^0(v_n;W)-2t.\] Subtract the displayed old collar energy. The difference on the left is \(\xi_n(\overline V)\ge\xi_n(V)\), so the resulting inequality is \[\xi_n(V) \le\mathcal E_n^0(w_n;W)-2t+o(1).\] By Lemma 25, the left side converges to \(\int_V\lvert \nabla u\rvert^2\), since \(V\) is a continuity set. Use (83), then shrink \(U\) to \(V\) through annuli with the same clearance from \(\mathop{\mathrm{supp}}\nabla\chi\). At each stage the annuli and cutoffs are fixed before \(n\) tends to infinity. Absolute continuity of the continuum energy and the zero mass of the boundary circles give \[ \int_V\lvert \nabla u\rvert^2\,\mathrm dx \le\int_V\lvert \nabla(u+t\chi)\rvert^2\,\mathrm dx-2t. \tag{84}\] Expanding this inequality yields \[0\le 2t\left(\int_V\nabla u\cdot\nabla\chi\,\mathrm dx-1\right) +t^2\int_V\lvert \nabla\chi\rvert^2\,\mathrm dx.\] It holds for every real \(t\), so the coefficient of \(t\) is zero: \[ \int\nabla u\cdot\nabla\chi\,\mathrm dx=1. \tag{85}\] For a cutoff equal to one near \(y\) and zero near \(x\), the replacement changes the sink value by \(t\) and the source value by zero. The source term in (74) then changes by \(+2t\); the expanded inequality gives flux \(-1\) at \(y\).

We verify that these fluxes determine the only possible singularities. Choose an outer coordinate disk about \(x\) whose closure excludes \(y\), and a fixed collar inside it surrounding a smaller disk about \(x\). For every sufficiently large \(n\), take a surrounding network circuit in that collar. Eventually \(\mathsf a_n\) is in the smaller disk and \(\mathsf b_n\) is outside the outer disk, hence outside the circuit. Lemma 24 bounds the voltages on that collar uniformly. Inside the circuit, \(Q_nu_n=0\) except at the positive source, where \(Q_nu_n=1\). The minimum principle therefore bounds every interior voltage from below by the minimum on the circuit. Passing to the limit bounds \(u\) from below near \(x\). For \(y\), choose an outer disk whose closure excludes \(x\) and a collar inside it surrounding a smaller disk about \(y\), and take a surrounding network circuit there. Eventually \(\mathsf b_n\) is inside its circuit and \(\mathsf a_n\) is outside the outer disk. The maximum principle then bounds \(u\) from above near \(y\).

We recall the planar isolated-singularity argument at the needed strength. If \(v\) is a positive harmonic function on a punctured disk, its circular mean \(M(r)\) satisfies \((rM'(r))'=0\). Thus \[M(r)=a\log(1/r)+b,\qquad a\ge0,\] where positivity forces the sign of \(a\). Harnack’s inequality on the fixed-shape annulus \(\{r/2<\lvert z\rvert<2r\}\) bounds the values on \(\lvert z\rvert=r\) by a constant times \(M(r)\), uniformly in small \(r\). This version of Harnack follows directly from the Poisson kernel on a finite chain of disks in the annulus. Hence \(v=O(1+\log(1/r))\). For each nonzero angular Fourier mode, harmonicity makes its radial coefficient a linear combination of \(r^j\) and \(r^{-j}\), \(j\ge1\). The logarithmic bound forces every \(r^{-j}\) coefficient to vanish. The remaining positive-power series is the harmonic Poisson extension from any fixed smaller circle. Consequently \[v(z)=a\log(1/\lvert z\rvert)+h(z),\] where \(h\) extends harmonically through the center. Applying this to \(u\) plus a constant near \(x\), and to \(-u\) plus a constant near \(y\), shows that \(u\) has only logarithmic or removable singularities.

For \(a\log(1/\lvert z\rvert)\), a radial cutoff equal to one near zero and zero outside has \[\int\nabla\!\bigl(a\log(1/\lvert z\rvert)\bigr)\cdot\nabla\chi\,\mathrm dx =-2\pi a\int\chi'(r)\,\mathrm dr=2\pi a.\] The harmonic remainder contributes zero. Equation (85) therefore sets the coefficient at \(x\) to \(1/(2\pi)\), and the sink flux sets the coefficient at \(y\) to \(-1/(2\pi)\).

For comparison, the round zero-mean Green kernel has, up to its additive constant, \[G_{\rm rd}(z,w) =-\frac1{2\pi}\log\operatorname{ch}(z,w).\] Away from \(w\), a radial calculation on the unit sphere gives \(-\Delta_z G_{\rm rd}(z,w)=-1/(4\pi)\); the preceding local flux is one at \(w\). Thus its distributional equation is \(-\Delta_zG_{\rm rd}(z,w)=\delta_w-(4\pi)^{-1}\,\mathrm d\mathrm{vol}_{\rm rd}\). The difference \(G_{\rm rd}(\cdot,x)-G_{\rm rd}(\cdot,y)\) has exactly the two fluxes and singular coefficients just found. Subtract it from \(u\). The difference extends harmonically across \(x,y\), and a harmonic function on the compact sphere is constant. Its value at \(z_0\) gives precisely (75).

Every subsequence has therefore a further locally uniform subsequence with the same limit \(U_{x,y;z_0}\). This proves (76) along the entire extraction. Evaluating at \(\mathsf c_n\) gives the asserted four-point value. The reasoning applied to arbitrary varying pole and evaluation vertices with the stated distinct limits, so it proves the claimed pathwise quantifier. ◻

Proof. The projected rank and jump inequalities are the spanning-forest and displaced-grid estimates in (OpenAI 2026b, sec. 10). The fixed-code estimate is (OpenAI 2026b, Proposition 10.15); its proof charges a left and right fresh slot for each monochrome rank and gives the coefficient \(\vartheta=1/2+2/\gamma^2\). The decorated address count is (OpenAI 2026b, Lemma 10.13): distinct depths give \(\sum_p m_{k_p}\le T_{\rm occ}\), and the terminal base list supplies every base’s coarser address. The regular-slot deletion bound and grid-partition estimate are the corresponding lemmas in the same section. Its sparse-diagonal remark gives the list and jump order of limits; the extra-ancestor estimate in the proof of (OpenAI 2026b, Lemma 11.6) gives the growing-enclosure condition. In these formulas a generation has log width \(\ell_R\); this accounts for every conversion between sums over generations and log lengths. ◻

Proof. Choose fixed address parameters within Proposition 27 so that \[\delta=\vartheta-1-\varepsilon-\frac{C_0+\eta}{\ell_R}>0.\] Fix \(k,M\) and a log stride \(L>2\ell_R\). Sample an independent translation of a stride-\(L\) mesh in \(\mathcal I_N\). At each seed choose a maximum-cardinality \(H\)-packing truncated at \(M\), with all access witnesses. Use a fixed measurable ordering of the finite reachable set, and keep the independent mesh phase as side information in the code count. Round a seed \(y\) up to generation \(\lceil y/\ell_R\rceil\), discarding the final incomplete generation if needed. The seed depths are distinct because \(L>2\ell_R\). The terminal base-list augmentation adds at most one probe and \(h_{\mathrm b}\) fixed entries, so \(P_N=O(N/L)\) and \(K=O(kM+h_{\mathrm b})\), with constants allowed to depend on the fixed enclosing depth. Only then sample the chronological address-grid displacement. These lists have exactly the order and the form in Proposition 27.

