{"id":"2b59ee68-c029-4e32-bcbe-cebc650dc7b1","arxiv_id":"2605.24705","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Explicit metrics on S^d and weighted R^d for d≥4 violate Milman's spectral comparison conjectures, ruling out 1-Lipschitz transport maps; the abstract's extra dimension-2 and hemisphere results are not in the text.","lead":"Using explicit perturbations of the round sphere and cylinder-ended weighted metrics, this paper constructs counterexamples to Milman's conjectured spectral comparisons and 1-Lipschitz transport maps in dimensions d≥4. The abstract also claims dimension-two and hemisphere results, but the full text contains no proofs or references for them.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract promises settlement in every dimension d≥2, but the submitted full text proves only the d≥4 counterexamples; the d=2 IMCF construction and d=3 [LWX26] counterexample are absent.","rationale":"The reader's weakest-assumption analysis identifies exactly the right vulnerability: the paper's headline depends on ingredients absent from the submitted full text. My independent pass over the d≥4 proofs found no clear mathematical error—the Ricci estimates, Rayleigh bounds, orthogonality, and min-max arguments in Theorems 1 and 2 are internally consistent, and the positive quantities δ_U and δ_b are correctly computed once the full warped-product Ricci formula and the 1/δ factor in a0' are accounted for. Thus the body's counterexample theorems appear sound. The central claim, however, is the complete settlement of the spherical spectral comparison for all d≥2, which also requires the dimension-two IMCF construction and the dimension-three counterexample from [LWX26]. Neither is present, and the references are absent. This is not a mere stylistic omission: without those sections, the strongest statement in the abstract is unsupported. An honest verdict is CONDITIONAL: accept only the d≥4 theorems unless the missing material is supplied. Since this matches the reader's assessment, no verdict change is warranted.","tokens_in":16722,"tokens_out":20781,"duration_ms":192868,"concrete_test":"Inspect the arXiv source package for any Section 5, appendix, or supplementary file containing the inverse mean curvature flow argument, the references [LWX26], [BJ21], [CM98], [FFGZ26], and the d≥5 Ric_g≥0 / ∇^2V≥g construction. If these are absent, ask the author to supply or locate them; unless the missing d=2 and d=3 arguments are provided and verified, the 'settles every dimension' headline cannot be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The submitted full text (Sections 1–4) establishes Theorem 1 and Theorem 2 for d≥4. I checked the main estimates—the Ricci lower bounds via Lemma 3.1, the Rayleigh-quotient expansion and orthogonality argument in Theorem 1, and the torpedo construction, CD(1,∞) verification, and test-function computations in Theorem 2—and found no fatal gap in these d≥4 arguments; the formulas for δ_U and δ_b check out after restoring the correct factors of δ. The load-bearing problem is the abstract's headline: 'settles the spherical spectral comparison in every dimension d≥2.' That assertion requires (i) a dimension-two contracting transport map constructed by inverse mean curvature flow, (ii) the dimension-three counterexample [LWX26], and (iii) the d≥5 refinement with Ric_g≥0 and ∇^2V≥g separately. None of these appears in the body; the bibliography omits [BJ21], [CM98], [LWX26], and [FFGZ26], and there is no Section 5. If those ingredients are absent or erroneous, the strongest claim—complete resolution in all dimensions and the affirmative hemisphere case—is not established, even though the d≥4 counterexamples may stand. The paper's actually supported contribution is therefore strictly weaker than the abstract claims.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs counterexamples to two spectral-comparison conjectures of Milman in the style of Caffarelli's contraction theorem. For every d≥4, Theorem 1 gives a smooth metric g on S^d with Ric_g ≥ ρg but λ_K(S^d,g) < λ_K(S^d,g^ρ_can), and Theorem 2 gives a CD(1,∞) weighted manifold (R^d,g,μ) with λ_{d+2}(R^d,g,μ)<2. The proofs use warped-product Ricci computations, explicit test functions, and min–max arguments. The abstract additionally claims a dimension-two contracting transport map constructed by inverse mean curvature flow, a dimension-three counterexample by Lin–Wang–Xu, and a resolution of the Beck–Jerison hemisphere conjecture; these ingredients do not appear in the submitted text (Sections 1–4 only).","tokens_in":17022,"tokens_out":4394,"duration_ms":40558,"significance":"If the d≥4 results are correct, they disprove Milman's Conjecture 3 and Conjecture 1* in