{"id":"4d3c8383-e3a2-4f3a-a632-bd1f30e3b8d2","arxiv_id":"2510.26755","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Optimal-exponent stability estimates with universal dimensional constants for the Bahn–Ehrlich and Cavalletti–Mondino Lorentzian isoperimetric inequalities.","lead":"This paper proves sharp quantitative stability versions of Lorentzian isoperimetric inequalities: if a hypersurface almost maximizes enclosed volume for its area, it must be close to a hyperboloid. The estimates carry optimal exponents and explicit dimension-dependent constants, giving tools for mathematical relativity and cosmology.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract promises a Hausdorff stability estimate that is not stated or proved anywhere in the body; the advertised scope is unverified.","rationale":"The central stability theorem and corollaries are supported by detailed proofs and appear mathematically sound; the CM corollaries do not actually import [CM25] as a black box because Proposition 1.2 and Theorem 3.3 reprove the needed special case. The load-bearing gap is the disconnect between the abstract's promise of a Hausdorff stability estimate and the body, which contains no such statement. This warrants a conditional verdict: the advertised claim must either be proven in the paper or removed from the abstract. The reader's CONDITIONAL verdict is right, but the identified weakest assumption should shift from the CM dependency to the missing Hausdorff section.","tokens_in":21762,"tokens_out":30592,"duration_ms":231378,"concrete_test":"Search the manuscript for the terms 'Hausdorff' and 'Bahn–Ehrlich distance'; if no theorem or definition appears after §5, request the missing section or an amended abstract. If the authors supply the missing statement and proof, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's final sentences advertise a main result: 'in a fixed conical Minkowski spacetime, we use a Lipschitz bound ... to upgrade our quantitative control to a Hausdorff stability estimate ... formulated in terms of a distance defined by Bahn and Ehrlich.' The full text contains no such theorem, no definition of the Bahn–Ehrlich distance, and no Hausdorff-type metric: after §5 the paper proceeds to appendices on complementary stability results. This is an omitted advertised result, not a stylistic gap, so the paper's stated scope is not established. The reader's stated weakest assumption about dependence on [CM25] is less serious than it first appears: Proposition 1.2 derives δ_CM from δ_BE and E using only definitions and Bernoulli's inequality, and Theorem 3.3 proves δ_BE≥0, so Corollaries 1.3–1.4 are self-contained for the stated class of achronal Lipschitz hypersurfaces. A minor constant-factor check in the §4.5 application of Lemma 4.2 is worthwhile but does not threaten the exponent-2 conclusion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves quantitative stability versions of Lorentzian isoperimetric inequalities for achronal Lipschitz hypersurfaces in conical Minkowski spacetimes. Theorem 1.1 gives a quadratic bound on the Fraenkel asymmetry in terms of the Bahn–Ehrlich deficit, A_F(S)^2 ≤ 16(n+1)^2/n δ_BE(S), with exponent 2 optimal. Proposition 1.2 relates the BE deficit to the Cavalletti–Mondino deficit and a relative volume excess, leading to a linear stability estimate for the CM inequality (Corollary 1.3), and a quadratic stability estimate for a refined CM deficit (Corollary 1.4). The paper also gives self-contained proofs of the underlying BE inequality via Hölder/Bernoulli and a geometric simplex argument, a quantitative Minkowski inequality (Lemma 4.2), and sharpness examples (Section 5). Appendices provide complementary stability results and an improved constant for the refined CM inequality.","tokens_in":22008,"tokens_out":32625,"duration_ms":231727,"significance":"If the identified issues are fixed, this is a valuable contribution: it introduces quantitative Lorentzian isoperimetric stability with explicit dimension-dependent constants, a quadratic exponent matching the Euclidean case for the BE inequality, and a linear exponent for the special CM inequality with a quadratic refinement. The proofs are elementary and largely self-contained, and the paper provides machine-checkable-style derivations from first principles. The sharpness examples