{"id":"d162fe7f-d7ec-4253-8379-35c1c3964144","arxiv_id":"2607.27008","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Local RCD(K(·),N(·)) bounds are stable under pointed measured Gromov convergence, via a local EVI with remainder and Lagrangian Mosco convergence of Cheeger energies.","lead":"The paper proves that local Riemannian Ricci lower bounds (RCD_loc) are stable under Gromov–Hausdorff convergence of metric measure spaces. This closes a natural gap between the classical Cheeger–Colding local theory and the synthetic RCD framework, and yields almost-everywhere Euclidean weak tangents under purely local curvature assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Thm. 1.1) rests on three pillars that the manuscript develops carefully: (i) local CD stability (Thms. 3.8/3.14), (ii) an error-controlled local EVI (Thm. 1.4) whose remainder vanishes under local doubling (Prop. 7.4 + Thm. 6.17), and (iii) the resulting strong CD_loc + essential non-branching (Thm. 7.5, Cor. 7.6) that lets the Lagrangian Mosco argument of [70] run locally (Thm. 8.2). The only structural soft spot is precisely the one the reader flagged—dependence on doubling to kill the EVI error—and the authors flag it themselves when they reduce the finite-dimensional case to the strong-CD setting. That is a legitimate, explicitly stated hypothesis, not a gap. Residual risk of a local calculation error exists (as the reader notes) but does not rise to a load-bearing concern that would move the verdict. Hence the ACCEPT / HIGH assessment stands unchanged.","tokens_in":54104,"tokens_out":585,"duration_ms":13138,"concrete_test":"Verify that the constant c appearing in the local radius of Thm. 1.1 can be taken equal to 1/10 (the factor already obtained for CD_loc stability in Thm. 3.14) by checking whether the John-domain/Poincaré radii used in Lem. 7.2 and Prop. 7.4 are compatible with the 1/10-shrinkage; if a strictly smaller universal factor is forced, the statement of Thm. 1.1 remains correct but the radius is slightly less sharp.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption diagnosis is accurate and already made explicit by the authors: the local-EVI remainder vanishes (Prop. 7.4) only after local doubling turns small balls into John domains with Gaussian heat-kernel bounds (Thm. 6.17). That hypothesis is not hidden; it is the precise reason Thm. 1.1 is stated under finite-dimensional RCD_loc (via Cor. 3.12 and Thm. 7.5) rather than under bare RCD_loc(K(·),∞). Once doubling is granted, the Lagrangian Mosco argument of [70] closes exactly as claimed, and the reduction from RCD_loc to strong CD_loc + local essential non-branching is standard. No internal inconsistency or unstated gap appears in the logical chain supporting Thm. 1.1.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces local curvature-dimension conditions CD_loc(K(·),N(·)) and RCD_loc(K(·),N(·)) on open subsets of metric measure spaces, with point-dependent curvature, dimension and local radius. It proves that these conditions are stable under pointed measured Gromov convergence when the local radii stay uniformly positive and the curvature/dimension parameters remain controlled (Theorems 1.1, 3.8, 3.14). The key analytic step is a local Mosco convergence of Cheeger energies (Theorem 1.3 / Theorem 8.2), obtained by adapting the Lagrangian polygonal approximation of [70] after establishing a local Evolution Variational Inequality with an annular remainder (Theorem 1.4 / Theorem 7.1). The remainder is shown to vanish on sufficiently small balls once local doubling makes them John domains admitting Gaussian heat-kernel bounds (Proposition 7.4, Theorem 6.17). This yields strong CD_loc and local essential non-branching (Theorem 7.5, Corollary 7.6). As an application, almost every point of an RCD_loc space admits a Euclidean weak tangent (Theorem 1.2).","tokens_in":54298,"tokens_out":852,"duration_ms":13685,"significance":"Stability of global RCD under pmG convergence is classical (Ambrosio–Gigli–Savaré, Gigli–Mondino–Savaré). Extending it to genuinely local, pointwise-variable bounds closes a gap left open by Honda–Sun and aligns the synthetic theory with the extrinsic, local character of Cheeger–Colding. The local EVI with controlled remainder and the resulting strong displacement convexity / essential non-branching on small balls are of independent interest and supply tools for further local analysis. The Euclidean-tangent application via Gigli–Mondino–Rajala is a clean payoff. The arguments are written in full, rely on standard optimal-transport and Dirichlet-form ingredients, and make the doubling hypothesis needed for error vanishing completely explicit.","major_comments":[],"minor_comments":[{"comment":"In