We first bound their occupation for this fixed \(L\). Markov’s inequality in part (i) gives a set of grid displacements of fixed positive conditional probability on which \(\sum D_k\le C_KN/L+O(1)\). Intersect it with the regular-slot event, retaining a fixed positive probability. Use \(e_k\ge m_k-h_{\mathrm b}-D_k\) in (89) and sum the codes by (90). All terminal and decoration terms at fixed \(K,L\) are \(O_K(N/L)+O(1)\). Let \(\mathsf F_t(\Xi)\) be the existence of a realized regular code with this jump bound and occupancy \(t\). Uniformly in the displacement, \[ \log\mathbb P_m^W(\mathsf F_t(\Xi)) \le -\delta\ell_Rt+ \bigl[\vartheta h_{\mathrm b}b+O_K(L^{-1})\bigr]N+O(1). \tag{92}\] The constants in the leading term are independent of \(K\). This calculation retains the terminal errors explicitly; the asymptotic occupancy statement with \(L=L_m\to\infty\) alone would not justify the present order of limits.

For two displacements, each cell of one grid meets at most two cells of the other, hence \(T_{\rm occ}(\Xi)\le 2T_{\rm occ}(\Xi')\). If the maximum occupancy over displacements is large, it is at least half as large on every displacement, including the fixed positive fraction just used. Integrate (92) over the grid and sum over \(t\). Then apply (88). It follows that, for a constant \(C_1\) independent of \(K\), the total log occupation is at most \[[C_1+O_K(L^{-1})]N\] with \(\mathbb Q_m\)-probability tending to one. The same bound applies to the occupation restricted to \(\mathcal I_N\).

Here is the conversion to packing, which is the mechanism of (OpenAI 2026b, Lemma 11.4). A list chosen at a seed \(y\) remains reachable throughout the coarser interval \([y-L,y]\). Use generation \(\lceil x/\ell_R\rceil\) boxes on each continuous log cell, trimming incomplete endpoint cells. They are active throughout \([y-L,y]\) for a seed rounded as above, and their integrated occupation is at most \(\ell_R T_{\rm occ}\). If these boxes at level \(x\) have diameter at most \(c_{\rm box}r_m(x)\), then its chosen terminals stay in distinct boxes throughout that interval whenever \(H e^{-L}>2c_{\rm box}\). Thus the truncated seed count contributes at least \(L\) times that count to integrated occupation, apart from the two end intervals and a fixed-width endpoint strip. Averaging the independent mesh translation turns the seed sum into the uniform integral on \(\mathcal I_N\). On the occupation-exception event, bound the truncated packing by \(M\). Dividing by \(\lvert \mathcal I_N\rvert\), the end error is at most \(O(M(L+\ell_R)/\lvert \mathcal I_N\rvert)\). First let \(m\to\infty\) for fixed \(H,L,k,M\); then take \(H\to\infty\) and \(L=L(H)\to\infty\) slowly with \(He^{-L}>2c_{\rm box}\). The term \(O_K(L^{-1})\) vanishes for each fixed \(k,M\), and the limiting bound is independent of them. This proves (91). ◻

Proof. Choose fixed \(1<\alpha<\beta<a_0\) and apply Lemma 28 with one selected base and \(\mathcal I_N=[\alpha N,\beta N]\). For every \(x\in\mathcal I_N\), the proposed jumps divided by \(r_m(x)\) are at most \(C_0e^{-(a_0-\beta)N}\), while the proposed separation divided by \(r_m(x)\) is at least \(c_0e^{(\alpha-1)N}\). For a fixed enlargement of \(d\), the proposed list is therefore contained in \(\mathcal R_{m,k}(x)\) and is an \(H\)-packing for every fixed \(H\), eventually uniformly in \(x\). Select the base and only \(M\) terminals with their witnesses before the grids. On the asserted event the truncated packing equals \(M\) for every fixed \(H,M\) and all large \(m\). Taking the limits in (91) and then choosing \(M>C_{\rm pack}\) is a contradiction. ◻

Proof. The base is marked at every generation, so \(b_k=1\) and \(m_k\ge1\). At a generation where its component has a rank cycle, that component contributes one rank beyond a union spanning tree. Part (i) of Proposition 27 therefore gives \[e_k\ge m_k-D_k \quad\text{there},\qquad e_k\ge m_k-1-D_k \quad\text{at every other generation}.\] On a regular grid with integrated jump error \(o(N)\), the event in the statement consequently implies \[ \ell_RR_{\rm rk}\ge\ell_RT_{\rm occ}-aN-o(N), \qquad \ell_RT_{\rm occ}\ge BN+O(1). \tag{93}\] Here the first loss counts the generations before \(aN\) and the exceptional high generations. Extra components separated by jumps have already been charged to \(\sum D_k\). A permitted growing enclosure shifts the log origin by \(O(\log\widehat C_m)=o(N)\); its extra ancestors are included in the same error.

Fix \(R\) and the errors so that the \(\delta\) in the proof of Lemma 28 is at least \((\vartheta-1)/2\). Combining (89), (90), and (93), with the sparse errors \(o(N)\), gives total log weight at most \[-\delta\ell_RT_{\rm occ}+\vartheta aN+o(N) \le -\frac{\delta}{2}\ell_RT_{\rm occ} +\left(\vartheta a-\frac{\delta B}{2}\right)N+o(N).\] Choose \(B\) with \(\delta B/2>2\vartheta\) by a fixed margin. The first negative term makes the sum over \(T_{\rm occ}\) geometric, and the remaining exponent is \(-cN+o(N)\). Multiplication by \(e^{\kappa_m}=e^{o(N)}\) preserves convergence to zero.