and the improved constant in Appendix B are useful additions. However, the advertised Hausdorff stability estimate is missing, and the constant in Theorem 1.1 is not established as written; these must be addressed before the paper can be considered complete.","major_comments":[{"comment":"The abstract's final sentences advertise a main result: a Lipschitz bound upgrades the quantitative control to a Hausdorff stability estimate formulated in terms of a distance defined by Bahn and Ehrlich. No such theorem, definition, or proof appears in the body; §1.2 (structure) also omits it. This is not a stylistic gap but an omitted advertised result. The authors should either add the statement and proof (including the definition of the Bahn–Ehrlich distance) or remove the claim from the abstract.","section":"Abstract / §1.2"},{"comment":"The displayed inequality after 'Applying Lemma 4.2 to the second inequality' omits a factor 2^{-1/(n+1)}. With a=V(C(B1)), b=V(C(B2)), Lemma 4.2 multiplied by (n+1)(σ/2)^{1/(n+1)} gives (n+1)σ^{1/(n+1)}2^{-1/(n+1)} n/(4(n+1)^2) max{a,b}^{-(n+2)/(n+1)} |a-b|^2, not σ^{1/(n+1)} n/(4(n+1)) V^{-(n+2)/(n+1)} |a-b|^2. Consequently the proof yields A_F^2 ≤ 16·2^{1/(n+1)}(n+1)^2/n δ_BE, not the stated constant in Theorem 1.1. The quadratic exponent and the strategy are intact; please correct the constant in the theorem or the proof.","section":"§4.5, application of Lemma 4.2"}],"minor_comments":[{"comment":"The definition of Ω2 should read {f>t0} ∪ E2; the text currently has {f<t0} ∪ E2 twice, which is inconsistent with the subsequent disjointness and measure statements.","section":"§4.5"},{"comment":"The equality V(C(S)ΔB_t0(M)) = V(C(B1)) - V(C(B2)) has the wrong sign; it should be |V(C(B1)) - V(C(B2))| (or V(C(B2)) - V(C(B1))). Since the quantity is squared, the argument is unaffected.","section":"§4.5"},{"comment":"The asymptotic expansions for δ_CM and δ*_CM omit the (inf φ)^2 term that arises from expanding 1/dist(O,S_ε). Moreover, if the second-order term of E is kept, the coefficient of ∫φ^2 in δ*_CM is n/2 rather than n as displayed. The boundedness of the quotients in (19) is unaffected, but the formulas should be corrected.","section":"§5"},{"comment":"The notation B_t(M) is used in the introduction's definition (2) but defined only later in §2.2. A forward pointer or a brief definition at first use would improve readability.","section":"§2.2 / introduction"}],"recommendation":"major_revision","confidential_remarks":"The main reason for major revision is the missing Hausdorff stability estimate advertised in the abstract; the paper's actual content does not contain it. The constant-factor gap in §4.5 is easily fixable, but the abstract must be reconciled with the body. The remaining mathematical core is sound and novel."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this for the BE/CM stability results, not for the Hausdorff estimate the abstract advertises. The main theorems are new: quadratic stability for the Bahn–Ehrlich inequality with optimal exponent, linear stability for the Cavalletti–Mondino special case, and a refined CM deficit that restores quadratic behavior. The constants are explicit and dimension-only, and the Section 5 examples show the exponents are sharp. That part is solid.\n\nThe paper also reproves the BE inequality from first principles (Hölder/Bernoulli and a geometric simplex argument), so the results are self-contained for the stated class of achronal Lipschitz hypersurfaces. The reader's worry about reliance on [CM25] is overblown: Proposition 1.2 derives δ_CM from δ_BE and the relative volume excess using Bernoulli, and Theorem 3.3 proves δ_BE≥0, so Corollaries 1.3–1.4 do not import an unproved inequality. The compact reduction and level-set decomposition are clean.\n\nSoft spots. The arXiv abstract promises a Hausdorff stability estimate 'formulated in terms of a distance defined by Bahn and Ehrlich' and a Hausdorff-type metric on Cauchy hypersurfaces. The full text has no such theorem, no definition of that distance, no such metric. The body's abstract stops at the self-contained proofs. That scope mismatch has to be fixed—either add the result or amend the abstract. In §4.5, the application of Lemma 4.2 appears to drop a factor 2^{-1/(n+1)} in the lower bound; it doesn't affect the exponent, but the explicit constant 16(n+1)^2/n may be too small by that factor. Worth a careful check. Minor: the definitions of the deficits and asymmetry for the generalized hypersurfaces are clear but scattered; a consolidated statement would help.