{"id":"b2c52105-4a98-489e-bb37-94d69e183abd","arxiv_id":"2505.09681","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Martinet and Engel sub-Riemannian structures do not satisfy the measure contraction property for any curvature bound K and dimension N, so MCP-type Ricci lower bounds are not universal beyond step two.","lead":"This paper proves that the measure contraction property, a synthetic Ricci curvature lower bound studied in optimal transport, can fail in sub-Riemannian geometry beyond step two. The failure occurs in the Martinet and Engel structures, and the authors develop a quotient technique that spreads the failure to many other Carnot groups and, generically, to high-dimensional rank-three distributions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unconditional Martinet counterexample is sound, but the headline higher-step and generic MCP-failure statements rest on the open minimizing Sard conjecture for Carnot groups and on omitted finite verifications.","rationale":"The reader's weakest-assumption analysis correctly identifies the open minimizing Sard conjecture as the main load-bearing condition. The Martinet failure itself is a direct computation and does not depend on this conjecture; however, the paper's broadest consequences for Carnot groups beyond step two, for generic structures, and for ideal structures inherit the conjecture through Theorem 3.5 and the quotient machinery. The paper is transparent about this in the theorem statements, but the abstract's generic and ideal-structure claims are stronger than what is presently unconditional. I found no internal contradiction in the Martinet Jacobian computation; the double-limit argument can be justified by a diagonal sequence, and the pointwise elliptic asymptotics suffice for that purpose. The remaining issue is the conditional status of the propagation results, together with the explicitly omitted finite computations in Theorem 7.6 and the reliance on unpublished preliminary Sard verifications in Remark 1.15. These are addressable but real, so the reader's CONDITIONAL verdict is appropriate and should not be changed.","tokens_in":58319,"tokens_out":21710,"duration_ms":233035,"concrete_test":"Take a starred group in Table 1 whose Sard status is not already known, for example 2457A. Re-run the algebraic Martinet-quotient criterion of Theorem 6.13(iii) using the Lie brackets from [LT22], then attempt to verify the minimizing Sard property for this group with the endpoint-map and wave-front techniques of [BV20, BNV22]. If Sard can be verified, remove the asterisk; if Sard is found to fail while the Martinet quotient exists, Theorem 7.2's conclusion for that group is not established and the conditional scope of the paper must be narrowed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Martinet counterexample (Theorem 9.6) is independent of the Sard conjecture, and I do not see a fatal flaw in its Jacobian-ratio computation. The load-bearing uncertainty is in the propagation layer. Theorem 7.2, Theorem 8.1, Theorem 8.2 and Corollary 7.8(ii) assume the minimizing Sard property for all Carnot groups of step at most s, or for all Carnot groups in Theorem 8.1. This is exactly Agrachev's Problem 3 and the Rifford-Trélat Conjecture 1; it enters through Theorem 3.5 to obtain the δ-essentially non-branching hypothesis needed in the quotient theorems. If some Carnot group in the relevant range violates the conjecture, the corresponding MCP-failure conclusions are not established. The unconditional part (Corollary 7.3) covers only step-3, filiform, and rank-2 step-4 groups. In the proof of Theorem 7.6 the Table 2 verification is explicitly 'omitted', and Remark 1.15 defers the step-4/5 Sard cases to preliminary computations, so several table entries remain conditional as printed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the measure contraction property (MCP) in sub-Riemannian geometry. Its main unconditional result is that the Martinet Carnot homogeneous space fails MCP(K,N) for all K in R and N in [1,∞), proved via the explicit optimal synthesis of Martinet and an asymptotic expansion of the Jacobian of the exponential map. The authors also introduce a weakened non-branching condition, δ-essential non-branching, which is implied by the minimizing Sard property, and prove stability of the local MCP under quotients by discrete and compact abelian isometric group actions. These tools are combined in a factorization argument showing that MCP descends along quotients of Carnot homogeneous spaces, conditional on the minimizing Sard property for Carnot groups. Applications include failure of MCP for Carnot groups admitting a Martinet quotient, with unconditional consequences for step-3, filiform, and rank-2 step-4 groups, and a generic failure statement in high dimension that is conditional on the same open Sard conjecture.","tokens_in":58539,"tokens_out":10619,"duration_ms":118746,"significance":"The unconditional Martinet counterexample is a substantial contribution: it disproves the