This estimate was for regular grids. The cycle event itself depends on the grid, so its intersection with a positive fraction of grids is not automatic. Since \(M_{\rm term}=o(N)\), part (iv) provides \(q_m=o(N)\) and \(\alpha_m\to0\) such that \(\mathbb P_\Xi(\Delta(\Xi)>q_m\mid\text{markers})\le\alpha_m\). Let \(E^*\) be the corresponding cycle event for the grid-independent partition, allowing \(q_m\) additional exceptions. If \(E(\Xi)\) is the event in the statement, then \[\mathbb Q_m(E(\Xi))\le\mathbb Q_m(E^*)+\alpha_m.\] For a second independent displacement \(\Xi'\), the conditional probability of being regular with jump error \(o(N)\) and having \(\Delta(\Xi')\le q_m\) is at least \(1/2-\alpha_m\). On \(E^*\), this implies the cycle event for \(\Xi'\) with another \(q_m=o(N)\) exceptions. Hence \[(1/2-\alpha_m)\mathbb Q_m(E^*) \le\mathbb Q_m\{E'(\Xi'),\ \Xi'\text{ regular}\}.\] The preceding exponential estimate applies to the right side. This proves the lemma. ◻

Proof. The simultaneous saturation and all-seed construction in (OpenAI 2026b, sec. 11) uses precisely (91), the sparse cutoffs, and the preselection event. Its order is as follows. First sample the independent log-mesh phase, taking \(L_m=q_m\ell_R\) with integer \(q_m\to\infty\) as in the source; changing the chosen length by at most \(\ell_R\) preserves all displayed asymptotic conditions. Run the truncated recipe at every complete-cell endpoint. Let \(P_N^{\rm seed}=\Theta(N/L_m)\) count these cells and let \(\Lambda_m\) bound one truncated seed list. Before the chronological grid is drawn, round the seed depths as in the source, leaving them distinct, and apply the terminal base-list augmentation at \(J_R\). Coding then uses \(P_N\le P_N^{\rm seed}+1\) and \(K_m=O(\Lambda_m+h_{\mathrm b})\); the two sparse ratios remain vanishing because \(h_{\mathrm b}\) is fixed. The independent mesh phase is held as side information in this count. Next draw the independent chronological grid, form one forest from all the lists, and round and freeze origins inside their coded terminal cells. Then choose uniformly among the original \(P_N^{\rm seed}\) complete cells, including fallback cells, and coarsen independently. This law differs from the stated uniform single-seed law by \(O(L_m/N)\) in total variation; the fallback probability at that sampled cell vanishes along the sparse diagonal. There is no conditioning that every cell succeeded. Conditional on the independent phases, the chosen seed-generation trial is uniform among \(P_N^{\rm seed}q_m=\Theta(N/\ell_R)\) trials, including empty and deleted trials. For each of finitely many fixed residue classes of sufficiently spaced generations, the retained enlarged raw slots are disjoint after deletion. The deletion probability tends to zero by taking the discrete limit first and then the fixed ancestor padding to infinity. The address occupancy estimate gives an expected code length \(O(N)\) under (88). Entropy allocation in each residue class, followed by their finite sum, gives bounded mean conditional entropy for this trial. Its preliminary selected-slot comparison is only a statement about continuum contour paths and finite height differences. It gives the stationary finite-line system without asserting a law for discontinuous lattice outputs. (OpenAI 2026b, Lemma 11.12) supplies the simultaneous actual return links and whole-gap margins. (OpenAI 2026b, Lemma 11.24) supplies the identical original and auxiliary patch, including full stars and full preimages. The inner and outer time-preimage bounds in (ii) are verified from this patch in the proof of (OpenAI 2026b, Proposition 11.30). The patch construction first bounds all relevant partner coordinates in a centered range \([-M,M]\) about their enrolled origins. The retained rational cuts are chosen about those origins, and a finite recipe truncation bounds the absolute values of their endpoint coordinates in view units. Finally (OpenAI 2026b, Lemma 11.27) constructs the countable retained-data recipes and their open fillability. The fixed-duration ordinary completion is the separate construction immediately preceding (OpenAI 2026b, Proposition 11.30). The source construction samples the class anchors after its fixed-recipe observation vector. All tight quantities can first be bounded by fixed constants with an arbitrary fixed loss; its finite data and sparse diagonals are then exhausted. ◻

Proof. Suppose such selected laws exist. Mark and label the alleged unavailable bases before the independent seed mesh, address grid, and view sampling, and fix an arbitrary deterministic label \(\iota\) at that stage. Let \(\mathsf S_m\) denote the selected sampled-trial law on the common array and its successive extensions by the stated construction kernels for base \(\iota\). Let \(\mathsf A_m^{\mathrm{acyc}}\) be the retained indicator from Lemma 31, belonging to the common trial marginal, that base \(\iota\)’s combined center component is acyclic, and write \(\mathsf A^{\mathrm{acyc}}\) for its extracted limit. The auxiliary extensions may depend on \(\iota\); no independence between extensions for different labels is used.

Use Lemma 31 and restrict first to finitely many recipe indices and fixed duration, line-count, and margin bounds, losing an arbitrary fixed probability. For each fixed recipe, before its class anchors are drawn, record \[Y_{m,s}=\mathop{\mathrm{EL}}(\mathcal P_{m,s}),\qquad 1\le s\le q,\] in raw unit-edge units, where \(\mathcal P_{m,s}\) is its full family of paths between the recorded vertex ports in the recorded confinement graph. Use \(+\infty\) for an empty family and compactify these coordinates. This is a function of the same exact incidence record and fixed intrinsic instruction, as required by Proposition 6.

For each fixed recipe and fixed truncation, consider the compatible anchored selected subprobabilities constructed below, retaining \(\mathsf A_m^{\mathrm{acyc}}\) as an extra coordinate. Let \(\nu\) be any weak limit of their pushforwards to normalized view coordinates, after the nuisance trial, unit, and parity labels are projected away at the step specified below. Write \(c_0\) for the limiting intrinsic pre-anchor input, \(a\) for its class anchors, and \(y=(y_s)_{s=1}^q\) for the extended-real electrical output in \([0,\infty]^q\), retained through compactification. Let \(F(c_0,a)\) be the fixed recipe’s anchored feasibility event in the retained limiting view of Lemma 31; its strict margins give the eventual lattice crossing interpretation. Our target is \[ \nu\{\mathsf A^{\mathrm{acyc}}=1,\ F(c_0,a),\ \max_{1\le s\le q}y_s=+\infty\}=0. \tag{95}\] This is a null statement for an unnormalized subprobability; no acyclic, compatibility, or successful-recipe probability is divided out.

