{"id":"75885ca5-d483-4775-88b7-0f44fb6a1f58","arxiv_id":"2511.13320","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Cheeger p-energies and BV total variations are lower-semicontinuous along pointed-measure Gromov–Hausdorff limits of essentially non-branching CD(K,N) and MCP(K,N) spaces, with a 2^N factor in the MCP case.","lead":"This paper proves that Sobolev and bounded-variation energies pass to Gromov–Hausdorff limits of curved metric-measure spaces, even without the usual Riemannian assumptions. It gives the first lower-semicontinuity results of this kind for measure-contraction spaces and for finite-dimensional curvature-dimension spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"CD(K,N) case is inferred to satisfy the strong-CD hypotheses of Propositions 4.5/4.8 via [45], but the required uniqueness/compression estimate is never stated or proved; if [45] does not deliver it, The main theorems are unproven.","rationale":"The reader's conditional verdict points to the same load-bearing spot: the transition from ordinary CD(K,N) to the strong-CD-type uniqueness/compression estimates used in Propositions 4.5 and 4.8. My analysis confirms this is the most fragile point. The rest of the proof is coherent: Proposition 3.1 gives a clean Lagrangian characterization, and the polygonal interpolation strategy is plausible; no circularity or fatal contradiction is apparent. The concern is whether [45] and [23] really supply the needed uniqueness and density bounds under the stated hypotheses. If they do, the gap is purely expository and a revision can close it. If they do not, the main theorems lack a proof. I also noticed a small sign error in (4.12): the exponent on M should be -(q-1)/2, not +(q-1)/2; this is easily corrected and actually makes the bound stronger, so it is not load-bearing. Given the paper is long, has no formal verification, and leans on several external results, CONDITIONAL remains the appropriate verdict.","tokens_in":37138,"tokens_out":17856,"duration_ms":160854,"concrete_test":"Check the precise hypotheses of [45, Theorem 5.8] against Definition 2.16: does it imply uniqueness of every optimal dynamical plan in OptGeo_q(ρ0m,ρ1m) for all bounded densities on a q-essentially non-branching CD_q(K,N) space (or MCP(K,N) space), with finite or σ-finite m? Then rewrite Propositions 4.5 and 4.8 under the hypothesis 'q-essentially non-branching CD_q(K,N)' instead of 'strong CD_q(K,∞)' and re-derive line (4.13). If uniqueness is not available for all q or for all optimal plans, exhibit a counterexample or add the missing assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main semicontinuity results for finite-dimensional CD(K,N) spaces rely on Propositions 4.5 and 4.8, which are stated for strong CD_q(K,∞) spaces (or, in Prop 4.8, for q-essentially non-branching CD_q(K,N) with finite mn). Theorem 1.1 assumes only CD(K,N), q-essential non-branching for all q, and mn(Xn)<∞. The bridge is the final sentence of Remark 2.12: property (2.18) is asserted to hold for q-essentially non-branching CD_q(K,N) spaces 'thanks to [45]'. But no formal statement or proof of this replacement is given. In Proposition 4.5, the key estimate (4.13) is explicitly derived 'exploiting the strong curvature dimension condition' via Theorem 2.11 plus Remark 2.12. For CD_q(K,N), Theorem 2.11 only produces one optimal plan with the density bound; the proof needs the particular plan η_{i,n} to be that plan, i.e. uniqueness of optimal dynamical plans for absolutely continuous marginals. Uniqueness is a nontrivial property, and [45] proves it under hypotheses not spelled out here. The same gap affects Proposition 4.8 and the MCP case in Proposition 5.2, where uniqueness from [45, Thm 5.8] and the density estimate from [23, Prop 9.1] are asserted without proof. If [45] requires a stronger notion of essential non-branching — e.g. all optimal plans concentrated on non-branching geodesics, rather than existence of one — then Theorems 1.1 and 1.2 collapse at this point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes Mosco-convergence type lower semicontinuity results for p-Cheeger energies and total variations along pointed-measure Gromov–Hausdorff converging metric measure spaces satisfying synthetic curvature bounds. The main theorems are: for q-essentially non-branching CD(K,N) spaces with finite reference measures, L^p-weak convergence f_n → f_∞ implies Ch_p(f_∞) ≤ liminf Ch_p(f_n) for every p∈(1,∞), and L^1-weak convergence implies |Df_∞|(X_∞) ≤ liminf |Df_n|(X_n); for q-essentially non-branching MCP(K,N) spaces the same conclusions hold with a factor 2^N on the right-hand side. The proofs combine a Lagrangian characterization of Sobolev and BV functions (Proposition 3.1) with polygonal geodesic interpolation built on the varying spaces, using curvature-dimension conditions to control compression and kinetic-energy