{"id":"7d53428a-d028-4ce2-a532-3b6125d924c3","arxiv_id":"2607.29305","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For every d≥3 and every λ≥0, the Neumann counting function of the unit d-ball satisfies N(λ) ≥ w_d λ^d, so Pólya's conjecture holds for all Euclidean balls.","lead":"This paper proves Pólya's conjecture for Neumann eigenvalues of the Laplacian on Euclidean balls in every dimension at least three: the eigenvalue counting function never falls below Weyl's leading volume term. It closes the higher-dimensional Neumann case left open in the authors' earlier work.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Potential uncovered subinterval if Λ*_d is floor, not ceiling, in §5's computer-assisted gap-filling.","rationale":"The reader identified the computer-assisted part as the weakest point, but focused on code availability and reproducibility. The more load-bearing issue is internal to the proof: the interval actually verified by Procedure MAIN may be shorter than the gap that needs to be filled. The text defines Λ*_d with a symbol that renders as parentheses; Table 1's integer values are consistent with either floor or ceiling. The proof of Theorem 1.1 requires the inequality for every λ, including the half-open interval between the last checked eigenvalue and the upper endpoint of the gap. Since N(λ) is constant on eigenvalue-free intervals, the missing check is a finite computation, so the issue is concrete and likely fixable. This is an internal consistency concern, not a disagreement with external consensus. The recommended verdict is CONDITIONAL: accept only after confirming the ceiling interpretation or extending the verification. I agree partially with the reader: they targeted §5 but did not identify this specific subinterval problem.","tokens_in":37309,"tokens_out":19406,"duration_ms":182087,"concrete_test":"Determine from the source/PDF whether Λ*_d is \\lceil ... \\rceil or \\lfloor ... \\rfloor. If it is floor, rerun Procedure MAIN for each d = 3,...,12 with Λ = ceil( (τ*_d)^3 d^{3/2} ) (using a certified rational upper bound for the root τ*_d) and recompute the maximum in (5.4). The concern is settled if the maximum remains < 1 for all d, or equivalently if MAIN with Λ equal to a certified upper bound for (τ*_d)^3 d^{3/2} returns no extra eigenvalue violating (5.4). If the symbol is already ceiling, no change is needed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.1 splits the frequency range at Λ*_d = ⌊(τ*_d)^3 d^{3/2}⌋ (if the symbol in §5 is floor, which the text suggests). The analytic results cover λ > (τ*_d)^3 d^{3/2} and λ ≤ (28/25)^3 d^{3/2}. For 3 ≤ d ≤ 12 the remaining gap is ((28/25)^3 d^{3/2}, (τ*_d)^3 d^{3/2}). The computer-assisted step invokes MAIN(d, Λ*_d, 1/100), whose output contains only zeros p'_{d,m,k} ≤ Λ*_d. Consequently, any eigenvalue frequency in (Λ*_d, (τ*_d)^3 d^{3/2}) is not checked. For λ just below such an eigenvalue, N(λ) equals N(Λ*_d), but the required lower bound w_d λ^d is larger than w_d (Λ*_d)^d; the check (5.4) does not cover this. Even if no eigenvalue lies in that interval, the interval itself extends to the upper endpoint where the large-λ inequality is only shown for strictly larger λ. The reported margins (~0.7–0.9 in Table 1) suggest the extended check would pass, but as written the proof may leave a subinterval unverified. If Λ*_d is instead ceil, this concern evaporates; the extracted manuscript text is ambiguous, so this must be checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Pólya's conjecture for the Neumann Laplacian on the unit ball in R^d for every d≥3: N^N_{B_d}(λ) ≥ w_d λ^d for all λ≥0, with w_d = 1/(2^d Γ(d/2+1)^2). The proof combines three ingredients: a large-λ argument comparing zeros of derivatives of ultraspherical Bessel functions with zeros of ordinary Bessel derivative zeros and reducing to weighted lattice-point estimates; new variational estimates for low-lying eigenvalues using dimension-dependent test functions; and a rigorous computer-assisted enumeration of the remaining finite ranges for d=3,...,12 using exact rational arithmetic and certified enclosures. Together with the authors' earlier work [FLPS23], this would establish the Neumann Pólya conjecture