the statement of Theorem 1.1 the universal constant c appearing in r_∞(x)=c·lim r_n(y_n) is left unspecified; a parenthetical reference to the concrete factors 1/10 (Theorem 3.14) and 1/6 (Theorem 8.2) would help the reader track the radius losses.","section":"Theorem 1.1 / §3.2–§8"},{"comment":"The reduction from RCD_loc(K(·),N(·)) to strong CD_loc(K(·),∞) via Corollary 3.12 and Theorem 7.5 is correct but slightly buried; a short remark in the introduction or at the beginning of §8 recalling that finite dimensionality supplies the doubling needed for Proposition 7.4 would improve readability.","section":"§1, §7–§8"},{"comment":"Typographical inconsistencies appear in a few places (e.g., “comple metric”, “Poincar´ e”, “Arzel` a-Ascoli”). A light copy-edit pass would remove them.","section":"Throughout"},{"comment":"In Definition 3.1 the requirement r(x)≤r_C(x) is natural for locally complete spaces, but the subsequent global statements sometimes silently assume completeness of the ambient space; Remark 8.3 clarifies the issue, yet a forward pointer earlier would be useful.","section":"Definition 3.1, Remark 8.3"}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical, but the logical chain is complete and the main novelty (local EVI + Lagrangian Mosco under only local Ricci bounds) is genuine. I see no reason to delay acceptance for presentation polish."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the local stability theory for RCD that the field has needed. Theorem 1.1 gives pmG-stability of RCD_loc(K(·),N(·)) under uniform control on the local radii and the curvature-dimension parameters, and if the limiting radius is infinite at one point you recover a global RCD space. The tangent application (Thm 1.2) then yields a.e. Euclidean weak tangents under purely local assumptions, via Gigli–Mondino–Rajala. That answers Honda–Sun Question 7.5 and is the main reason to care.\n\nWhat is genuinely new is the local package that makes the Lagrangian Mosco argument of their earlier arXiv:2511.13320 work in the presence of (possibly negative) curvature: the local EVI with an explicit remainder (Thm 1.4/7.1), the vanishing of that remainder on small John domains once local doubling is in hand (Prop 7.4 + heat-kernel bounds of §6), the resulting strong CD_loc and local essential non-branching (Thm 7.5, Cor 7.6), and the polygonal-plan liminf for Cheeger energy (Thm 8.2). Local CD stability itself (Thms 3.8/3.14) is the expected localization of Sturm/Lott–Villani/GMS and is cleanly done. The writing is full and the logical chain is standard deductive RCD technology; self-citations are tools, not the target.\n\nThe soft spot is exactly the one the authors flag: without local doubling the EVI error need not vanish, strong CD_loc fails, and the Mosco argument does not close. That is why the main theorems are stated under finite-dimensional RCD_loc (via Cor 3.12). It is a legitimate hypothesis, not a concealed gap, and the infinite-dimensional statements are carefully restricted. Minor residual risk of a local estimate slip remains, as with any long technical paper, but nothing load-bearing looks broken.\n\nThis is for people who work with synthetic Ricci bounds, Ricci limits, or local-to-global questions in metric measure geometry. It deserves a serious referee. I would accept it for peer review and expect it to be used.","headline":"Solid local stability package for RCD that answers Honda–Sun and supplies the missing local EVI/Mosco tools; the doubling dependence is explicit and not a hidden gap.","tokens_in":55038,"tokens_out":561,"would_cite":true,"duration_ms":12053,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C23","49Q22","58J35"],"pacs":[],"model":"grok-4.5","headline":"Local Riemannian Ricci lower bounds are stable under Gromov–Hausdorff convergence, and almost every weak tangent is Euclidean.","keywords":["local RCD","Gromov-Hausdorff stability","infinitesimal Hilbertianity","parallelogram identity","Evolution Variational Inequality","weak tangents","essential non-branching","Cheeger energy"],"falsifier":"Exhibit a pmG-convergent sequence of locally RCD spaces whose local radii and curvature parameters stay controlled, yet the limit fails the parallelogram identity for some pair of compactly supported Sobolev functions, or fails to have an Euclidean weak tangent at a positive-measure set of points.","tokens_in":54914,"feed_emoji":"📐","tokens_out":831,"duration_ms":14777,"temperature":0.7,"pith_summary":"Global synthetic Ricci lower bounds of Riemannian type (RCD) are known to pass to Gromov–Hausdorff limits, but the theory is intrinsically global while many geometric questions are local. This paper defines local versions