constants.","tokens_in":37579,"tokens_out":15573,"duration_ms":137298,"significance":"If the main results are correct, they provide the first sharp lower semicontinuity/Mosco-convergence statements for Cheeger energies in the non-Riemannian, finite-dimensional CD(K,N) and MCP(K,N) settings, extending earlier results that were restricted to RCD(K,∞) or strong CD_q(K,∞) spaces. The paper also gives a self-contained Lagrangian characterization of Sobolev spaces (Proposition 3.1) and a detailed polygonal-interpolation machinery that may be of independent interest. The use of the q-independence theorem of Akdemir–Colinet–McCann–Cavalletti–Santarcangelo is elegant and allows all p to be treated by the same mechanism. However, as discussed below, a load-bearing step in the CD(K,N) case is only asserted via an external reference and is not formally stated or proved in the manuscript.","major_comments":[{"comment":"Theorem 5.1 and Proposition 4.5 are stated for strong CD_q(K,∞) spaces. The proof of Theorem 1.1(i) obtains CD_q(K,N) from Theorem 2.14, but CD_q(K,N) does not by itself provide the strong-CD property needed in Proposition 4.5. The only bridge is the final sentence of Remark 2.12, which asserts that property (2.18) holds for q-essentially non-branching CD_q(K,N) spaces 'thanks to [45]', without stating or proving the corresponding theorem. This estimate is used in an essential way in (4.13) to control Comp(η_{i,n}); without it the polygonal construction does not yield property (iii) of Proposition 4.5. Since uniqueness of the optimal dynamical plan for absolutely continuous marginals is nontrivial and depends on the precise non-branching definition, the claim cannot be checked by the reader. Please state the exact result from [45] that supplies (2.18), verify its hypotheses against those","section":"§5.2, proof of Theorem 1.1(i); Remark 2.12; Proposition 4.5"},{"comment":"The BV part of Theorem 1.1 has the same gap. Proposition 4.8 invokes Proposition 4.6, which is proved under strong CD_{q_k}(K,∞) (or CD_{q_k}(K,∞) when K≥0). For K<0, the CD(K,N) setting does not satisfy the hypotheses of Proposition 4.6 unless the uniqueness/density assertion of Remark 2.12 is valid. The proof should either apply a version of Proposition 4.6 whose hypotheses are verified after Theorem 2.14, or replace the appeal by a direct argument using the asserted (2.18). As written, the deduction is not complete.","section":"§5.2, proof of Theorem 1.1(ii); Proposition 4.8"},{"comment":"The MCP case also relies on an unstated uniqueness theorem. Proposition 5.2 invokes [45, Theorem 5.8] for uniqueness of optimal dynamical plans and then uses [23, Proposition 9.1] for the density estimate. The hypotheses of [45, Theorem 5.8] — in particular, any qualitative non-degeneracy condition on the reference measure and the exact class of essentially non-branching spaces — are not checked against the assumptions of Theorem 1.2. Since uniqueness is what upgrades the density bound to the particular plan used in Proposition 5.3, the theorem should be stated explicitly and its hypotheses verified. This is likely fixable by a precise citation, but as written it leaves a gap in the MCP proof.","section":"§5.3, Proposition 5.2"}],"minor_comments":[{"comment":"The word 'Poligonal' in the section heading and in Proposition 4.5 should be 'Polygonal'.","section":"§4, headings"},{"comment":"In the definition of τ^{(t)}_{K,N}(θ), the last displayed case involving sinh should be 'if K<0', not 'if K>0'.","section":"§2.5, Definition 2.13"},{"comment":"The exponent in the bound is written as e^{K_- Lip(η)^2/M^2}, while the proof obtains e^{(K_-/12) Lip(η)^2/M^2}. The proof's factor 1/12 should be carried through or intentionally absorbed; as written the statement is weaker than the proof and may confuse.","section":"Proposition 4.8(iii)"},{"comment":"The notation η(Γ_k) appears where η_k(Γ_k) is meant; also in Proposition 4.6 the expression for π_k is typographically garbled.","section":"§5.3, proof of Proposition 5.3"},{"comment":"The sentence 'The argument is analogous to that of Theorem 1.2 (in fact, Theorem 5.1)' should presumably read 'Theorem 1.1' instead of 'Theorem 1.2'.","section":"§5.3, proof of Theorem 1.2(i)"}],"recommendation":"major_revision","confidential_remarks":"The central gap in the CD(K,N) part appears repairable: the authors likely intend to import a uniqueness/density-estimate theorem from Kell's paper [45]. However, the manuscript as submitted does not state the theorem or verify its hypotheses, and the issue is load-bearing for Theorem 1.1 with K<0. The MCP part has a similar but slightly more explicit reliance on [45, Thm 5.8] and [23, Prop 9.1]. I recommend major revision rather than rejection because the overall strategy is sound and the missing piece is a precise, verifiable citation/proof rather than an evident counterexample."