\n\nThis paper is for people in quantitative geometric inequalities, Lorentzian geometry, and mathematical relativity who need ready-to-use area bounds. It deserves a serious referee. I'd send it to review, with a request to fix the abstract/body gap and re-check the constant.","headline":"Solid quantitative stability results, but the arXiv abstract promises a Hausdorff estimate the paper does not deliver; fix that and the paper is referee-ready.","tokens_in":22463,"tokens_out":26348,"would_cite":true,"duration_ms":170107,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves sharp quantitative stability for Lorentzian isoperimetric inequalities in conical Minkowski spacetimes: the Fraenkel asymmetry is controlled quadratically by the Bahn–Ehrlich deficit, linearly by the Cavalletti–Mondino def","keywords":["quantitative stability","Lorentzian isoperimetric inequality","Fraenkel asymmetry","conical Minkowski spacetime","achronal hypersurface","reverse isoperimetric inequality","hyperboloid rigidity","Hausdorff stability"],"falsifier":"Fix a conical Minkowski spacetime, choose a smooth mean-zero φ on a ball in H, and consider S_ε = graph(1 + εφ) as in the sharpness section. Compute lim_{ε→0} δ_BE(S_ε)/A_F(S_ε)^2. If this ratio diverges to +∞ for any φ, the quadratic bound with a finite dimension-only constant is false; if it tends to 0, the claimed optimality of the exponent 2 fails. The paper asserts the ratio stays bounded for all such φ and is positive for suitable φ, so a single explicit computation to the contrary would refute Theorem 1.1.","tokens_in":21652,"feed_emoji":"📐","tokens_out":8340,"duration_ms":84170,"temperature":0.7,"pith_summary":"The paper establishes optimal stability estimates for reverse isoperimetric inequalities in Lorentzian geometry. In a conical Minkowski spacetime, every achronal Lipschitz hypersurface with finite cone volume satisfies a quadratic bound on its Fraenkel asymmetry in terms of the Bahn–Ehrlich isoperimetric deficit, and the exponent 2 is shown to be optimal. The related Cavalletti–Mondino inequality has a stronger linear stability behavior, also optimal, but subtracting a relative volume excess from its deficit restores the familiar quadratic stability. The proofs are self-contained for the Bahn–Ehrlich inequality and reduce the quantitative question to a sharpened convexity estimate of Minkowski type. If correct, these results say that nearly optimal achronal hypersurfaces are nearly hyperboloids, with a rate that cannot be improved.","feed_headline":"Lorentzian isoperimetric inequality is quadratically stable","feed_subtitle":"In conical Minkowski spacetimes, squared Fraenkel asymmetry controls the isoperimetric deficit with sharp exponents.","key_machinery":"The key mechanism is the radial graph representation of an achronal Lipschitz hypersurface: S = S_f = {f(x)x : x ∈ π(S)} for some domain in the unit hyperboloid H, with ln f being 1-Lipschitz with respect to the hyperbolic metric. This converts cone volume and area into explicit integrals, V(C(S)) = (1/(n+1))∫ f^{n+1} dμ and A(S) = ∫ f^n √(1−|∇ln f|^2) dμ. The stability proof decomposes the domain into sub- and superlevel sets of f, applies a quantitative Minkowski-type convexity inequality (Lemma 4.2) to the two pieces, and uses a comparison lemma showing the geometric Fraenkel asymmetry is equivalent, up to a factor 2, to an L1-distance between f^{n+1} and a constant. A compact exhaustion","core_discovery":"The central discovery is that the Lorentzian isoperimetric inequality bounding the area of an achronal hypersurface by the volume of its past cone admits a sharp quantitative form: for every achronal Lipschitz hypersurface S in a conical Minkowski spacetime, A_F(S)^2 ≤ 16(n+1)^2/n · δ_BE(S), and the exponent 2 is optimal. When the analogous deficit is measured as δ_CM(S) = (n+1)V(C(S))/(A(S)·dist(O,S)) − 1, the stability becomes linear, A_F(S) ≤ 2(n+1)δ_CM(S), also with optimal exponent. Defining a refined deficit δ*_CM(S) = δ_CM(S) − E(S), where E(S) is the relative volume excess between the cone over S and the past hyperboloid at distance dist(O,S), recovers quadratic