general, and via the contraction principle also the corresponding transport-map conjectures. The proofs in Sections 3–4 are detailed and internally consistent: the Ricci lower bounds, Rayleigh-quotient expansions, and orthogonality arguments all check out. This is a substantial contribution. However, the paper's headline claim—that the spherical spectral comparison is settled in every dimension d≥2—is not supported by the submitted manuscript, which proves only the d≥4 counterexamples. The d=2 and d=3 ingredients are absent, so the strongest advertised conclusions are unverified.","major_comments":[{"comment":"The abstract states that the work 'settles the spherical spectral comparison in every dimension d≥2' and answers the Beck–Jerison conjecture. The submitted full text contains only Sections 1–4, proving Theorems 1 and 2 for d≥4. There is no Section 5, no inverse mean curvature flow argument for dimension two, and no mention or proof of the dimension-three counterexample [LWX26]. The bibliography omits [BJ21], [CM98], [LWX26], and [FFGZ26], all cited in the abstract. The 'settles every dimension' claim therefore collapses unless the missing material is supplied. This is a load-bearing overstatement; the manuscript's actually supported contribution is the d≥4 counterexamples.","section":"Abstract and Introduction"},{"comment":"The abstract further claims: 'In dimensions d≥5, the weighted counterexamples can be chosen to satisfy Ric_g≥0 and ∇_g^2 V ≥ g separately.' Theorem 2 in the body establishes only the CD(1,∞) condition, i.e. Ric_g + ∇^2 V ≥ g. No result in Sections 1–4 proves or even states the separate inequalities Ric_g≥0 and ∇^2 V ≥ g. If this refinement is true, it must be stated and proved; otherwise it should be removed from the abstract.","section":"Abstract and §4"},{"comment":"The inequality for Rε(h) on E_{k−1} in (3.5) is asserted with the factor sqrt(1+ε). It is correct: gε ≥ g0, so |∇h|²_{gε} ≤ |∇h|²_{g0}, and dvol_gε = sqrt(1+ε q²) dvol_g0 ≤ sqrt(1+ε) dvol_g0, while the denominator is ≥ ∫h² dvol_g0. The text would benefit from stating this justification; currently the reader must supply the metric comparison. This is a presentation issue rather than an error.","section":"§3.1 (Theorem 1, Step (iii))"},{"comment":"The proof of Theorem 2 uses a non-C¹ test function b_A and then treats its derivative in the weak sense. This is legitimate because b_A ∈ W^{1,2}(R^d,gε,νε), as noted in the text. However, the manuscript should explicitly indicate that the Rayleigh quotient and the Dirichlet-energy orthogonality are understood with weak derivatives; the current wording (e.g. 'all appearances of B'_ε = b'_A below are understood in the weak sense') is adequate but could be stated earlier and more prominently, since it is essential for the admissibility of Bε.","section":"§4 (Theorem 2)"}],"minor_comments":[{"comment":"The title on the first page contains typos: 'TRANSPOR T' and 'CUR VED SP ACES' should be 'TRANSPORT' and 'CURVED SPACES'.","section":"Title/header"},{"comment":"The references [BJ21], [CM98], [LWX26], and [FFGZ26] are cited in the abstract but are missing from the reference list. All cited works must appear in the bibliography.","section":"Bibliography"},{"comment":"In the definition of ε₀, the denominator is written without parentheses: 'ε_0 := 2k+d−2 / 1/2 (k−1)(k+d−2)+ 25/(12(d−1)) k(k+d−1)'. This should be formatted unambiguously, e.g. ε₀ = (2k+d−2) / [ (1/2)(k−1)(k+d−2) + (25/(12(d−1))) k(k+d−1) ].","section":"§3 (Theorem 1)"},{"comment":"The constants δ_U and δ_b are introduced and then used immediately; the reader must track that d≥4 is used to make them positive. The text does this, but the display for δ_U has several missing superscripts and spacing: 'δd−2', 'δd+1' etc. should be written as δ^{d−2}, δ^{d+1} for readability.","section":"§4 (Theorem 2)"},{"comment":"In Lemma 4.1 the proof that η(−s)=1−η(s) is correct, but the sentence 'Since ψ is even' should also note that this is what makes the integral of η over [−1,1] equal 1. This is a minor clarity point.","section":"§4 (Lemma 4.1)"}],"recommendation":"major_revision","confidential_remarks":"The d≥4 results are sound and valuable, but the manuscript as submitted is not the paper promised in the abstract. The missing d=2 and d=3 components are not optional embellishments; they constitute the 'settles every dimension' claim. I would suggest asking the author to either supply the missing Section 5 and references, or revise the abstract and introduction to state the d≥4 counterexamples accurately. If the missing sections are supplied, the paper could be acceptable; as it stands, the strongest claims are unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the actual mathematical content in the body—Theorems 1 and 2 for d≥4—is coherent and, as far as I checked, correct. The Ricci lower bounds via the warped-product formulas, the Rayleigh quotient estimates for the test functions, and the min-max argument all line up. The trick of using a slight S^1-direction distortion to produce a low eigenvalue below the rescaled round sphere is neat, and the torpedo metric with a cylindrical end plus Gaussian weight is a clean way to get a CD(1,∞) space with λ_{d+2}<2. I found no fatal gap.