{
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  "source": "vibemath",
  "problem_id": "vibemathed:polya-conjecture-neumann-balls",
  "title": "Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher",
  "canonical_statement": "Pólya conjectured in 1954 that the Weyl-law expression bounds the eigenvalue counting function of the Laplacian. The paper proves the Neumann case for Euclidean balls in dimensions three and higher, extending the authors' earlier planar and Dirichlet results. Key difficulty: estimating zeros of derivatives of ultraspherical Bessel functions rather than of Bessel functions themselves.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:polya-conjecture-neumann-balls:event",
      "problem_id": "vibemathed:polya-conjecture-neumann-balls",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-31T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:polya-conjecture-neumann-balls:attempt"
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      "problem_id": "vibemathed:polya-conjecture-neumann-balls",
      "model": "ChatGPT + Claude (several models)",
      "provider": null,
      "attempted_at": "2026-07-31T00:00:00.000Z",
      "human_collaborators": [
        "Nikolay Filonov",
        "Michael Levitin",
        "Iosif Polterovich",
        "David A. Sher"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper carries an AI usage disclosure: several models of ChatGPT and Claude were used for mathematical discussions and editorial assistance, and all AI-assisted arguments and computations were independently checked by the authors, who take full responsibility. The disclosure does not separate which arguments were AI-assisted, so the lowest tier applies.",
      "failed_routes": [],
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      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.29305",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher",
          "url": "https://vibemathed.com/problem/polya-conjecture-neumann-balls",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "independence": "unknown"
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      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
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      "note": "No independent review, and the AI disclosure is general rather than pointing at specific results. The authors state they independently checked every AI-assisted argument and computation. The proof also uses conventional computer-assisted arguments, which are not the same thing. Preprint, not refereed.",
      "sources": [
        {
          "label": "VibeMathed: Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher",
          "url": "https://vibemathed.com/problem/polya-conjecture-neumann-balls",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.29305",
          "kind": "primary-mathematical-source",
          "license": null
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      "url": "https://arxiv.org/abs/2607.29305",
      "kind": "primary-mathematical-source",
      "license": null
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    {
      "label": "VibeMathed: Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher",
      "url": "https://vibemathed.com/problem/polya-conjecture-neumann-balls",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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  "recommended_exposure": "result_only",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}