stability with the sa","pith_inferences":["Because the stability constants are cone-independent, a natural conjecture is that analogous quantitative control holds for spacelike Cauchy graphs in more general warped Robertson–Walker spacetimes where the radial graph representation and a reverse triangle inequality remain available.","The transition from linear to quadratic stability when subtracting E(S) suggests that the volume-excess term acts as a symmetry-breaking correction; one could test whether the stability exponent is governed by how the distance-to-boundary enters the volume normalization.","A direct extension would be to study the family of deficits δ_α = δ_CM − αE(S) and identify the value of α that optimizes the stability exponent or the constant; δ*_CM corresponds to α = 1.","The sharpness construction with perturbations r_ε = 1 + εφ shows δ_CM ~ ε, δ*_CM ~ ε^2, and A_F ~ ε; computing these asymptotics numerically for several explicit φ would provide a concrete check of the claimed exponents."],"forward_implications":["Every achronal Lipschitz hypersurface with finite cone volume and finite projected measure satisfies A_F(S)^2 ≤ 16(n+1)^2/n · δ_BE(S); in particular, the estimate covers Cauchy hypersurfaces in any conical Minkowski spacetime.","The stability constant depends only on the dimension, not on the shape of the cone, in contrast to the Euclidean relative isoperimetric stability inside convex cones.","For the Cavalletti–Mondino deficit the linear bound A_F(S) ≤ 2(n+1)δ_CM(S) is sharp, so near-optimal hypersurfaces are forced toward hyperboloids at a faster rate than in the Bahn–Ehrlich case.","Subtracting the relative volume excess gives a refined deficit δ*_CM = δ_CM − E(S) with quadratic stability; the refinement lies between the Bahn–Ehrlich and Cavalletti–Mondino deficits and inherits sharpness and rigidity.","As noted in the paper, the stability estimates yield improved upper area bounds for acausal hypersurfaces in cosmological and black-hole-type settings covered by the Cavalletti–Mondino inequality.","The abstract also advertises a Hausdorff-type stability estimate for Cauchy hypersurfaces, obtained by upgrading the quantitative control using a Lipschitz bound supplied by the causal structure."],"fun_headline_variants":["Optimal stability for Lorentzian isoperimetric inequality","Sharp quadratic stability in conical Minkowski spacetimes","Linear to quadratic: Lorentzian isoperimetric stability refined","Lorentzian isoperimetry: optimal quadratic stability","Quadratic stability sharpened for Lorentzian isoperimetry"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The linear and refined quadratic stability results for the Cavalletti–Mondino inequality rely on the imported isoperimetric inequality δ_CM(S) = (n+1)V(C(S))/(A(S)·dist(O,S)) − 1 ≥ 0 for every achronal Lipschitz hypersurface; the paper does not reprove this, so if its hypotheses are more restrictive than stated, the advertised stability bounds would hold only on a narrower class of hypersurfaces.","fun_headline_variants_meta":{"raw":{"variants":["Optimal stability for Lorentzian isoperimetric inequality","Sharp quadratic stability in conical Minkowski spacetimes","Linear to quadratic: Lorentzian isoperimetric stability refined","Lorentzian isoperimetry: optimal quadratic stability","Quadratic stability sharpened for Lorentzian isoperimetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000705,"raw_usage":{"total_tokens":3033,"prompt_tokens":781,"completion_tokens":2252,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":2172}},"tokens_in":525,"tokens_out":2252,"duration_ms":15640,"temperature":1.0,"reasoning_tokens":2172,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:04:24.031047+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a conical Minkowski spacetime, choose a smooth mean-zero φ on a ball in H, and consider S_ε = graph(1 + εφ) as in the sharpness section. Compute lim_{ε→0} δ_BE(S_ε)/A_F(S_ε)^2. If this ratio diverges to +∞ for any φ, the quadratic bound with a finite dimension-only constant is false; if it tends to 0, the claimed optimality of the exponent 2 fails. The paper asserts the ratio stays bounded for all such φ and is positive for suitable φ, so a single explicit computation to the contrary would refute Theorem 1.1.","supporting_citations":[],"review_version":2}