In an ordinary completion of the fixed duration, feasibility has a countable family of rational route witnesses. Work first with one such witness: choose compact endpoint balls inside its open ports and a compact connecting route inside its recorded open confinement. Finitely many ordinary charts off the auxiliary marks cover it. Theorem 11 gives a path law with finite squared traffic there. Full preimages and the strict inner/outer margins put its endpoints in the recorded ports and every used edge in the recorded graph for all large indices. The countable intersection covers every feasible witness. Therefore the limit of \(Y_{m,s}\) is finite on feasibility in every jointly extracted ordinary completion. This uses only the fixed-test electrical conclusion, for arbitrary diverging units and a fixed duration; neither its threshold nor its route constant has to be uniform over recipes.

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

Write \(\mathsf R_m^0(dc_0,dy)\) for the independent raw-block reference law of this pre-anchor input and output, integrated against the same retained trial and parameter marginal as \(\mathsf S_m\), and set \(\mathsf S_m^0(B)=\mathsf S_m\{G_m^0,(c_{0,m},Y_m)\in B\}\). The finite domination part of Lemma 8, followed by the bounded all-seed trial entropy, gives \[ \mathsf R_m^0(B_m)\longrightarrow0 \quad\Longrightarrow\quad \mathsf S_m^0(B_m)\longrightarrow0. \tag{96}\] This is tight absolute continuity of the compatible subprobability law; no division by the compatibility mass occurs. The comparison sees the lattice output \(Y_m\), whereas the preliminary comparison in Lemma 31 only concerned continuum contours.

After the entire pre-anchor selected sample, including its full fixed observation vector, draw the finite class-anchor vector \(a_m\). Its translation may depend on \(Y_m\). First restrict to finitely many anchor counts and compact height ranges, keeping these restrictions unnormalized. Its conditional density given that entire sample is bounded, as in (OpenAI 2026b, Lemma 11.25). Choose a positive smooth reference anchor density, with its scaled feasible-lattice version \(\varrho_{m,\theta_m}(da)\). The lattice law is expressed relative to covolume-normalized counting measure; on the compact truncation its density is uniformly bounded below. On those ranges the restriction of the actual conditional anchor law is therefore bounded above by \(C\varrho_{m,\theta_m}\). The same bound given \((c_{0,m},Y_m)\) follows by conditional expectation. The reference anchors are independent of \(Y_m\) conditional on the recorded \(\theta_m\).

Here is the sectional argument, applied without normalizing \(\mathsf S_m^0\). Let \(\mathsf S_m^a\) be the joint selected subprobability after adjoining \(a_m\), restricted to \(G_m^0\) and the chosen anchor-count and height ranges, and put \[\mathsf R_m^a(dc_0,dy,da) =\mathsf R_m^0(dc_0,dy)\varrho_{m,\theta(c_0)}(da).\] For an event \(B_m\) with \(u_m=\mathsf R_m^a(B_m)\to0\), let \(p_m(c_0,y)=\varrho_{m,\theta(c_0)} \{a:(c_0,y,a)\in B_m\}\). Markov’s inequality gives \(\mathsf R_m^0\{p_m>\sqrt{u_m}\}\le\sqrt{u_m}\), while the conditional anchor bound gives \[\mathsf S_m^a(B_m) \le \mathsf S_m^0\{p_m>\sqrt{u_m}\}+C\sqrt{u_m} \longrightarrow0\] by (96). This is the proof of (OpenAI 2026b, Lemma 11.25), including its scaled-lattice form. It allows the actual anchors to depend on the output. Apply it separately to each fixed finite recipe vector; the countable null intersection and the preceding finite recipe truncation handle the later recipe selection.

Before taking the kernel limit, project away the trial index and raw unit/parity labels after expressing the retained paths, shifts, durations, and anchors in normalized view units. Tight absolute continuity survives this pushforward. The fiberwise anchor reference laws are lattice discretizations of one fixed smooth law \(\varrho\); on each normalized compact they converge to \(\varrho\) uniformly over the allowed unit and parity parameters as the minimum unit diverges. The normalized reference factorization of (OpenAI 2026b, Theorem 7.10) and Proposition 6 therefore gives the independent-anchor limit law \(\mu_0(dc_0)K_{c_0}(dy)\varrho(da)\). The positive ordinary filling argument, applied to augmented input \((c_0,a)\), gives the retained subprobability law \[f(c_0,a)\,\mu_0(dc_0)\varrho(da)K_{c_0}(dy), \qquad f(c_0,a)>0\quad\text{on }F.\] As in (30), \(f\) integrates the strict omitted filling and its input-dependent density weight, so it has no \(y\) argument. The ordinary finiteness proved above yields \[0=\int f(c_0,a)\mathbf1_{F(c_0,a)} K_{c_0}\{\max_s y_s=\infty\}\, \mu_0(dc_0)\varrho(da).\] Positivity of \(f\) makes the anchored feasibility/infinite-cost event null for the independent-anchor reference product. The post-anchor tight absolute continuity just proved transfers this null event to the selected anchored limit on compatibility. Every successful selected recipe has \(G_m^0\) by its construction from the actual compatible raw instructions. The projection of \(\nu\) restricted to \(\{\mathsf A^{\mathrm{acyc}}=1\}\) is a submeasure bounded by this compatible anchored marginal, so (95) follows without a kernel or independence assertion for the indicator. Only the independent-anchor reference and ordinary filling laws use the displayed output kernel \(K_{c_0}\); the selected anchored law is used through tight absolute continuity.

All comparisons through the anchor step are finite lattice comparisons at fixed recipe, guard, duration, shift, anchor-count, and height truncations. Their effective slot lengths diverge because \(r_m(bN)\to\infty\). Apply them in each joint extraction with \(r_m(X)\) and its recorded unit parameters; the normalized reference construction and the scaled-lattice anchor argument apply to these varying laws. The ordinary finiteness assertion holds for every such extraction, so the argument also covers mixtures over \(X\), as in (OpenAI 2026b, Proposition 7.18). The anchor restrictions are exhaustible under the selected law: on a fixed line-count and centered-oscillation truncation, the completion assigns class heights \(H_A=(4K_0+10)s(A)+u_A\), with \(1\le s(A)\le k\) and \(\lvert u_A\rvert<1\), where \(K_0\) bounds the retained centered oscillations. Bounding \(k\) and \(K_0\) bounds these normalized heights; bounded density alone is not used for their tightness. Fix the finite recipe list and the other stated truncations. By the strict certificate limits in (OpenAI 2026b, Lemma 11.27), (95) gives vanishing unnormalized selected mass for the finite-threshold cost failure of a successful retained recipe, first as \(m\to\infty\) and then as the cost threshold tends to infinity. Only afterward exhaust the finite recipe and truncation losses. No slot meets the center, and feasibility has no omitted filling as an argument. This is the restricted tightness of \(\sum_sY_{m,s}\) used below.

