{"id":"1d505e62-58d8-49b0-bacd-e6ed4636b26c","arxiv_id":"2607.28328","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For origin-symmetric Gaussian domains, the harmonic mean of the first N Neumann Ornstein–Uhlenbeck eigenvalues is maximized uniquely by the centered ball of equal Gaussian measure.","lead":"Among origin-symmetric domains of fixed Gaussian measure, the sum of the reciprocals of the first N Neumann eigenvalues of the Ornstein–Uhlenbeck operator is minimized by the centered ball. The result closes the Gaussian version of a long-studied harmonic-mean spectral inequality.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly identifies origin symmetry as the essential structural hypothesis rather than a flaw, and correctly assesses the proof as complete within the stated class A_N. My second pass finds no additional load-bearing vulnerability: the Gaussian weight prevents translation recentering, the raywise rearrangement produces exactly the same trace-free defect controlled by Lemma 3.2, and the equality-case analysis (5.16)–(5.21) closes rigorously for Lipschitz domains. The appendix on boxes further corroborates that centring is spectrally active. Consequently the ACCEPT verdict stands without adjustment.","tokens_in":14030,"tokens_out":441,"duration_ms":8352,"concrete_test":"Independently verify the energy identity (2.15) from the weak form of (2.2) with the boundary conditions (2.3)/(2.5), then recompute the scalar coefficients a,c,d appearing in (5.13) and (5.15); confirm that they satisfy the hypotheses of Lemma 3.2 with λ=μ_1(B_R) and that equality forces Z=0 and Y(θ)=Y_R a.e.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.1) is supported by a complete, self-contained argument. The three ingredients—radial monotonicity of g (Lemma 2.1 and (2.12)), the trace-free matrix convexity lemma (Lemma 3.2), and the weighted raywise bathtub/tangent estimates (Lemmas 4.1–4.2)—combine cleanly in §5. Origin symmetry is used only for simultaneous mean-zero of the trial fields P_i (5.3) and is stated openly; the equality case is upgraded from almost-everywhere radial sections to open-set identity via the Lipschitz exterior-cone property. No hidden gap, circularity, or unsupported estimate appears in the chain from the Ritz trace bound (3.2) through the matrix sandwich (5.13)–(5.15) to the final comparison.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves a sharp N-term reciprocal-sum inequality for the first N nontrivial Neumann eigenvalues of the Ornstein–Uhlenbeck operator on origin-symmetric domains Ω in the Gaussian class A_N: ∑_{k=1}^N 1/μ_k(Ω) ≥ N/μ_1(B_R), with equality if and only if Ω equals the centred ball B_R of equal Gaussian measure. The argument transplants the radial first eigenfunctions of the ball via a C^1 extension G, forms the coupled mass and stiffness matrices of the N trial fields P_i, obtains a trace-free angular defect matrix Z from the raywise Gaussian volumes, and controls tr(K^{-1}M) by a finite-dimensional convexity lemma. Radial monotonicity of g, weighted bathtub/tangent estimates, and an equality-case upgrade from a.e. sections to open-set identity via the Lipschitz exterior-cone condition complete the proof. An appendix treats rectangular boxes by tensorization.","tokens_in":14197,"tokens_out":642,"duration_ms":12451,"significance":"The result closes the Gaussian counterpart of the Ashbaugh–Benguria reciprocal-sum conjecture, after the (N−1)-term estimate of Gao–Wang and the recent Euclidean and space-form resolutions by He–Li–Tang and You–Zhang. Origin symmetry is a genuine structural hypothesis (the weight is not translation-invariant), and the paper makes this explicit while still recovering the full N-term bound and the Gaussian Szegő–Weinberger inequality as a corollary. The proof is self-contained, with clean equality analysis and an informative appendix on centring for boxes. If correct, it is a natural and substantial contribution to spectral shape optimisation under Gaussian measure.","major_comments":[],"minor_comments":[{"comment":"In the introduction, the phrase “the missing last reciprocal term was recovered only recently by He, Li and Tang [12]” could briefly note that their arXiv identifier appears with a 2026 date, consistent with the other recent citations, so that the chronological claim is unambiguous for readers.","section":"§1"},{"comment":"Display (5.2) for |∇P_i|^2 is standard but could be given a one-line derivation (or a reference to the analogous Euclidean identity) for readers less familiar with the Weinberger trial fields.","section":"§5"},{"comment":"The appendix is valuable; a single sentence in Remark 6.1 cross-referencing Proposition A.1 would make the motivational role of centring even clearer.","section":"Appendix A / Remark 6.1"},{"comment":"Typographical consistency: “Ornstein–Uhlenbeck” versus “Ornstein--Uhlenbeck” and occasional missing spaces before citations could be standardised in copy-editing.