for balls in all dimensions d≥2.","tokens_in":37688,"tokens_out":20421,"duration_ms":197671,"significance":"This is a substantial result: the Neumann case of Pólya's conjecture for Euclidean balls has been open in higher dimensions, and the paper gives a coherent strategy with separated analytic and computational components. The paper is unusually transparent: the large-λ estimates, the low-λ variational bounds, and the finite verification are cleanly separated; the computational part uses exact rational arithmetic, explicit enclosure lemmas, and stated certification procedures. The novel dimension-dependent variational test functions in §4 are an interesting ingredient in their own right. The proof relies on a number of technical results imported from the authors' own earlier papers [FLPS23, FLPS24b], and the finite verification depends on externally supplied data; these are normal features of this type of argument but require careful checking.","major_comments":[{"comment":"The finite-gap verification is incomplete as written. First, (5.4) takes its maximum only over the enumerated zeros i=2,...,K_d. For λ in the interval [p'_{K_d}, Λ*_d] the counting function is constant equal to N_{B_d}^N(Λ*_d), and Pólya's inequality at λ=Λ*_d requires N_{B_d}^N(Λ*_d) ≥ w_d Λ*_d^d; this is not implied by the check at the last enumerated zero unless that zero lies within the tolerance of Λ*_d. A virtual endpoint term with p'=Λ*_d and n=N(Λ*_d)+1 must be added to (5.4). Second, the definition of Λ*_d is ambiguous: if Λ*_d is the floor of τ*_d^3 d^{3/2}, then MAIN(d, Λ*_d, 1/100) enumerates only zeros up to Λ*_d, while Theorem 3.1 starts only at λ > τ*_d^3 d^{3/2}, leaving the interval (Λ*_d, τ*_d^3 d^{3/2}) unverified. The authors should state explicitly that Λ*_d is the ceiling (or otherwise close this interval) and add the endpoint check. The margins in Table 1 suggest t","section":"§5, Eq. (5.4); §2.3"},{"comment":"The exact rational certificates justifying (5.4) are not included in the paper; Table 1 gives only decimal approximations, and Appendix D describes the algorithm but not its full output. The Mathematica script is available only on an author's personal website, without a version identifier or permanent archive. Since the theorem depends on these computations, the manuscript should include the exact output, or the reproducing script and printout should be archived as supplementary material with a stable DOI or commit hash. As submitted, the computer-assisted component cannot be independently checked by the reader.","section":"§5 and Data availability statement"}],"minor_comments":[{"comment":"The 'explicit verified rational approximations' showing T61(1)>10^{-4} and T13(28/25)>10^{-3} are not displayed. Please include the exact values or point to a table where they can be found.","section":"§3.1, Theorem 3.1"},{"comment":"Numerical evaluations such as 𝔉_4≈1.07496 and 𝔉_{11}≈1.08168 are presented with '≈' symbols. If these are certified rational enclosures, this should be stated explicitly; otherwise they are not part of a rigorous proof.","section":"§4.4 and §C.9"},{"comment":"The line 'μ_2(B_d) ≤ d+2 ≤ w_d^{-2/d} d' appears to contain an extra 'd' after w_d^{-2/d}; please clarify the intended inequality.","section":"§4.2"},{"comment":"'derivaives' should be 'derivatives'.","section":"Keywords"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct and the identified gaps appear easily fixable: the endpoint check in (5.4) would pass with the reported margins, and using the ceiling instead of the floor for Λ*_d closes the remaining subinterval. The main issue is that the text must be made unambiguous and the exact computer-assisted certificates made publicly and permanently available. I would not reject the paper, but the current version should not be accepted without these revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem here is already proved in Li26, which the authors cite. So the novelty is not the result but the independent proof and the tools: dimension-dependent radial test functions, control of zeros of derivatives of ultraspherical Bessel functions, and a certified rational-arithmetic gap-filling algorithm. The analytic parts are thorough and I see no internal contradiction. The variational low-frequency section is a genuine new ingredient, and the large-λ estimates look carefully done.