By Lemma 10, choose a near-minimizing path law for each \(\mathcal P_{m,s}\) with squared traffic at most \(Y_{m,s}+1\). Sample those paths and select a simple surrounding circuit from their union. Every sample works by the recipe’s transverse pattern. If its traffic is \(\tau\) and the path traffics are \(\tau_s\), then \[\tau(e)\le\sum_{s=1}^q\tau_s(e),\qquad \sum_e\tau(e)^2\le q\sum_{s=1}^q\sum_e\tau_s(e)^2.\] On each fixed truncation, its unnormalized upper tail is controlled by the restricted cost estimate. Since \(e^{\zeta N}\to\infty\), the corresponding traffic-failure mass tends to zero.

The inner and outer preimage bounds in Lemma 31(ii) give the two disk conditions. The event \(bN-X\le\sqrt N\) has probability \(O(N^{-1/2})\). Outside it, \(\delta_m r_m(X)/r_m(bN)\to\infty\) after the fixed tightness truncation. Also \[\frac{M_mr_m(X)}{r_m(a_0N)} \le M_m e^{-(a-a_0)N}\longrightarrow0.\] Every uninterrupted visit in a closed circuit disk, including a run on its edges, belongs to the full outer preimage, so it has length less than \(\ell_m\). These conclusions apply to every circuit in the constructed law.

They also have the required support reach from the given base. If its combined component has \(K\) lines, a simple path of consecutive scalar-class connections from the base line to any retained line uses at most \(K-1\) actual return links. Interleaving jumps on those lines and a final jump to a retained edge occurrence uses at most \(2K+O(1)\) steps, each jump bounded by a tight multiple of \(r_m(X)\). The same finite return witnesses work for every occurrence in every retained interval. All path support is in the protected patch, whose complete time preimages are on those intervals. Since \(r_m(X)/r_m(a_0N)\le e^{-(a-a_0)N}\), the jumps are eventually at most \(\ell_m\), and tight \(K\) is eventually below the diverging \(D_m\). The master event and its complete-star mesh bound put the entire supported edges in the required confinement. Combining these vanishing disk and support errors with the restricted cost estimate and then exhausting the fixed losses gives \[\mathsf S_m\!\left( \{\mathsf A_m^{\mathrm{acyc}}=1\}\cap \{\text{base }\iota\text{ is unavailable}\} \right)\longrightarrow0.\] Here unavailable means failure of a law satisfying (a)–(c) and (94). Under the selected laws assumed for contradiction, base \(\iota\) is unavailable with probability one, hence \(\mathsf S_m\{\mathsf A_m^{\mathrm{acyc}}=1\}\to0\).

It remains to use the address charge. The bases in (i) are \(\ell_m\)-separated, whereas high-range cell widths are \(O(r_m(aN))=o(\ell_m)\); the same remains true along the permitted sparse enclosure diagonal. The later return links certify colored center-pair bits in the saturated graph. For each such bit the all-seed code retained an earlier chosen actual seed witness and jumps associating its endpoints to the corresponding original pre-grid seed centers. The association jumps cost \(o(N)\) split generations. A base has a tight seed offset from its original pre-grid seed center. On a fixed offset truncation, the independent-grid probability that they are separated after coarsening by \(U\) is \(O(\min\{1,e^{-U}\})\), whose average for \(U\sim\operatorname{Unif}[0,L_m]\) is \(O(L_m^{-1})\). Thus only \(o(N)\) further trial generations fail, in probability. Rounding an origin preserves its seed center’s terminal cell and all its ancestors. Outside these exceptions the centers, witness endpoints, and bases have the corresponding projected cells. Thus a rank cycle of a center component is a rank cycle in the projected base component. Adding further connected vertices cannot destroy it, since a union spanning tree gains at least one total monochrome rank per added vertex. Apply this argument separately to each fixed label \(\iota\) on the common trial array. Let \(E_{m,\iota}\) count generations \(k\ell_R\in[aN,bN]\) at which its projected base component lacks a rank cycle. After including the conversion exceptions above, averaging the vanishing acyclic probability over the uniform seed-generation trials and applying Markov’s inequality gives \(E_{m,\iota}/N\to0\) in probability under the common selected array law. Since \(h_{\mathrm b}\) is fixed, \(N^{-1}\sum_{\iota=1}^{h_{\mathrm b}}E_{m,\iota}\to0\) in that same probability. The number of generations exceptional for at least one label, divided by \(N\), is bounded by this normalized sum. Outside that union every projected base component has a rank cycle. Choose a deterministic \(\eta_m\downarrow0\) so that the exceptional count is at most \(\eta_mN\) with probability tending to one. This is the cost-independent witness conversion in the proof of the circuit-sparsity result in (OpenAI 2026b, sec. 12).

In regime (i) this contradicts (OpenAI 2026b, Proposition 10.18). In regime (ii) it contradicts Lemma 30. The latter uses only fixed \(B\) and \(r_m(BN)\to\infty\); the distance normalization and the narrow length-recursion restriction in (OpenAI 2026b) were not used. Removing the arbitrary finite truncation losses proves the proposition. ◻

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

By the dictionary of incidence occurrences in (OpenAI 2026a, Proposition 3.1 and Lemma 3.2), each original edge has the two contour steps of one matched burger/order pair: \(L\)-pairs represent primal-tree edges, while the quadrangles of \(R\)-pairs carry the complementary primal edges. The convention includes loops, bridges, and repeated incidences. If the up-step ends at \(u\) and its matching down-step ends at \(v\), their scalar baselines satisfy \[Z(u-1)=Z(v)\le \min_{u-1\le s\le v}Z(s).\] Thus \((u-1,v)\) is an actual scalar link; either step occurrence is within one time step of the corresponding endpoint. One link and at most two unit jumps join its two occurrences, including for loops and parallel edges. The units \(r_m(a_0N)\) diverge whenever a circuit is used, so these unit jumps are allowed. Joining through \(e\) therefore reaches every responsible time from the first within twice the support depth plus a fixed number of steps. The times are \(cT\)-separated. The simultaneous version of Lemma 29, with \(\zeta=1/40\), bounds their number by \(e^{N/40}\). This bounds the multiplicity of each edge in the left side.

