{"id":"2c669ca0-c2af-499a-8cbf-0d138f2a8689","arxiv_id":"2604.14600","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Large-p asymptotic limits of capacity, p-eigenvalue, and Maz'ya constant on noncompact Riemannian manifolds recover volume entropy (and rank at second order) under isoperimetric or Hadamard conditions.","lead":"The authors define large-p asymptotic limits of p-capacity, the first p-eigenvalue, and the Maz'ya constant on complete noncompact Riemannian manifolds and prove a hierarchy among them. Under isoperimetric, symmetry, or Hadamard assumptions these limits recover volume entropy, topological entropy of the geodesic flow, and—at second order—the rank.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the abstract-only limitation already flagged by the reader.","rationale":"The reader correctly treats the abstract as an announcement whose soundness cannot be certified without proofs, definitions, and the missing geometric hypotheses. No stronger load-bearing concern (hidden circularity, inconsistent inequalities, or misuse of entropy rigidity) is visible from the given text. The hierarchy itself is plausible and the applications are standard once equality holds. Therefore the UNVERDICTED / LOW-confidence status remains appropriate; no adjustment is warranted.","tokens_in":1986,"tokens_out":405,"duration_ms":3623,"concrete_test":"Once the full text appears, extract the precise statement of the centered-ball isoperimetric condition (or the “certain condition” for the second-order claim) and verify that the proof of equality with volume entropy invokes only that condition plus the already-established hierarchy; if the equality step uses an unstated curvature or volume-growth assumption, the identification weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states a coherent hierarchy V(M)≥C(Ω)≥Λ(M)=M(M)≥0 and equality with volume entropy (or topological entropy on Hadamard manifolds) under centered-ball isoperimetric or rotational-symmetry hypotheses, plus a second-order rank detection under an unspecified “certain condition.” These claims sit inside a standard program (p-capacity asymptotics, entropy rigidity) and exhibit no internal contradiction or circularity visible from the abstract. The only genuine soft spot is precisely the one already noted: the geometric hypotheses and the “certain condition” are not spelled out, so one cannot yet check whether the equalities are forced or merely conditional. That is an incompleteness of the available text, not a flaw in the announced argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript introduces large-p asymptotic geometric quantities associated with p-capacity, the first p-eigenvalue, and the Maz'ya constant on complete noncompact Riemannian manifolds. It asserts the hierarchy V(M) ≥ C(Ω) ≥ Λ(M) = M(M) ≥ 0, and claims that under a centered-ball isoperimetric condition or rotational symmetry these quantities coincide with volume entropy (or dimension). On Hadamard manifolds of nonpositive curvature the common value is further identified with the topological entropy of the geodesic flow, yielding via entropy rigidity a characterization of hyperbolic manifolds. A second-order refinement is announced: for a Hadamard manifold with compact quotient, under an unspecified “certain condition,” the first-order capacitary limit recovers volume entropy while a logarithmic second-order correction detects rank.","tokens_in":2185,"tokens_out":907,"duration_ms":11245,"significance":"If the announced equalities and the second-order rank detection hold under clearly stated geometric hypotheses, the work would supply analytic large-p characterizations of classical geometric and dynamical invariants (volume entropy, topological entropy, rank) and would link nonlinear potential theory with entropy rigidity. That would be a meaningful contribution to geometric analysis on noncompact manifolds. The hierarchy itself, even when strict, organizes several asymptotic quantities in a useful way. Because the full text is unavailable, these strengths remain conditional on the proofs and on the precise scope of the geometric hypotheses.","major_comments":[{"comment":"Abstract, hierarchy and coincidence claims: the equalities with volume entropy (and with topological entropy on Hadamard manifolds) are stated only under a “centered-ball isoperimetric condition,” rotational symmetry, or an unspecified “certain condition.” These hypotheses are load-bearing for the central identification claims and for the entropy-rigidity application. Without their precise statements, one cannot assess whether the equalities are forced or merely conditional, nor whether the hierarchy