\n\nSecond, the abstract overshoots the body badly. It claims a dimension-two contraction via inverse mean curvature flow, a positive hemisphere result, a settlement of the spherical spectral comparison in all dimensions d≥2, and a d≥5 version with Ric_g≥0 and Hess V≥g separately. None of that appears in the submitted text. The bibliography is missing the four references the abstract relies on ([BJ21], [CM98], [LWX26], [FFGZ26]), and there is no Section 5. So the submission is incomplete with respect to its own advertised contribution.\n\nThe soft spot is proportionate: the d≥4 results themselves stand, and they already refute Milman's conjectures in general—you don't need d=2 or d=3 for that. But the abstract's 'settles every dimension' and the hemisphere claim are load-bearing for the paper's overall narrative, and they are unsupported here. This is not a minor typo; the reader has to either see the missing arguments or have the claims withdrawn.\n\nWho's this for? Anyone working on spectral comparison, Caffarelli's contraction theorem, or optimal transport on curved spaces. The d≥4 counterexamples are a real contribution and deserve citation. But I'd advise the author to split or revise: either produce the missing sections or restrict the abstract to what's proved.\n\nRecommendation: send it to peer review. A serious referee should spend time on Sections 3 and 4; the burden should be on the author to reconcile the abstract with the body. If the missing parts fail to materialize, the paper can be accepted after the claims are trimmed to the d≥4 theorems.","headline":"The d≥4 counterexamples are solid and worth taking seriously, but the abstract promises results the body doesn't contain.","tokens_in":17519,"tokens_out":6647,"would_cite":true,"duration_ms":64162,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","58J50","35P15","53C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"In dimensions four and higher, smooth metrics on spheres and Gaussian-weighted spaces can have Laplace spectra below those of the round sphere and Gaussian model, which rules out any 1-Lipschitz optimal transport map between them.","keywords":["optimal transport","Laplace eigenvalues","spectral comparison","curvature-dimension condition","Ricci curvature","warped product metrics","contracting transport maps","min-max principle"],"falsifier":"For d=4, directly compute the Rayleigh quotient R_{g_ε}(f_k) from the explicit formula and check whether it stays below (ρ_ε/(d−1))k(k+d−1) for some k≥k_0; if not, Theorem 1's spectral drop fails. For Theorem 2, evaluate the Dirichlet-to-mass ratios of the d+2 test functions U_{ε,i} and B_ε on the explicit torpedo metric; if any ratio reaches 2, the inequality λ_{d+2}<2 fails. Both are finite calculations using the formulas in Sections 3 and 4.","tokens_in":16578,"feed_emoji":"🌐","tokens_out":11425,"duration_ms":111294,"temperature":0.7,"pith_summary":"The paper aims to show that a classical Gaussian contraction theorem does not extend to curved spaces. It constructs, in every dimension d≥4, a smooth metric on the sphere S^d and a weighted Euclidean metric on R^d such that the Laplace eigenvalues of the target are strictly below those of the corresponding model space (the round sphere with the same Ricci lower bound, and the standard Gaussian with λ_{d+2}=2). Because a 1-Lipschitz transport map would force the target spectrum to dominate the source spectrum, these spectral drops obstruct the existence of contracting transport maps, thereby falsifying a family of conjectures that generalize the Gaussian result. The body proves the d≥4 claims with two explicit constructions; the abstract additionally announces a dimension-two positive result via inverse mean curvature flow and a dimension-three counterexample from the literature, but those ingredients are not present in the included sections.","feed_headline":"Dimensions 4 and up break spherical contraction bounds","feed_subtitle":"Small metric wrinkles lower Laplace eigenvalues below the round sphere's, so no 1-Lipschitz transport map can exist.","key_machinery":"The engine is the product structure of the model spaces. The sphere is presented as a warped product over an S¹ fiber, and Euclidean