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Independent preprints claim Pólya’s Neumann bound for every Euclidean ball

Two independently developed preprints claim Pólya’s lower bound for the Neumann eigenvalue-counting function of every Euclidean ball. Together with the published Dirichlet result and Neumann-disk case, the claims would settle both Pólya inequalities for balls in every dimension at least two.

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Stable ID
math-polya-neumann-euclidean-balls-2026
Revision
math-polya-neumann-euclidean-balls-2026.v2
Field
Mathematics · Spectral theory and partial differential equations
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Provisional claimed proof
Last checked

AI role

Li discloses no AI role in the inspected manuscript. Filonov, Levitin, Polterovich and Sher say ChatGPT and Claude contributed to several technical lemmas and the design and implementation of their computational algorithm; they independently checked the work and accept responsibility.

Record details

Problem or result
Pólya’s conjectured lower bound for the Neumann Laplacian on Euclidean balls
Authors
Yutian Li; independently, Nikolay Filonov, Michael Levitin, Iosif Polterovich and David A. Sher
Institutions
Nanfang College, Guangzhou; Steklov Institute of Mathematics; University of Reading; Université de Montréal; DePaul University
Result date
Li submitted July 28 and revised through August 5; the independent preprint was submitted July 31, 2026

Why it matters

Pólya’s 1954 spectral bound is known for tiling domains and selected non-tiling domains, but higher-dimensional Neumann balls resisted the methods that settled the disk. Two groups now report different dimension-uniform arguments, including exact or computer-assisted finite checks.

Limits

Both manuscripts remain unreviewed, and concurrent independent proofs are not a substitute for specialist assessment. Li’s 72-page third version and the separate 39-page first version use technical special-function estimates and exact or computer-assisted checks. The scope is Euclidean balls, not arbitrary domains.

Sources

  1. Primary: Yutian Li, Pólya’s Conjecture for the Neumann Eigenvalues on Euclidean Balls
  2. Independent preprint: Filonov, Levitin, Polterovich and Sher, Pólya’s conjecture for higher-dimensional Neumann balls
  3. Published context: Filonov, Levitin, Polterovich and Sher, published Dirichlet-ball and Neumann-disk proof

Correction and revision history

  1. 2026-07-29 — Added as a provisional claimed proof after full-text review; the ball-only scope, first-version status and unreviewed finite calculations are explicit.
  2. 2026-08-08 — Reviewed Li’s substantially revised v3 and added a concurrent independent preprint by Filonov, Levitin, Polterovich and Sher. The active record now discloses the second group’s substantive AI contribution; status remains provisional.

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