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

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

For (ii), in a good run retain the chosen disks maximal for inclusion. Consecutive original interiors overlap, so the maximal interiors have connected intersection graph. Two overlapping incomparable Jordan disks have intersecting boundaries. Their boundary union is consequently a connected subgraph with total \(\lvert df\rvert\)-cost at most the sum of the chosen circuit costs. The run’s first vertex reaches the boundary of a disk containing it within \(\ell+1\) forward steps, and its last vertex does so within \(\ell+1\) backward steps, by the no-run condition. A hit is at a graph vertex, since the walk and circuit use the same embedded graph. Join the two hits in the boundary union. Concatenating these connections across the good runs leaves only the deleted intervals and at most two such exit lengths at each gap. Their number and lengths have the asserted bounds. If there is no good run, use the one unresolved endpoint pair. This is the additive-cost proof of (OpenAI 2026b, Lemma 12.2); it applies to the nonnegative edge costs \(\lvert df\rvert\). ◻

Proof. By Proposition 5, for \(\mu\)-almost every \(z\) the fiber \(\eta_2^{-1}(\{z\})\) has exactly one element among all times in \([0,1]\). The pushforward identity for \(\eta_1\) therefore implies that, for almost every \(t\), \(\eta_1(t)\) has such a unique second time; call it \(\tau(t)\). Fix one of these \(t\), also avoiding the countable union of step endpoints. Every subsequence of \(\tau_n(t)\) has a further convergent subsequence, with some limit \(s\). The two matched corners have the same projected vertex. The uniform tracking in (13) and the continuity of \(\eta_2\) give \[\eta_2(s)=\lim_n z_n\bigl(v_n^{(2)}(\sigma_n(\lfloor2nt\rfloor))\bigr) =\lim_n z_n\bigl(v_n^{(1)}(\lfloor2nt\rfloor)\bigr)=\eta_1(t)\] along this further subsequence. Singleton fibers force \(s=\tau(t)\). Thus the entire sequence converges. The all-times formulation of the singleton property is what also determines limits that could lie in a null set of second-tour times.

Extend \(\tau\) arbitrarily on the exceptional null set. It is measurable as an almost-everywhere pointwise limit of the measurable \(\tau_n\). For every continuous \(g\) on \([0,1]\), bounded convergence yields \[\int_0^1g(\tau(t))\,\mathrm dt =\lim_n\int_0^1g(\tau_n(t))\,\mathrm dt=\int_0^1g(s)\,\mathrm ds.\] Hence \(\tau\) preserves Lebesgue measure. Away from the two limiting endpoints, membership in \(J_n\) eventually agrees with membership in \(J\). For the composed indicators the only additional exceptional times are those for which \(\tau(t)\) is an endpoint; these have measure zero by the pushforward just proved. The indicators converge almost everywhere and are bounded by one, so they converge in \(L^2\). The assertion for finite products and differences follows in the same way.

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

Proof. Put \(\tau_n^{(1)}=\mathop{\mathrm{Id}}\) and \(\tau_n^{(2)}=\tau_n\). Assign a time to the first interval that covers it by setting \[\chi_{n,\ell}(t)= \mathbf 1_{I_{n,\ell}}(\tau_n^{(r_\ell)}(t)) \prod_{q<\ell}\bigl(1- \mathbf 1_{I_{n,q}}(\tau_n^{(r_q)}(t))\bigr).\] For large \(n\), these functions take values in \(\{0,1\}\) and sum to one almost everywhere. For a nonempty original block, let \[\bar f_{n,\ell,j,b}= \frac{1}{\lvert J_{n,\ell,j,b}\rvert} \int_{J_{n,\ell,j,b}}U_n^{(r_\ell)}f(t)\,\mathrm dt.\] Define the linear map on \(F=U_n^{(1)}f\) by \[ Q_{n,j}F= \sum_{\ell=1}^L\ \sum_{b:J_{n,\ell,j,b}\ne\varnothing} \bar f_{n,\ell,j,b}\, \chi_{n,\ell}\, \bigl(\mathbf 1_{J_{n,\ell,j,b}}\circ\tau_n^{(r_\ell)}\bigr). \tag{135}\] Thus the coefficient is always the average on the full original block; the priority rule only restricts the indicator on which this coefficient is used. There are at most \(L2^j\) summands with fixed indicator functions, proving the rank bound with \(C_0=L\).

Equation (130) and the disjoint priority assignment give \[\begin{align*} \lVert F-Q_{n,j}F\rVert_H^2 &\le \sum_{\ell=1}^L \int_{I_{n,\ell}} \lvert U_n^{(r_\ell)}f-P_{n,\ell,j}f\rvert^{2}\,\mathrm dt \\ &\le \sum_{\ell=1}^L C_\ell2^{-pj} \mathcal E_n^0(f;F_{n,\ell}) \le C_1 2^{-pj}\mathcal E_n^0(F), \end{align*}\] where \(C_1=\sum_\ell C_\ell\). The last inequality uses that every \(F_{n,\ell}\) is a subset of the original edge set, including its multiplicities. The map \(Q_{n,j}\) is allowed to take values outside \(\mathcal H_n\); only its rank and its approximation error are used.

Fix \(j\) and any sequence of indices and norm-bounded functions as in the statement. Relabel its indices by \(n\), and first extract a subsequence on which the finitely many absolute interval endpoints converge. By Cauchy–Schwarz and the isometry of \(U_n^{(r)}\), \[ \lvert \bar f_{n,\ell,j,b}\rvert \le \lvert J_{n,\ell,j,b}\rvert^{-1/2}\lVert F\rVert_H. \tag{136}\] Equation (131) bounds these coefficients uniformly for a norm-bounded sequence. Lemma 38 shows that each of the finitely many indicator factors in (135), and hence their product, converges strongly in \(H\). After a subsequence the finitely many scalar coefficients also converge. This proves the fixed-level compactness. A priority piece can have limiting length zero; the bound still holds because the coefficient was taken on the original block in (136).

For the asserted energy compactness, start with any sequence of indices and functions with both norms and energies bounded, again relabelling its indices by \(n\). For each successive integer \(j\ge J\), extract a subsequence on which \(Q_{n,j}F_n\) converges, and use a diagonal subsequence. Given \(\varepsilon>0\), choose \(j\) so large that the square root of the right side of (133) is less than \(\varepsilon/3\) uniformly along this sequence. For two sufficiently late terms, the difference of their level-\(j\) approximants is also less than \(\varepsilon/3\). The triangle inequality makes the diagonal sequence Cauchy in \(H\), proving strong convergence.