space as a rotationally symmetric warped product over S^{d−1}; both fibers carry many low-lying modes. The paper perturbs the fiber metric — enlarging the S¹ factor on the sphere, and making the sphere-slices of the Euclidean metric cylindrical with radius √(d−2) — and uses the Ricci formula for multiply warped products to control the lower curvature bound. The 'Contraction Principle' (an L-Lipschitz map forces λ_k(target) ≥ L^{−2} λ_k(source)) is the named theorem that converts a spectral drop into a transport obstruction. In the weighted case, a torpedo radiu","core_discovery":"At the core is an explicit spectral computation. On S^d, the warped metric g_ε = dt² + sin²t(1+ε sin⁴t)dθ² + cos²t g_{S^{d−2}} has Ricci curvature at least ρ_ε = d−1−25ε/12; test functions f_k = Re(z^k) give Rayleigh quotients below the round sphere's eigenvalue ρ_ε/(d−1)·k(k+d−1). On R^d, a rotationally symmetric 'torpedo' metric with cross-section radius √(d−2) and a Gaussian weight on the cylinder satisfies CD(1,∞), yet d+2 test functions have Rayleigh quotient below 2, forcing λ_{d+2}<2 while the Gaussian model has λ_{d+2}=2. The Contraction Principle turns each spectral drop into an obstruction to 1-Lipschitz transport maps.","pith_inferences":["The same warped-product mechanism should transfer to other symmetric model spaces, such as complex projective space or hyperbolic space: a small fiber perturbation should create low eigenvalue clusters below the symmetric-space threshold.","The weighted construction verifies the stronger conditions Ric_g≥0 and ∇²V≥g only for d≥5; whether d=4 can also satisfy both simultaneously is an open seam exposed by this paper.","Numerically evaluating the explicit Rayleigh quotients for d=4 would yield how small ε must be for the K-th eigenvalue to drop below the round value, turning the existence argument into a quantitative threshold.","If the abstract's dimension-two inverse mean curvature flow construction and dimension-three literature counterexample are later supplied in full, the 'every dimension' conclusion would connect these d≥4 results to a complete picture; the d≥4 counterexamples are independent of those additions."],"forward_implications":["The spherical spectral comparison conjectures are false in every dimension d≥4: the constructed metric has a strictly smaller Laplace eigenvalue than the round sphere with the same Ricci lower bound.","The Gaussian spectral comparison on weighted spaces is false in every dimension d≥4: the constructed CD(1,∞) space has λ_{d+2}<2, whereas the standard Gaussian has λ_{d+2}=2.","Consequently, no 1-Lipschitz map pushing the round-sphere volume or the standard Gaussian forward onto these targets can exist; the Contraction Principle would force the opposite eigenvalue inequality.","The product-structure mechanism (many small eigenvalues arising from fiber coordinates) is a general source of spectral obstructions, not a peculiarity of the Clifford torus."],"fun_headline_variants":["Spectral drops block contraction on spheres in d≥4","Curved-space transport maps can't always be 1-Lipschitz","High-dim spheres break contraction, dimension 2 holds","Spherical spectral comparison: counterexamples in d≥4"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The counterexamples stand or fall on the Ricci-curvature lower bound of the constructed warped metrics (and the curvature-dimension condition in the weighted case); if either curvature computation is off, the eigenvalue comparison against the round sphere or Gaussian model has no valid target.","fun_headline_variants_meta":{"raw":{"variants":["Spectral drops block contraction on spheres in d≥4","Curved-space transport maps can't always be 1-Lipschitz","High-dim spheres break contraction, dimension 2 holds","Spherical spectral comparison: counterexamples in d≥4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1201,"prompt_tokens":939,"completion_tokens":262,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":683,"completion_tokens_details":{"reasoning_tokens":192}},"tokens_in":683,"tokens_out":262,"duration_ms":3712,"temperature":1.0,"reasoning_tokens":192,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T04:59:39.556691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For d=4, directly compute the Rayleigh quotient R_{g_ε}(f_k) from the explicit formula and check whether it stays below (ρ_ε/(d−1))k(k+d−1) for some k≥k_0; if not, Theorem 1's spectral drop fails. For Theorem 2, evaluate the Dirichlet-to-mass ratios of the d+2 test functions U_{ε,i} and B_ε on the explicit torpedo metric; if any ratio reaches 2, the inequality λ_{d+2}<2 fails. Both are finite calculations using the formulas in Sections 3 and 4.","supporting_citations":[],"review_version":2}