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe real news is good: this is the first Mosco-convergence statement for p-Cheeger energies and BV functionals over pmGH-converging essentially non-branching CD(K,N) spaces, and the first stability result for MCP(K,N), with a 2^N loss. That moves past [35] (2-energy on CD(K,∞)) and [10] (RCD spaces). The Lagrangian equivalence (Proposition 3.1) and the polygonal interpolation machinery are written in real detail, with no visible circularity or fitting-to-data. The MCP half is abbreviated, but the strategy is the same as the CD half and the loss factor is explained by the weaker interpolation estimates.\n\nThe soft spot is exactly where the stress-test points. Proposition 4.5 is stated for strong CD_q(K,∞) spaces, and the crucial estimate (4.13) uses uniqueness of the optimal dynamical plan with the density bound. Theorem 1.1 only assumes CD(K,N) with q-essential non-branching and finite reference measures. The entire bridge is the last sentence of Remark 2.12, which says that (2.18) holds for q-essentially non-branching CD_q(K,N) 'thanks to [45]'. No statement or proof is given that [45] delivers uniqueness of the particular plan constructed in the polygonal interpolation, or the required density bound for that plan. If [45] only guarantees existence of some non-branching optimal plan, or needs a stronger non-branching notion, then (4.13) is not justified and Theorem 1.1(i) does not go through. The same gap affects Proposition 4.8 via Proposition 4.6 for K<0, and then the MCP analogs in Proposition 5.2/5.3, where uniqueness from [45, Thm 5.8] and the density estimate from [23, Prop 9.1] are cited without detail.\n\nI do not think this is a fatal flaw in the architecture. The paper is honestly organised; the missing piece is a stated technical input, not a hidden circularity. Either [45] actually provides it (then the theorems are correct) or the hypotheses need strengthening or the proof needs a different route. A referee can ask for exactly that in one round.\n\nThis paper deserves serious refereeing and would make a useful reading-group topic, especially to test the [45] bridge. I would send it out.","headline":"Solid, genuinely novel stability result with one load-bearing bridge: the CD(K,N) case is made to satisfy strong-CD-type uniqueness/density inputs via an asserted citation to [45] that is never stated or proved.","tokens_in":38087,"tokens_out":4919,"would_cite":true,"duration_ms":48468,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C23","49Q22","46E36"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes lower semicontinuity of Cheeger energies and BV total variations along pointed measured Gromov–Hausdorff convergence for essentially non-branching CD(K,N) and MCP(K,N) spaces, proving Mosco-convergence of the nonsmoot","keywords":["Mosco convergence","Cheeger energy","curvature dimension condition","measure contraction property","essentially non-branching","pmGH convergence","BV functions","optimal transport"],"falsifier":"Look for a pmGH-convergent sequence of q-essentially non-branching CD(K,N) spaces with finite reference measures and functions f_n with sup_n Ch_p(f_n) < ∞, f_n → f∞ weakly in L^p, yet Ch_p(f∞) > liminf_n Ch_p(f_n). The theorem asserts the reverse inequality, so any such example would falsify it. A concrete candidate to test would be a sequence of shrinking-neck surfaces with controlled Ricci lower bounds where a concentration of gradient occurs at the neck.","tokens_in":37046,"feed_emoji":"📐","tokens_out":6097,"duration_ms":47507,"temperature":0.7,"pith_summary":"This paper proves that the nonsmooth Sobolev and BV calculus is stable under Gromov–Hausdorff convergence of spaces carrying synthetic Ricci curvature lower bounds. Specifically, if a sequence of essentially non-branching CD(K,N) spaces converges in the pointed measured Gromov–Hausdorff sense, then any weak L^p limit of functions with bounded p-Cheeger energy has Cheeger energy no larger than the limiting energy; the same holds for the total variation of BV functions. The result extends to MCP(K,N) spaces at the cost of a factor 2^N. The proof works through a Lagrangian description of Sobolev and BV functions in duality with test plans, combined with polygonal geodesic interpolations that preserve compression and kinetic-energy estimates. A consequence is the continuity of Neumann eigenvalues along such convergent sequences.","feed_headline":"Curvature bounds make Cheeger energy lower semicontinuous","feed_subtitle":"Weak L^p limits on essentially non-branching CD(K,N) or MCP(K,N) spaces obey the expected energy drop; BV too.","key_machinery":"The engine is a Lagrangian characterization of W^{1,p} and BV: f has finite p-Cheeger energy iff its oscillations along q-test plans (q conjugate to p) are controlled by Comp(π)^{1/p} Ke_q(π)^{1/q} times the Cheeger constant; BV is similarly characterized by