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P\'{o}lya's conjecture for higher-dimensional Neumann balls

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Pith's one-line read The paper proves Pólya's conjecture for the Neumann Laplacian on Euclidean balls in every dimension d ≥ 3.

desk verdict 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. read the letter →

arxiv 2607.29305 v1 pith:CFTECAZI submitted 2026-07-31 math.SP

Nikolay Filonov , Michael Levitin , Iosif Polterovich , David A. Sher This is my paper · ORCID
classification math.SP MSC 35P1535P2033C1011P21
keywords Pólya'sconjectureNeumanneigenvaluesLaplacianWeyl'slawEuclideanballsultrasphericalBesselfunctionslattice-pointcountingcomputer-assistedproof
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [§5, Eq. (5.4); §2.3] 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
  2. [§5 and Data availability statement] 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.
minor comments (4)
  1. [§3.1, Theorem 3.1] 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.
  2. [§4.4 and §C.9] 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.
  3. [§4.2] 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.
  4. [Keywords] 'derivaives' should be 'derivatives'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorem is derived from published independent lemmas and explicit computations, not assumed.

full rationale

The paper does not reduce Theorem 1.1 to its own target statement. The spectral representation (2.2) is exact; the key zero-comparison (2.3) is Spigler's external bound; the lattice-point lower bound (3.2) cites [FLPS24b, Theorem 1.10] and [FLPS23, Proposition 3.1], which are published, independent results about Bessel-function zero counting, not equivalent to the Neumann Pólya inequality for balls. The large-frequency argument then combines explicit estimates for the auxiliary functions T_{d,j} with standard combinatorial identities (Appendix B) and external analytic tools such as Lampret's Wallis-ratio bound and the Lorch-Szego spacing theorem. The low-frequency part uses explicit variational test functions; the constants and the split at tau* = 28/25 are choices of proof strategy, not fitted values that are later renamed as predictions. The finite gap in dimensions 3 through 12 is closed by a direct certified computation: Procedure MAIN constructs rational enclosures for every relevant zero and then verifies inequality (5.4) exactly. This is an exhaustive computation, not an importation of the conclusion. The possible issues raised about the unarchived Mathematica script or the exact meaning of the floor/ceiling in Lambda*_d are correctness and verification concerns, not circularity: they do not exhibit an equation in which the theorem is assumed by construction. Self-citations appear, but they cite previously published, independently checkable technical statements rather than the present theorem, so they do not raise the circularity score.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are postulated. The proof relies on hand-selected thresholds and test functions, on standard and self-cited analytic estimates, and on an external certified computation for the finite gap.

free parameters (3)
  • τ* = 28/25
    Hand-chosen cutoff for the low-frequency regime in Theorem 4.1; selected so that the variational test functions and the finite gap computation work.
  • τ^− and τ^+ low-λ interval endpoints = 4/5 and 28/25 (33/40 for d=3)
    Endpoints of the positivity interval for the alternative test function (Proposition 4.9); chosen by explicit evaluation rather than derived from general principles.
  • Exponent and constants in test function ρ(x) = 3/2; 1 and 2/d
    The test function (4.13) was selected by hand to make the variational lower bound beat Weyl's term over [τ^−,τ^+]; this is a legitimate but ad hoc construction.
assumptions (4)
  • standard math Zeros p'_{d,m,k} interlace with Bessel zeros and satisfy p'_{d,m,k} < j'_{m+d/2-1,k} (Lemma 2.2, from [Sp78]).
    Used to derive the counting lower bound (3.1) and the enclosures in Appendix D.
  • domain assumption The lattice-point estimate (3.2) reducing Ñ^N to a weighted sum of G_λ (Theorem 1.10 of [FLPS24b]) is correct.
    Imported from the authors' prior work; the large-λ argument depends on it directly.
  • standard math Lorch–Szegő spacing bound j_{ν,k} - j_{ν,k-1} > π > 3.
    Ensures BesselZeroBrackets in Appendix D enumerates all zeros without omission.
  • ad hoc to paper Procedure MAIN and Table 1 in §5 correctly certify the finite ranges for d=3,...,12.
    The gap-filling rests on this external computation; code and printout are only on the authors' website, not archived in the arXiv source.

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Pith. "Pith review of P\'{o}lya's conjecture for higher-dimensional Neumann balls." pith.science (2026). https://pith.science/paper/CFTECAZI

@misc{pith2026260729305,
  author       = {Pith},
  title        = {Pith review of: P\'olya's conjecture for higher-dimensional Neumann balls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CFTECAZI}},
  note         = {Machine review of arXiv:2607.29305}
}
read the original abstract

We prove P\'olya's conjecture for the Neumann eigenvalues of the Laplacian on Euclidean balls in dimensions three and higher. The proof further develops the approach introduced in our earlier work on the two-dimensional case and on Dirichlet eigenvalues in arbitrary dimensions. The main difficulty in the higher dimensional Neumann case is that one has to estimate zeros of the derivatives of ultraspherical Bessel functions, rather than of the usual Bessel functions. For low-lying eigenvalues, we use variational estimates involving dimension-dependent test functions, which is a novel ingredient allowing us to control a larger dimension-scaled frequency range. Other components of the proof include phase-function bounds, lattice-point counting techniques, and computer-assisted arguments.

Figures

Figures reproduced from arXiv: 2607.29305 by the authors.

Figure 1
Figure 1. Summary of analytic results in (𝑑, 𝜏)-range. 4 6 8 10 12 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Remaining 𝜆 gaps. §3. Large 𝜆 via comparison with zeros of Bessel derivatives §3.1. Main result and approach As a consequence of (2.3) and (2.2), N N B𝑑 (𝜆) ≥ ∑︁∞ 𝑚=0 𝜅𝑑,𝑚#  𝑘 ∈ N : 𝑗 ′ 𝑚+ 𝑑 2 −1,𝑘 ≤ 𝜆  =: e N N B𝑑 (𝜆) (3.1) This allows us to relate the eigenvalue counting functions to a lattice point count. Specifically, due to [FLPS24b, Theorem 1.10], see also [FLPS23, Proposition 3.1], we have e N N B𝑑 (𝜆) ≥ j … view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Plots of the numerically-evaluated left-hand side of (4.7) as a function of 𝜏 for the constant test function. We are looking at the intervals of positivity. Lemma 4.5 Let 𝜌(𝑥) ≡ 1 and 𝑑 ≥ 4. Then the function 𝔉𝑑 (𝜆) is strictly decreasing in the variable 𝜆 ∈ (0, ∞). Fo…
Figure 5
Figure 5. Figure 5: Numerically evaluated left-hand side of (4.7) for the test function (4.13), shown as a function of 𝜏. The red lines show the positivity intervals guaranteed by Proposition 4.9. Lemma 4.8 Let 𝑑 ≥ 3 and 𝜌 be given by (4.13). Then the function𝔉𝑑 (𝜆) is unimodal in the var…
Figure 6
Figure 6. Figure 6: Summary of the results of §4 in (𝑑, 𝜏)-parameter space. §5. Filling the gap Whenever a finite gap was left in the analytic proofs in our previous papers [FLPS23, FLPS24a, FLPS26], we have filled it using verified rational approximations of trapezoidal floor sums. Here,…
Figure 7
Figure 7. Figure 7: Function ℎ(𝑡) and its derivatives. Writing 𝑢 = 1 − 𝑤/𝜆, this becomes 𝐺𝜆 (𝑧) < 𝜆 ∫ 𝜃 0 √ 2 𝜋 + [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

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