\n\nThe soft spot is the computer-assisted part in §5, and the stress-test note lands. Λ*_d is defined with a floor, and the large-λ analytic bound is only proved for λ > (τ*_d)^3 d^{3/2}. That leaves a subinterval (Λ*_d, (τ*_d)^3 d^{3/2}) that neither the analytic estimate nor the computational check covers. The margins in Table 1 are large enough that the inequality almost certainly still holds there, but as written the proof has a gap. Using the ceiling instead of the floor should fix it. The authors should also archive the Mathematica code with a commit hash; right now it is only on one author's website, which is weaker than the paper suggests.\n\nA separate point: the theorem is not new, but the paper explicitly acknowledges Li26 and presents an independent proof with different techniques. That is legitimate, and the technical contribution is substantial. The reliance on the authors' own prior papers is fine; those results are published and checkable.\n\nI would send this to a serious referee and ask for a revision that resolves the floor/ceil issue and provides the archived code. The analytic core is strong, the computational gap is narrow and probably fixable, and the paper deserves a proper referee rather than a desk rejection.","headline":"Independent, detailed proof of a theorem already covered by Li26; the new value is in the method, and the floor/ceiling gap in the computational section is real but likely patchable.","tokens_in":38136,"tokens_out":5118,"would_cite":true,"duration_ms":46654,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P15","35P20","33C10","11P21"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves Pólya's conjecture for the Neumann Laplacian on Euclidean balls in every dimension d ≥ 3.","keywords":["Pólya's conjecture","Neumann eigenvalues","Laplacian","Weyl's law","Euclidean balls","ultraspherical Bessel functions","lattice-point counting","computer-assisted proof"],"falsifier":"Run an independent certified interval-arithmetic search for all zeros p'_{d,m,k} ≤ Λ*_d for d=3,...,12 and compare with the paper's Table 1 and the associated archived printout; any missing triple or any value failing inequality (5.4) would falsify the proof. A direct numerical search for a pair (d,λ) with N^N_{B_d}(λ) < w_d λ^d would falsify the theorem itself.","tokens_in":37210,"feed_emoji":"🔵","tokens_out":7408,"duration_ms":73722,"temperature":0.7,"pith_summary":"Pólya's conjecture for the Neumann Laplacian says the eigenvalue counting function is never below the leading Weyl term. The paper proves this for Euclidean balls in every dimension d ≥ 3, settling the Neumann case for balls in all dimensions when combined with the earlier two-dimensional result. The proof splits the spectrum into three regimes: a large-frequency lattice-point comparison involving zeros of Bessel derivatives, a low-frequency variational analysis with new dimension-dependent test functions, and a certified computer-assisted enumeration that fills the remaining finite gaps in dimensions 3 through 12. A reader should care because balls are the first non-tiling domains for which the full Neumann Pólya conjecture is now established in arbitrary dimension, and the proof introduces reusable techniques for special-function spectra.","feed_headline":"Neumann balls satisfy Pólya's lower bound in every dimension ≥3","feed_subtitle":"Large frequencies become lattice counts, low frequencies variational estimates; certified computation closes gaps in dimensions 3–12.","key_machinery":"The central object is the family of zeros p'_{d,m,k} of derivatives of ultraspherical Bessel functions P_{d,m}(x)=x^{-(d/2-1)}J_{m+d/2-1}(x); the Neumann eigenvalues of B_d are their squares, weighted by spherical-harmonic multiplicities κ_{d,m}. The argument's load-bearing bound is p'_{d,m,k} < j'_{m+d/2-1,k}, which lets the counting function be bounded below by a lattice-point count involving G_λ(z)=π^{-1}(√(λ²-z²)-z arccos(z/λ)). Around this, the proof assembles: monotone envelope functions T_d(τ) for large frequencies; a variational principle for the first radial eigenvalue in each angular