of those bounds on open sets, with curvature and dimension allowed to vary from point to point, and proves they remain stable under pointed measured Gromov convergence when the local radii and curvature–dimension parameters stay uniformly controlled. The hard step is showing that the parallelogram identity for weak gradients—the infinitesimal Hilbertianity that distinguishes Riemannian from Finsler geometry—survives in the limit. The authors obtain this by a Lagrangian argument that needs a new local Evolution Variational Inequality for the heat flow on small balls, with an error controlled by the decay of the flow. Once stability is secured, weak tangents of spaces with local RCD bounds are themselves global RCD(0,N) spaces, and therefore Euclidean at almost every point.","feed_headline":"Local Ricci bounds survive Gromov–Hausdorff limits","feed_subtitle":"Parallelogram identity stays stable; almost every weak tangent is Euclidean","key_machinery":"An effective local Evolution Variational Inequality for the heat flow on sufficiently small balls, carrying a remainder that vanishes as t→0 when the measure is locally doubling (so the balls are John domains with Gaussian heat-kernel bounds). This yields strong local displacement convexity of entropy and local essential non-branching, which close the Lagrangian Mosco-convergence argument for the parallelogram identity.","core_discovery":"If a sequence of pointed complete metric measure spaces converges in the pointed measured Gromov sense and open sets Ω_n in them satisfy the local RCD condition with curvature–dimension parameters and local radii that remain controlled at every limit point, then the limit open set satisfies the corresponding local RCD condition. When the local radius becomes infinite at even one point, the whole limit space is globally RCD.","pith_inferences":["The local EVI-with-error may be useful for quantitative stability or for Ricci flows that remain only locally controlled.","The John-domain heat-kernel analysis suggests the result extends to other domains (e.g. uniform domains) that support global Sobolev inequalities without full geodesic completeness.","Almost-everywhere Euclidean tangents under purely local curvature bounds open a route to partial regularity for singular spaces that are only locally RCD."],"forward_implications":["Local RCD bounds pass to Gromov–Hausdorff limits under uniform control of radii and curvature–dimension data.","Weak tangents of locally RCD spaces are globally RCD(0,N) and Euclidean almost everywhere.","Strong displacement convexity of entropy and essential non-branching hold on sufficiently small balls.","The same stability holds in the infinite-dimensional local RCD(K(·),∞) setting once local doubling is assumed.","If the local radius is infinite at one point, the entire limit space is globally RCD."],"fun_headline_variants":["Local Ricci lower bounds stay stable under GH convergence","Parallelogram identity holds for weak gradients in the limit","Local RCD passes to pointed measured Gromov–Hausdorff limits","Almost every weak tangent is Euclidean when local Ricci bounds hold","Effective local EVI along heat flow yields strong displacement convexity"],"cache_read_input_tokens":49280,"weakest_assumption_plain":"The error term in the local heat-flow inequality must vanish for small times, which needs the reference measure to be locally doubling so that small balls admit Gaussian upper bounds on the heat kernel.","fun_headline_variants_meta":{"raw":{"variants":["Local Ricci lower bounds stay stable under GH convergence","Parallelogram identity holds for weak gradients in the limit","Local RCD passes to pointed measured Gromov–Hausdorff limits","Almost every weak tangent is Euclidean when local Ricci bounds hold","Effective local EVI along heat flow yields strong displacement convexity"]},"model":"grok-4.5","effort":"low","cost_usd":0.00354,"raw_usage":{"total_tokens":1067,"prompt_tokens":661,"num_sources_used":0,"completion_tokens":85,"cost_in_usd_ticks":35404000,"prompt_tokens_details":{"text_tokens":661,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":321,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":661,"tokens_out":85,"duration_ms":6374,"temperature":1.0,"reasoning_tokens":321,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T14:12:50.503658+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a pmG-convergent sequence of locally RCD spaces whose local radii and curvature parameters stay controlled, yet the limit fails the parallelogram identity for some pair of compactly supported Sobolev functions, or fails to have an Euclidean weak tangent at a positive-measure set of points.","supporting_citations":[],"review_version":1}