expectation that MCP holds for all Carnot homogeneous spaces and identifies non-Lipschitzness of the distance as the source of failure. The δ-essentially non-branching condition and the quotient stability theorems are of independent interest and are developed in detail, with the proofs largely self-contained. The paper is honest about the main hypotheses. However, several of the headline applications, including the generic failure theorem, are conditional on the open minimizing Sard conjecture for Carnot groups, and a number of finite verifications in the low-dimensional classification are omitted. The unconditional core is strong enough to merit publication after revision.","major_comments":[{"comment":"The propagation of MCP-failure from Martinet to Carnot groups of step at least 3 and to generic sub-Riemannian structures assumes the minimizing Sard property for all Carnot groups of step at most s (or for all Carnot groups in Theorem 8.1). This is exactly Agrachev's Problem 3 and the Rifford--Trélat Conjecture 1, an open problem. The assumption is stated clearly in the theorems, but the abstract and the introduction present the generic failure as an established fact ('this actually happens generically'). Please reformulate the affected statements as conditional theorems, and ensure that the unconditional content, namely Corollary 7.3 and its precursors, is explicitly separated from the Sard-conditional results.","section":"Section 7.1, Theorem 7.2; Section 8, Theorems 8.1--8.2"},{"comment":"The classification of indecomposable Carnot groups of dimension at most 7 is not fully verifiable from the manuscript. The proof states that the subspaces h2,h3 verifying Theorem 6.13 are given in Table 2 and that 'the computations are omitted', and later 'We omit the computations' for the Goh--Legendre checks. Remark 1.15 defers the step-4 and step-5 Sard verifications to 'preliminary computations' not included in the paper. Because Theorem 7.6 is a classification result, these omitted verifications are load-bearing. Please include the computations, either in the text or in an appendix/supplement, or explicitly mark every table entry whose status depends on such omitted verification as conditional.","section":"Section 7.3, proof of Theorem 7.6 and Table 2; Remark 1.15"}],"minor_comments":[{"comment":"The second displayed definition in Step 3 repeats the symbol ν_left; it should be ν_right. This makes the mixing construction hard to follow.","section":"Section 3.2, proof of Theorem 3.6, Step 3"},{"comment":"The red/green color coding is essential to the statement; please add symbols or hatching so that the distinction survives non-color printing.","section":"Table 1 and Theorem 7.6"},{"comment":"The sufficient condition (38) is quoted from [AG01, Lemma 4]; adding the precise reference inside the proof of Remark 6.14 would help the reader verify the dimension count.","section":"Remark 6.14"},{"comment":"The claim that rank-2 step-4 Carnot groups always admit a quotient to the Martinet structure is asserted as 'easy to check' without any argument; a one-line verification would make the unconditional corollary self-contained.","section":"Corollary 7.3, third bullet"},{"comment":"The expression 'y(t) = A cn ωt + φ | m' should read 'y(t) = A cn(ωt + φ | m)' for consistency with the Jacobi elliptic function notation used in the surrounding equations.","section":"Section 9, after equation (47)"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and transparent about its assumptions. The main concern for the editor is that the conditional nature of the Sard-dependent results is not reflected in the abstract, where the generic failure is stated without qualification. I see no issue with novelty or scope. The unconditional Martinet counterexample and the quotient machinery justify publication after the requested clarifications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuine advance and it deserves peer review. The unconditional Martinet counterexample looks right — the Jacobian-ratio computation in Theorem 9.6 is explicit, the limiting argument is coherent, and the reliance on the classical optimal synthesis from ABCK97 is appropriate. The quotient machinery (δ-essentially non-branching, discrete and compact quotient stability, the factorization theorem for Carnot homogeneous spaces) is genuinely new and seems carefully proved. The new proof of MCP for the Grushin plane from the Heisenberg quotient is a nice dividend.\n\nThe soft spots are real but not fatal. The headline applications beyond the Martinet/Engel examples — Theorem 7.2, Theorem 8.1, Theorem 8.2, Corollary 7.8(ii) — assume the minimizing Sard property for all Carnot groups up to the relevant step (or for all Carnot groups in Theorem 8.1). That is exactly Agrachev's Problem 3 and the Rifford–Trélat Conjecture 1. If that conjecture fails for some group in range, those conclusions do not follow. The unconditional content is Corollary 7.3: step-3 groups with Martinet quotients, filiform groups, and rank-2 step-4 groups. That is already a substantial unconditional result, but the reader should not take away an unconditional failure of MCP for all higher-step Carnot groups.