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is ready for acceptance. The reader’s and skeptic’s assessments align with a direct reading of the proof chain; no load-bearing gap appears. Fit for a strong analysis/spectral-theory journal is clear."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This closes the full N-term Gaussian reciprocal-sum inequality for Neumann Ornstein–Uhlenbeck eigenvalues under the natural origin-symmetry hypothesis. Gao–Wang already had the (N−1)-term version; the Euclidean and space-form N-term proofs used the same coupled-matrix convexity idea. What is new is the genuine adaptation to the drifted operator with finite-mass radial Gaussian weight, where you cannot translate the Weinberger centre.\n\nThe argument is carefully written and complete. Radial monotonicity of the ball eigenfunction (Lemma 2.1), the trace Ritz bound, the trace-free matrix convexity lemma with equality case, and the weighted raywise bathtub/tangent estimates all line up cleanly in §5. Equality is upgraded from a.e. radial sections to open-set identity via the Lipschitz exterior-cone property. The appendix on boxes is useful: it shows why centring is spectrally active rather than a harmless normalisation, and it motivates the symmetry hypothesis without overclaiming.\n\nSoft spots are minor and openly stated. Origin symmetry is essential so that the N trial fields are simultaneously mean-zero; without it the coupled Ritz matrix does not start, and the Gaussian weight is not translation-invariant. Compactness of the Gaussian Sobolev embedding is assumed (with a sufficient extension-operator condition), which is standard for unbounded domains in this setting. No free parameters, no circularity, and the self-citations supply background lemmas rather than restating the target. Citations sit cleanly in the Ashbaugh–Benguria / Gao–Wang / He–Li–Tang / You–Zhang line.\n\nThis is for people working on spectral shape optimisation and Gaussian isoperimetry. A serious referee should see it. I would accept it for peer review and expect it to hold up.","headline":"Solid completion of the Gaussian N-term harmonic-mean inequality under origin symmetry; the adaptation is real and the proof chain is clean.","tokens_in":14841,"tokens_out":442,"would_cite":true,"duration_ms":8240,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","35J70","49R05"],"pacs":[],"model":"grok-4.5","headline":"Among origin-symmetric Gaussian domains of fixed measure, the centred ball uniquely minimises the harmonic mean of the first N Neumann eigenvalues of the Ornstein–Uhlenbeck operator.","keywords":["Ornstein–Uhlenbeck operator","Neumann eigenvalues","Gaussian measure","harmonic mean","isoperimetric inequality","raywise rearrangement","Hermite operator","Szegő–Weinberger"],"falsifier":"Exhibit an origin-symmetric admissible domain Ω with γ_N(Ω)=γ_N(B_R) for which the sum of the first N reciprocal Neumann eigenvalues is strictly smaller than N/μ_1(B_R), or find equality for some Ω that is not the ball.","tokens_in":14889,"feed_emoji":"⚪","tokens_out":963,"duration_ms":17905,"temperature":0.7,"pith_summary":"This paper settles a sharp reciprocal-sum inequality for the first N positive Neumann eigenvalues of the Ornstein–Uhlenbeck operator on origin-symmetric domains in Gaussian space. For any such domain with the same Gaussian measure as a centred Euclidean ball, the sum of the reciprocals of those eigenvalues is at least as large as N times the reciprocal of the ball’s first positive eigenvalue, with equality only for the ball itself. The result completes the Gaussian counterpart of the Ashbaugh–Benguria conjecture that was recently proved in Euclidean space and in space