can be strict outside those regimes.","section":"Abstract (hierarchy and applications)"},{"comment":"Abstract, second-order refinement: the claim that the logarithmic second-order correction detects rank on a Hadamard manifold with compact quotient is conditioned on an unspecified “certain condition.” This condition is essential to the rank-detection theorem; its absence from the abstract (and the unavailability of the full text) prevents verification that the second-order asymptotic is well-defined and that it genuinely separates rank from volume entropy.","section":"Abstract (second-order refinement)"},{"comment":"Full text unavailable: no proofs, error estimates, or precise definitions of V(M), C(Ω), Λ(M), M(M) are accessible. The hierarchy and coincidence statements cannot be checked for derivation gaps, domain of validity, or dependence on curvature/volume-growth assumptions. A proper technical assessment of the central claims is therefore impossible on the present material.","section":"Manuscript (full text)"}],"minor_comments":[{"comment":"Notation for the four asymptotic quantities (V, C, Λ, M) is introduced only by name in the abstract; a one-line schematic definition of each large-p limit would improve readability even at the abstract level.","section":"Abstract"},{"comment":"The phrase “or the dimension” in the coincidence statement is ambiguous (topological dimension, Hausdorff dimension, or asymptotic dimension?); clarification would help.","section":"Abstract"},{"comment":"The abstract does not indicate whether the results are new relative to existing large-p capacity/eigenvalue asymptotics literature; a brief positioning sentence would aid the reader.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"Only the abstract is available for this review (arXiv:2604.14600). The recommendation is therefore “uncertain” rather than a substantive accept/revise/reject decision. Once the full text is supplied, the load-bearing geometric hypotheses and the “certain condition” for rank detection should be checked first; if they are standard and the proofs are complete, the paper may well merit minor or major revision rather than rejection. No circularity or internal contradiction is visible from the abstract alone."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this is an abstract-only claim of a clean hierarchy of large-p asymptotic quantities built from p-capacity, the first p-eigenvalue and the Maz'ya constant: V(M) ≥ C(Ω) ≥ Λ(M) = M(M) ≥ 0. Under a centered-ball isoperimetric condition or rotational symmetry they equal volume entropy (or dimension); on Hadamard manifolds they also equal topological entropy of the geodesic flow, giving a hyperbolic characterization via entropy rigidity; a logarithmic second-order term is said to detect rank under an unspecified “certain condition.”\n\nWhat looks new is the systematic packaging and the hierarchy itself, plus the second-order rank claim. The ingredients are classical, but tying them together so that the same large-p limits recover both volume entropy and topological entropy, and then feeding that into entropy rigidity, is a useful organizational move inside geometric analysis. Circularity burden is low: the new objects are defined from analytic data and compared to independently defined geometric invariants. No internal contradiction is visible from the abstract, and the stress-test note is right that the only real soft spot is incompleteness of the available text, not a flaw in the announced logic.\n\nThe soft spots are exactly the ones the abstract flags: the geometric hypotheses (centered-ball isoperimetric, rotational symmetry, “certain condition”) are not spelled out, so one cannot yet see whether the equalities are forced or merely conditional, nor how sharp the second-order rank detection is. Without proofs, error estimates or precise statements, soundness cannot be verified. That is an abstract-only limitation, not evidence of a load-bearing gap.\n\nThis is for people who work on p-capacity asymptotics, volume entropy, and entropy rigidity on noncompact or Hadamard manifolds. A serious referee should see the full paper; the program is standard enough and the claims coherent enough that it deserves peer review rather than a desk reject. I would not cite it yet and would not bring the abstract alone to reading group, but I would look at the full text if it appears.","headline":"Abstract-only packaging of large-p limits of capacity/eigenvalue/Maz'ya into a hierarchy that recovers volume and topological entropy (and rank at second order) under standard geometric hypotheses; coherent but uncheckable without