\n\nTwo smaller issues. In Theorem 7.6 the Table 2 verification is explicitly 'omitted', and Remark 1.15 defers the step-4 and step-5 Sard cases to preliminary computations that are not in the paper. Those table entries are conditional as printed. The authors should either include the computations in a supplementary file or mark the asterisks more prominently in the abstract and introduction. This is presentation, not a flaw in the mathematics.\n\nThe citation pattern looks honest: the Martinet proof builds on external optimal synthesis rather than assuming the target conclusion, and the Sard dependence is stated in the paper, not hidden. I disagree with any reading that calls this circular. The paper is also careful to distinguish what is proved from what is conditional, which is exactly what you want in a paper that overturns a widely held expectation.\n\nWho this is for: anyone working on synthetic curvature bounds in sub-Riemannian geometry, and anyone interested in quotient stability of measure contraction properties. It deserves a serious referee. My recommendation: send it to peer review, with the request that the conditional status be made explicit in the abstract and the omitted table verifications supplied. I would cite the Martinet and Engel failure results myself.","headline":"Solid Martinet/Engel counterexample plus a new quotient toolbox; the sweeping generic conclusions rest on an open conjecture, so read the conditional statements carefully.","tokens_in":59050,"tokens_out":1663,"would_cite":true,"duration_ms":18268,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C17","53C23","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Martinet structure fails the measure contraction property for every K and N.","keywords":["measure contraction property","sub-Riemannian geometry","Carnot groups","Martinet structure","synthetic Ricci curvature","quotients by isometric group actions","essential non-branching","minimizing Sard property"],"falsifier":"Compute the reduced Jacobian ratio |JR(omega t, m)| / |JR(omega, m)| for a fixed t in (0,1) with m close to 1 and omega large: the paper's expansion shows the double limit is 0, so inequality (51) fails for every finite N, decisively falsifying MCP(0,N) for the Martinet structure. For the propagation results, a concrete counterexample to the minimizing Sard conjecture in a Carnot group of step at most s would invalidate the conditional corollaries.","tokens_in":58141,"feed_emoji":"📐","tokens_out":5001,"duration_ms":48863,"temperature":0.7,"pith_summary":"This paper establishes that the measure contraction property (MCP), a synthetic lower bound on Ricci curvature phrased in optimal-transport terms, can fail in sub-Riemannian geometry once the step is at least three. The failure already occurs in the Martinet structure, the simplest non-Lipschitz Carnot homogeneous space, and it propagates to the Engel group and beyond. The proof introduces a new stability result: the local MCP descends to quotients by isometric group actions under a weakened non-branching condition that follows from the open minimizing Sard conjecture. If the paper is right, then synthetic Ricci lower bounds of MCP type are not a universal feature of sub-Riemannian and Carnot homogeneous spaces beyond step two, and failure is generic in high dimension.","feed_headline":"Measure contraction fails in step-3 Martinet geometry","feed_subtitle":"A quotient argument turns this into generic failure of synthetic Ricci bounds in high-dimensional sub-Riemannian structures.","key_machinery":"The operative mechanism is the delta-essentially non-branching condition: non-branching is demanded only for Wasserstein geodesics between absolutely continuous measures and finite sums of Dirac masses. Unlike the classical essentially non-branching condition, this variant is implied by the star-minimizing Sard property in sub-Riemannian spaces, so it becomes available under a standard conjectural hypothesis. The second piece is a factorization theorem: any quotient between Carnot homogeneous spaces decomposes into finitely many local metric measure isometries and quotients by compact abelian groups, each step preserving the local MCP.","core_discovery":"The central discovery is that the Martinet structure does not satisfy MCP(K,N) for any K in the real line and any N in [1, infinity). The argument computes the Jacobian of the sub-Riemannian exponential map from the optimal synthesis of the Martinet flat case and shows that a ratio of reduced Jacobians tends to zero as the elliptic parameter m approaches 1 and the frequency omega tends to infinity, violating the inequality that the MCP would impose. Around this core, the paper builds a quotient calculus: delta-essentially non-branching metric measure spaces preserve the local MCP under discrete and compact abelian isometric quotients, and every quotient between Carnot homogeneous