Finally let \(i<\dim \mathcal H_n\) be sufficiently large and set \[j(i)=\left\lfloor\log_2\frac{i}{2C_0}\right\rfloor.\] Choose \(i_0\) so that \(j(i)\ge J\) for \(i\ge i_0\). The span of the first \(i+1\) orthonormal eigenvectors of \(B_n\), including its constant mode, has dimension greater than \(\mathop{\mathrm{rank}}Q_{n,j(i)}\le i/2\). It contains a unit vector \(F\) killed by \(Q_{n,j(i)}\). Its energy is at most \(\nu_{n,i}\), so (133) gives \[1\le C_1 2^{-pj(i)}\nu_{n,i},\qquad \nu_{n,i}\ge C_1^{-1}\left(\frac{i}{4C_0}\right)^p.\] This is (134) with \(c_*=C_1^{-1}(4C_0)^{-p}\). Padded eigenvalues are infinite and satisfy the same bound. ◻

Proof. The nonzero eigenvalues of \(K_n\), in decreasing order and with zeros appended, are \(\kappa_{n,i}=\nu_{n,i}^{-1}\), \(i\ge1\), where \((+\infty)^{-1}=0\). Its kernel is doubly centered because \(K_n\) is self-adjoint and kills constants. For \(N\ge i_0\), assumption (ii) gives the uniform Hilbert–Schmidt tail bound \[ \sum_{i>N}\kappa_{n,i}^2 \le c^{-2}\sum_{i>N}i^{-2p}=:r_N^2, \qquad n\ge n_0, \tag{138}\] where \(r_N\to0\) because \(2p>1\).

Suppose first that \(M_n=\lVert K_n\rVert_{\mathrm{op}}\) is unbounded. On a subsequence it tends to infinity, so we may assume \(M_n\ge1\). Choose orthonormal eigenvectors \(e_{n,i}\) for the positive eigenvalues and put \(\alpha_{n,i}=\kappa_{n,i}/M_n\). Every \(\alpha_{n,i}\) belongs to \([0,1]\). By a diagonal extraction, assume these numbers converge to \(\alpha_i\) for every fixed \(i\). If \(\alpha_i>0\), the vector \(e_{n,i}\) exists for large \(n\) and \[\mathfrak e_n(e_{n,i})=\nu_{n,i} =\frac{1}{M_n\alpha_{n,i}}\longrightarrow0.\] Assumption (i), and a further diagonal extraction over these indices, give strong limits \(e_{n,i}\to e_i\). If \(\alpha_i=0\), its rank-one term has Hilbert–Schmidt norm \(\alpha_{n,i}\to0\) and needs no eigenvector limit. It follows that each fixed finite truncation \[\sum_{i\le N}\alpha_{n,i}\,e_{n,i}\otimes e_{n,i}\] converges in Hilbert–Schmidt norm. Here \(e\otimes e\) denotes the rank-one operator \(F\mapsto\langle F,e\rangle_H e\), and a term with zero eigenvalue is omitted. For the positive limiting coefficients this uses \(\lVert e_{n,i}\otimes e_{n,i}-e_i\otimes e_i\rVert_{\mathrm{HS}} \le2\lVert e_{n,i}-e_i\rVert_H\).

After division by \(M_n\ge1\), the squared tail is still at most \(r_N^2\) by (138). The convergent finite truncations and this uniform tail show that \(k_n/M_n\) has an \(L^2\) limit \(k^*\). Hilbert–Schmidt convergence implies operator-norm convergence, so the operator with kernel \(k^*\) has norm one. On the other hand, assumption (iii) implies \(\mathfrak D(k_n/M_n)\to0\) almost everywhere, since its unnormalized numerator has a finite almost-everywhere limit. Boundedness of \(\mathfrak D\) also gives convergence in \(L^2\) to \(\mathfrak Dk^*\). Both imply convergence in measure, whose limit is unique. Thus \(\mathfrak Dk^*=0\). Centering passes to an \(L^2\) limit, so (137) gives \(k^*=0\), contradicting its operator norm one. This proves boundedness of \(M_n\).

Take now any subsequence without dividing by \(M_n\). The bounded inverse norms allow a numerical diagonal extraction \(\kappa_{n,i}\to\kappa_i\) for every \(i\). A positive \(\kappa_i\) gives \(\mathfrak e_n(e_{n,i})=1/\kappa_{n,i}\) bounded, so assumption (i) applies to its eigenvectors. A further diagonal extraction over every index with \(\kappa_i>0\) gives simultaneous strong limits \(e_{n,i}\to e_i\). A zero limiting coefficient again makes its rank-one term vanish. The same truncation argument using (138) supplies a further \(L^2\) limit \(k^*\) of the kernels. It is doubly centered. Assumption (iii), boundedness of \(\mathfrak D\), and uniqueness of limits in measure now give \(\mathfrak Dk^*=\mathfrak Dk\). Equation (137) implies \(k^*=k\). Every subsequence has a further subsequence with this same \(L^2\) limit, proving the full claimed convergence. ◻

Proof of Theorem 1. Start with any sequence of sizes tending to infinity and pass to a common representation extraction as above. On its probability-one event, (140) holds. Let \(\kappa_i\), \(i\ge1\), be the positive eigenvalues of \(K\) in decreasing order, with multiplicity. Proposition 37 says that \(K_h\) is injective on the centered subspace of \(L^2(\mu)\), which is infinite-dimensional. Thus every \(\kappa_i\) is positive and \[\kappa_i=\nu_i^{-1},\qquad \nu_i=2\Lambda_i(h),\quad i\ge1.\] For a nonnegative compact self-adjoint operator \(T\) on \(H\), let \(\kappa_i(T)\) denote its positive eigenvalues in decreasing order, with multiplicity and with zeros appended after a finite list. The variational formula is \[\kappa_i(T)= \inf_{\substack{F\subset H\\ \dim F=i-1}} \ \sup_{\substack{u\in F^\perp\\ \lVert u\rVert_H=1}} \langle Tu,u\rangle_H,\qquad i\ge1,\] where \(F\) ranges over linear subspaces. When \(\kappa_i(T)>0\), the spectral theorem gives the upper bound by choosing the span of the first \(i-1\) eigenvectors, and the lower bound because \(F^\perp\) meets the span of the first \(i\) eigenvectors nontrivially for every such \(F\). When \(\kappa_i(T)=0\), positivity gives the lower bound and a trial subspace containing the positive range gives the upper bound. For two such operators \(T,S\), their quadratic forms on any unit vector differ by at most \(\lVert T-S\rVert_{\mathrm{op}}\). Comparing the two suprema for each \(F\), and then the two infima, therefore gives \[\lvert \kappa_i(T)-\kappa_i(S)\rvert\le\lVert T-S\rVert_{\mathrm{op}}.\] Applying this to \(K_n,K\) yields \[\lvert \kappa_{n,i}-\kappa_i\rvert \le\lVert K_n-K\rVert_{\mathrm{op}} \le\lVert K_n-K\rVert_{\mathrm{HS}}\longrightarrow0 \qquad(i\ge1).\] The characterization applies with multiplicities and with zeros appended to the finite-rank spectra. Since \(\kappa_i>0\), for each fixed \(i\) the number \(\kappa_{n,i}\) is positive for all sufficiently large \(n\), and reciprocation gives \(\nu_{n,i}\to\nu_i\). The constant eigenvalue is zero on both sides. With the conventions of Lemma 2 and Proposition 37, \[ n\lambda_{n,i}=\frac{\nu_{n,i}}2 \longrightarrow\frac{\nu_i}2=\Lambda_i(h) \quad\text{for every }i\ge0. \tag{141}\] Zero padding for the inverses is exactly the prescribed infinite padding for the finite generator spectra. This proves coordinatewise convergence, including every multiplicity, with acceleration exactly \(n\).