proofs.","tokens_in":2741,"tokens_out":538,"would_cite":false,"duration_ms":3988,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C20","58J50","53C23","37D40"],"pacs":[],"model":"grok-4.5","headline":"Large-p asymptotic capacity and eigenvalue limits recover volume entropy and characterize hyperbolic manifolds","keywords":["p-capacity","first p-eigenvalue","Maz'ya constant","volume entropy","topological entropy","Hadamard manifolds","entropy rigidity","rank"],"falsifier":"Exhibit a complete noncompact Riemannian manifold that satisfies neither the centered-ball isoperimetric condition nor rotational symmetry and for which the large-p capacitary or eigenvalue limit is strictly larger than the volume entropy, or a Hadamard manifold with compact quotient where the logarithmic second-order correction fails to equal the rank.","tokens_in":2878,"feed_emoji":"📐","tokens_out":645,"duration_ms":4399,"temperature":0.7,"pith_summary":"This paper introduces three large-p asymptotic geometric quantities on complete noncompact Riemannian manifolds, built from p-capacity, the first p-eigenvalue, and the Maz'ya constant. It proves they form a hierarchy V(M) ≥ C(Ω) ≥ Λ(M) = M(M) ≥ 0 and that, under a centered-ball isoperimetric condition or rotational symmetry, they all equal the volume entropy (or the dimension). On Hadamard manifolds of nonpositive curvature they further coincide with the topological entropy of the geodesic flow, so that entropy rigidity yields a characterization of hyperbolic manifolds. A second-order logarithmic correction is shown to detect the rank of a Hadamard manifold with compact quotient. The point is that purely analytic large-p limits of capacity and spectrum can replace classical volume-growth or dynamical entropy as geometric invariants.","feed_headline":"Large-p capacity limits recover volume entropy","feed_subtitle":"Analytic asymptotics of p-capacity and p-eigenvalues characterize hyperbolic manifolds and detect rank","key_machinery":"The large-p asymptotic limits of p-capacity, the first p-eigenvalue and the Maz'ya constant (denoted roughly V, C, Λ, M). These limits form a hierarchy that collapses to volume entropy under isoperimetric or symmetry hypotheses and to topological entropy on Hadamard manifolds.","core_discovery":"The large-p asymptotic quantities associated with p-capacity, the first p-eigenvalue and the Maz'ya constant satisfy the hierarchy V(M) ≥ C(Ω) ≥ Λ(M) = M(M) ≥ 0, and under a centered-ball isoperimetric condition or rotational symmetry they coincide with volume entropy (or dimension); on Hadamard manifolds they also equal topological entropy of the geodesic flow, giving a characterization of hyperbolic manifolds via entropy rigidity; a logarithmic second-order correction detects rank.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Large-p capacity limits recover volume entropy and rank","p-capacity asymptotics coincide with volume entropy","Analytic p-limits equal topological entropy on Hadamard manifolds","Large-p quantities detect hyperbolic manifolds via entropy rigidity","Logarithmic capacity correction detects manifold rank"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The identification of the new asymptotic quantities with volume entropy (and the second-order detection of rank) requires a centered-ball isoperimetric condition, rotational symmetry, or an unspecified certain condition on a Hadamard manifold with compact quotient.","fun_headline_variants_meta":{"raw":{"variants":["Large-p capacity limits recover volume entropy and rank","p-capacity asymptotics coincide with volume entropy","Analytic p-limits equal topological entropy on Hadamard manifolds","Large-p quantities detect hyperbolic manifolds via entropy rigidity","Logarithmic capacity correction detects manifold rank"]},"model":"grok-4.5","effort":"low","cost_usd":0.004072,"raw_usage":{"total_tokens":1240,"prompt_tokens":746,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":40720000,"prompt_tokens_details":{"text_tokens":746,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":418,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":746,"tokens_out":76,"duration_ms":3581,"temperature":1.0,"reasoning_tokens":418,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T20:09:14.346021+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a complete noncompact Riemannian manifold that satisfies neither the centered-ball isoperimetric condition nor rotational symmetry and for which the large-p capacitary or eigenvalue limit is strictly larger than the volume entropy, or a Hadamard manifold with compact quotient where the logarithmic second-order correction fails to equal the rank.","supporting_citations":[],"review_version":2}