spaces factorizes into such steps. Applying this calculus, the failure propagates upward: any Carnot group admitting a Martinet quotient, including the Engel group and all free step-3 groups, fails every MCP(K,N), conditional on the minimizing Sard property, and generic rank-3 structures in high dimension fail as well.","pith_inferences":["The paper's dichotomy suggests that a strong Goh-Legendre abnormal geodesic may be the dividing line: local Lipschitz regularity, subanalyticity of the squared distance, and the MCP all fail together, although a direct connection is not yet established.","The conditional negative results could be made unconditional by resolving the minimizing Sard conjecture for Carnot groups; preliminary step-4 and step-5 computations mentioned in the paper point in that direction.","The quotient technique could be sharpened on the remaining unclassified groups such as N6,2,6 or N6,3,1a: proving MCP failure for either would force failure for several other groups in the low-dimensional table.","The same factorization argument appears adaptable to sub-Finsler geometry, where analogous measure-contraction questions remain open."],"forward_implications":["The Engel group, free Carnot groups of step 3, filiform Carnot groups of step at least 3, and rank-2 step-4 Carnot groups fail MCP(K,N) for every K and N; several of these failures are unconditional because the minimizing Sard property is known there.","Sub-Riemannian structures with pre-medium-fat distribution can fail the MCP, answering a question raised in the cited literature.","Ideal sub-Riemannian structures of rank greater than 3 and sufficiently high dimension generically fail the MCP, even though they have no nontrivial abnormal geodesics.","Weaker variants of the MCP, including an entropic version and the quasi curvature-dimension condition QCD, also fail for the same spaces.","The Grushin plane satisfies the MCP with sharp Heisenberg constants by a computation-free quotient argument, matching earlier direct estimates."],"supporting_citations":[{"why":"Supplies the optimal synthesis, conjugate and cut times, and the reduced Jacobian formula for the Martinet structure used in Theorem 9.6.","marker":"[ABCK97]"},{"why":"Provides the template for descending curvature-dimension and MCP bounds to quotients by compact isometric group actions.","marker":"[GGKMS18]"},{"why":"Supplies the optimal-map and mixing arguments adapted to prove the local quotient theorem under delta-essential non-branching.","marker":"[CM17]"},{"why":"Introduced the star-minimizing Sard property and the lemma used to show that this property implies delta-essential non-branching.","marker":"[BMR24]"},{"why":"Proves MCP for real-analytic Lipschitz sub-Riemannian structures, the positive baseline that the Martinet counterexample overturns.","marker":"[BR20a]"},{"why":"Introduces pre-medium-fat distributions, asks whether they satisfy the MCP, and proves the minimizing Sard property for them.","marker":"[Rif23]"},{"why":"Provides the genericity of absence of abnormal geodesics for rank at least 3, used in the generic failure theorem.","marker":"[CJT06]"},{"why":"Gives the dimension threshold at which generic tangent cones admit a Martinet quotient, feeding the high-dimensional failure results.","marker":"[AG01]"}],"fun_headline_variants":["Martinet structure breaks measure contraction property","MCP failure spreads via quotients in sub-Riemannian geometry","Generic sub-Riemannian spaces fail synthetic Ricci bounds","Step-3 Martinet geometry: measure contraction fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The propagation of MCP failure from the Martinet structure to Carnot groups and generic structures assumes the minimizing Sard property for all Carnot groups of step at most s, an open conjecture; the Martinet failure itself does not rely on this assumption.","fun_headline_variants_meta":{"raw":{"variants":["Martinet structure breaks measure contraction property","MCP failure spreads via quotients in sub-Riemannian geometry","Generic sub-Riemannian spaces fail synthetic Ricci bounds","Step-3 Martinet geometry: measure contraction fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000361,"raw_usage":{"total_tokens":1949,"prompt_tokens":941,"completion_tokens":1008,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":945}},"tokens_in":557,"tokens_out":1008,"duration_ms":10080,"temperature":1.0,"reasoning_tokens":945,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:26:45.524916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the reduced Jacobian ratio |JR(omega t, m)| / |JR(omega, m)| for a fixed t in (0,1) with m close to 1 and omega large: the paper's expansion shows the double limit is 0, so inequality (51) fails for every finite N, decisively falsifying MCP(0,N) for the Martinet structure. For the propagation results, a concrete counterexample to the minimizing Sard conjecture in a Carnot group of step at most s would invalidate the conditional corollaries.","supporting_citations":[],"review_version":1}