Comp(π) Lip(π). The paper then builds M-polygonal geodesic plans on the approximating spaces whose compression and kinetic energies asymptotically match those of a given limit test plan, using density estimates for optimal dynamical plans that follow from essential non-branching (uniqueness of optimal plans) plus CD or MCP. The CD case uses the independence of CD_q on q (Theorem 2.14) to cover all p simultaneously.","core_discovery":"Along a pmGH-converging sequence of q-essentially non-branching CD(K,N) spaces with finite reference measures, Lp-weak convergence f_n → f∞ forces f∞ into the Sobolev class with Ch_p(f∞) ≤ liminf Ch_p(f_n), for every p∈(1,∞); the same inequality holds for L1-weak convergence and the BV total variation. For MCP(K,N) spaces, the identical statement holds with the right-hand side multiplied by 2^N. This establishes Mosco-convergence of the Cheeger energies in these sweeping classes, covering possibly infinite-dimensional and non-Riemannian limits, and it is the first such stability result under the measure contraction property.","pith_inferences":["Because the proof estimates mass and energy on geodesic polygons in bounded regions, it likely extends to localized (non-global) curvature bounds; a natural test is whether the same conclusion holds for local CD or for spaces with a lower bound only on a uniformly shrinking neighborhood of the support.","The factor 2^N in the MCP theorem seems to be an artifact of the compression estimates; a sharper constant (perhaps 1) may hold for essentially non-branching MCP spaces, and examples such as n-dimensional Heisenberg groups (which are non-branching and satisfy MCP) could be used to probe this.","Without essential non-branching, branching geodesics could break the uniqueness of optimal dynamical plans and the density estimates; constructing a branching CD(K,N) example where semicontinuity fails would mark the exact limit of the method.","The Lagrangian test-plan formulation decouples the dual exponent from the curvature condition; this suggests the same scheme may prove Mosco-convergence for other first-order functionals defined via test plans, such as p-weak upper gradients."],"forward_implications":["The p-Cheeger energies Mosco-converge for all p∈(1,∞) along pmGH-converging q-essentially non-branching CD(K,N) spaces with finite measures.","The BV total variation is lower semicontinuous along the same convergence, so sets of finite perimeter and isoperimetric quantities pass to the limit in the expected direction.","For MCP(K,N) spaces, the same semicontinuity holds with a factor 2^N, without requiring finite reference measures, so σ-finite and non-compact limits are covered.","As a direct by-product, the Neumann spectrum of the Cheeger–Laplacian is continuous along such convergent sequences (claimed in the abstract).","The infinite-dimensional and non-Riemannian settings are included, going beyond the Riemannian RCD framework used in earlier work."],"fun_headline_variants":["Cheeger energy stability on CD and MCP spaces","Lower semicontinuity of Cheeger energy with curvature","Mosco-convergence of Cheeger energies in nonsmooth settings","Weak limits preserve Cheeger energy inequality","2^N factor in Cheeger liminf for MCP spaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that each approximating space is essentially non-branching — geodesics cannot fork at an intermediate time — and, in the CD case, that each reference measure is finite; if branching occurs or measures are not finite, the density estimates that replace strong curvature-dimension conditions are unavailable and the proof collapses.","fun_headline_variants_meta":{"raw":{"variants":["Cheeger energy stability on CD and MCP spaces","Lower semicontinuity of Cheeger energy with curvature","Mosco-convergence of Cheeger energies in nonsmooth settings","Weak limits preserve Cheeger energy inequality","2^N factor in Cheeger liminf for MCP spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000937,"raw_usage":{"total_tokens":3779,"prompt_tokens":611,"completion_tokens":3168,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":355,"completion_tokens_details":{"reasoning_tokens":3098}},"tokens_in":355,"tokens_out":3168,"duration_ms":20366,"temperature":1.0,"reasoning_tokens":3098,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T21:49:38.943544+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a pmGH-convergent sequence of q-essentially non-branching CD(K,N) spaces with finite reference measures and functions f_n with sup_n Ch_p(f_n) < ∞, f_n → f∞ weakly in L^p, yet Ch_p(f∞) > liminf_n Ch_p(f_n). The theorem asserts the reverse inequality, so any such example would falsify it. A concrete candidate to test would be a sequence of shrinking-neck surfaces with controlled Ricci lower bounds where a concentration of gradient occurs at the neck.","supporting_citations":[],"review_version":1}