For the heat traces, use \(e^{-t\infty}=0\) to write \[H_n(t)=1+\sum_{i\ge1}e^{-t\nu_{n,i}/2},\qquad H_h(t)=1+\sum_{i\ge1}e^{-t\nu_i/2}.\] Fix \(0<\delta<T<\infty\). For \(N\ge i_0\), the high-index bound (134) gives, for all sufficiently large \(n\), \[ \sup_{\delta\le t\le T}\sum_{i>N}e^{-t\nu_{n,i}/2} \le\sum_{i>N}e^{-\delta c_* i^p/2}\longrightarrow0 \quad\text{as }N\to\infty. \tag{142}\] The summability follows, for example, by comparison with the integral of \(e^{-\delta c_*x^p/2}\). Passing to the limit in each fixed index in (134) gives the same bound for \(\nu_i\) and the same estimate for the continuum tail. For a fixed finite set of indices, (141) gives uniform convergence of the corresponding exponential terms on \([\delta,T]\), since for finite \(x,y\ge0\), \[\sup_{\delta\le t\le T}\lvert e^{-tx/2}-e^{-ty/2}\rvert \le\frac T2\lvert x-y\rvert.\] Combining the finite sums and (142) proves \(H_n\to H_h\) uniformly on \([\delta,T]\), hence in \(C_{\mathrm{loc}}((0,\infty))\). Finiteness and measurability of the limiting heat trace were established in Proposition 37.

It remains to express the metric and measure convergence in the requested intrinsic topology. This uses the same coordinate sphere as the inverse kernel above. Write \(\Gamma_n\) and \(\Gamma\) for the two graphs in (12), and let \(\varepsilon_n\to0\) bound their Hausdorff distance in the maximum product metric of round distance and ordinary distance on \([0,\infty)\). By uniform continuity of \(D_h\) on \(\mathbb S^2\times\mathbb S^2\), its modulus \[\omega_h(r)=\sup\bigl\{\lvert D_h(x,y)-D_h(x',y')\rvert: d_{\mathrm{round}}(x,x')+d_{\mathrm{round}}(y,y')\le r\bigr\}\] tends to zero as \(r\downarrow0\). Approximating each point of \(\Gamma_n\) by a point of \(\Gamma\) gives \[ \sup_{u,v\in V_n} \lvert a_nd_n(u,v)-D_h(z_n(u),z_n(v))\rvert \le\varepsilon_n+\omega_h(2\varepsilon_n)\longrightarrow0. \tag{143}\] Approximating \((x,x,0)\in\Gamma\) in the opposite direction also shows that \(z_n(V_n)\) is \(\varepsilon_n\)-dense in the round sphere.

The pushed measures \(\bar\mu_n=(z_n)_*\mu_n\) converge weakly to \(\mu\) by Proposition 5. On the compact round sphere this is convergence in Prokhorov distance. Choose \(\rho_n\downarrow0\) larger than that distance and than \(\varepsilon_n\). The coupling characterization of Prokhorov distance gives a coupling of \(\bar\mu_n\) and \(\mu\) assigning mass at least \(1-\rho_n\) to pairs at round distance at most \(\rho_n\). Lift it to a coupling of \(\mu_n\) and \(\mu\) by the finite conditional distribution of a vertex given its projection \(z_n(v)\). The relation \[\mathcal R_n=\{(v,x)\in V_n\times\mathbb S^2: d_{\mathrm{round}}(z_n(v),x)\le\rho_n\}\] is a correspondence, by round density, and the lifted coupling gives it mass at least \(1-\rho_n\). Its distortion for \(a_nd_n\) and \(D_h\) is at most \[\varepsilon_n+\omega_h(2\varepsilon_n)+\omega_h(2\rho_n),\] by (143). This tends to zero. For completeness, if a correspondence \(\mathcal R\) between spaces \(X,Y\) has distortion less than \(2\xi\), the cross-distance \[d(x,y)=\inf_{(x',y')\in\mathcal R} \bigl(d_X(x,x')+\xi+d_Y(y',y)\bigr)\] together with the original distances gives a metric on their disjoint union. The distortion inequality verifies the triangle inequalities that cross between the spaces. Related pairs have distance at most \(\xi\), so the Hausdorff distance is at most \(\xi\). A coupling giving the relation mass at least \(1-\rho\) makes the Prokhorov distance at most \(\max\{\xi,\rho\}\). Apply this with \(\xi=\tfrac12\operatorname{dis}(\mathcal R_n)+1/n\) and \(\rho=\rho_n\). We obtain \[(V_n,a_nd_n,\mu_n)\longrightarrow(\mathbb S^2,D_h,\mu_h) \quad\text{in Gromov--Hausdorff--Prokhorov distance}.\] The normalization \(a_n\) here is exactly the deterministic diameter normalization in Theorem 1; its identification with the scale in the companion distance graph was proved in Lemma 4.

All three convergences have been proved on this one representation with the same \(\mu_h\), \(D_h\), and conformal sphere. The conformal or anticonformal sign retained by Proposition 5 preserves the round Dirichlet integral, so it also preserves the spectrum defined from this sphere. The continuum spectral readouts are measurable functions of its area in that coordinate by Proposition 37. Thus the joint limiting law on this extraction is the one in Theorem 1, with its stated clock.

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