sector with two explicit test functions; and a rigorous bisection/enclosure algorithm in exact rati","core_discovery":"The paper's central claim is that for every dimension d≥3 and every λ≥0, the Neumann counting function of the unit ball B_d satisfies N^N_{B_d}(λ) ≥ w_d λ^d, where w_d = 1/(2^d Γ(d/2+1)^2) is exactly the leading coefficient in Weyl's law. Since the Dirichlet analogue was already known for balls, this completes Pólya's conjecture for both boundary conditions on Euclidean balls in all dimensions. The proof is built on the fact that the Neumann eigenvalues of a ball are the squares of zeros p'_{d,m,k} of derivatives of ultraspherical Bessel functions, with multiplicities equal to dimensions of spaces of spherical harmonics. For large λ, the inequality p'_{d,m,k} < j'_{m+d/2-1,k} turns the count","pith_inferences":["Beyond the paper: the same G_λ lattice-count mechanism, with zeros of ordinary Bessel derivatives as comparison objects, should transfer to other rotationally symmetric Neumann problems whose eigenfunctions are Bessel-type, such as spherical shells or sectors; this is a natural test of the method's reach.","Beyond the paper: the dimension-dependent variational test functions suggest that the low-frequency control can be pushed further; optimizing them could shrink or eliminate the computer-assisted gap, perhaps yielding a fully analytic proof for all d.","Beyond the paper: since w_d is the exact Weyl constant, Theorem 1.1 implies that for balls the leading asymptotic term is also a uniform lower bound; this sharpens the expected form of remainder estimates and might inform conjectures about optimal constants in Weyl-type inequalities for other domains."],"forward_implications":["For every unit ball B_d with d≥2 and every λ≥0, the counting inequality N^N_{B_d}(λ) ≥ w_d λ^d holds; equivalently, each Neumann eigenvalue satisfies μ_n(B_d)^d ≤ ((n−1)/w_d)^2.","Euclidean balls become the principal non-tiling example for which both Pólya conjectures — Dirichlet and Neumann — are fully settled in arbitrary dimension.","For dimensions d≥61 the proof is entirely analytic: the monotone envelope T_d is positive already at τ=1, so no gap-filling computation is needed there.","The certified enumeration closes the finite ranges in dimensions 3 through 12, so the theorem is unconditional provided those certified computations are accepted."],"fun_headline_variants":["Pólya's Neumann bound proven for all balls in dimensions ≥3","Neumann balls match Weyl law: Pólya's conjecture solved in d≥3","Ultraspherical Bessel zeros settle Pólya's Neumann conjecture","All ball Neumann eigenvalues obey Pólya's bound in d≥3","Pólya's conjecture for Neumann balls confirmed in all higher dimensions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the computer-assisted enumeration in Section 5 correctly lists every eigenvalue below the cut-off for dimensions 3 through 12; if a single zero is missed or a rational bound in the supporting table is wrong, the finite gap remains open.","fun_headline_variants_meta":{"raw":{"variants":["Pólya's Neumann bound proven for all balls in dimensions ≥3","Neumann balls match Weyl law: Pólya's conjecture solved in d≥3","Ultraspherical Bessel zeros settle Pólya's Neumann conjecture","All ball Neumann eigenvalues obey Pólya's bound in d≥3","Pólya's conjecture for Neumann balls confirmed in all higher dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000945,"raw_usage":{"total_tokens":3841,"prompt_tokens":683,"completion_tokens":3158,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":427,"completion_tokens_details":{"reasoning_tokens":3059}},"tokens_in":427,"tokens_out":3158,"duration_ms":22737,"temperature":1.0,"reasoning_tokens":3059,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T09:32:47.828436+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent certified interval-arithmetic search for all zeros p'_{d,m,k} ≤ Λ*_d for d=3,...,12 and compare with the paper's Table 1 and the associated archived printout; any missing triple or any value failing inequality (5.4) would falsify the proof. A direct numerical search for a pair (d,λ) with N^N_{B_d}(λ) < w_d λ^d would falsify the theorem itself.","supporting_citations":[],"review_version":1}