forms: earlier Gaussian work had the (N−1)-term version, and this paper recovers the missing last term. A sympathetic reader cares because the Gaussian setting behaves differently from classical isoperimetry—half-spaces do not maximise the first eigenvalue—so a clean, equality-characterised harmonic-mean bound for the whole first eigenspace is a genuine structural statement about spectral shape under Gaussian measure.","feed_headline":"Centred ball wins the Gaussian harmonic-mean race","feed_subtitle":"Origin-symmetric domains cannot beat the ball on the sum of the first N reciprocal Neumann eigenvalues","key_machinery":"A coupled N-dimensional Ritz (trace) argument on the transplanted trial functions P_i(x)=G(|x|) x_i/|x|, combined with Gaussian raywise rearrangement: angular imbalance appears as a symmetric trace-free matrix Z whose contribution is killed by a finite-dimensional convexity inequality on the eigenvalues of the stiffness and mass matrices.","core_discovery":"For N≥2 and an origin-symmetric connected Lipschitz domain Ω in the admissible class A_N (compact Gaussian Sobolev embedding, 0<γ_N(Ω)<1), if B_R is the origin-centred ball with the same Gaussian measure, then the sum from k=1 to N of 1/μ_k(Ω) is at least N/μ_1(B_R), with equality if and only if Ω equals B_R.","pith_inferences":["Removing origin symmetry remains open: the paper’s own remarks suggest half-spaces are not the maximisers, so the unrestricted Gaussian optimiser for the reciprocal sum is still unknown.","The same coupled-matrix-plus-raywise-rearrangement pattern may extend to other radial log-concave weights once a simultaneous-admissibility condition replaces translation.","Tensorisation on product domains could yield sharp harmonic-mean bounds in mixed Gaussian–Euclidean cylinders without full radial symmetry."],"forward_implications":["The inequality immediately implies the Gaussian Szegő–Weinberger bound μ_1(Ω)≤μ_1(B_R) for origin-symmetric domains, with equality only for the ball.","Equality characterisation forces any origin-symmetric equality case to be exactly the centred ball, not merely equimeasurable up to null sets.","Among rectangular boxes of fixed Gaussian measure, the centred cube uniquely maximises μ_1, confirming that centring is spectrally active rather than a free normalisation.","The full N-term Gaussian reciprocal-sum problem is closed in the origin-symmetric class, matching the recently settled Euclidean and space-form cases."],"fun_headline_variants":["Centred ball uniquely maximises harmonic mean of Neumann eigenvalues","Only the ball hits the sharp Gaussian bound on reciprocal Neumann sums","Origin-symmetric domains lose to the ball on first-N eigenvalue harmonic mean","Ball minimises sum of first N reciprocal Neumann eigenvalues in Gaussian space","Sharp inequality pins centred ball as sole Gaussian harmonic-mean champion"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The domain must be symmetric through the origin; without that symmetry the N trial functions are not all orthogonal to constants at once, and the coupled matrix argument does not start.","fun_headline_variants_meta":{"raw":{"variants":["Centred ball uniquely maximises harmonic mean of Neumann eigenvalues","Only the ball hits the sharp Gaussian bound on reciprocal Neumann sums","Origin-symmetric domains lose to the ball on first-N eigenvalue harmonic mean","Ball minimises sum of first N reciprocal Neumann eigenvalues in Gaussian space","Sharp inequality pins centred ball as sole Gaussian harmonic-mean champion"]},"model":"grok-4.5","effort":"low","cost_usd":0.004974,"raw_usage":{"total_tokens":1422,"prompt_tokens":835,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":49744000,"prompt_tokens_details":{"text_tokens":835,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":511,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":835,"tokens_out":76,"duration_ms":9634,"temperature":1.0,"reasoning_tokens":511,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T11:10:05.080022+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit an origin-symmetric admissible domain Ω with γ_N(Ω)=γ_N(B_R) for which the sum of the first N reciprocal Neumann eigenvalues is strictly smaller than N/μ_1(B_R), or find equality for some Ω that is not the ball.","supporting_citations":[],"review_version":1}