{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:schiffer-conjecture",
  "title": "Schiffer's Conjecture and the Pompeiu Problem",
  "canonical_statement": "If a smooth bounded domain in $\\mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Pompeiu posed an equivalent integral-equation form in 1929; Schiffer's 1957 reformulation via Neumann eigenfunctions is the version on Yau's 1982 list (Problem 80), and Williams proved the two formulations logically equivalent for simply connected domains in 1976. Cao-Labora and de Dios Pont construct infinitely many planar domains with large $N$-fold symmetry that are not balls and admit such an eigenfunction, disproving Schiffer's conjecture; applying Williams' classical reduction to the same domains (their Corollary 1.2) disproves Pompeiu's problem as well.",
  "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:schiffer-conjecture:event",
      "problem_id": "vibemathed:schiffer-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-05T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:schiffer-conjecture:attempt"
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      "id": "vibemathed:schiffer-conjecture:attempt",
      "problem_id": "vibemathed:schiffer-conjecture",
      "model": "GPT-5.6, Claude Opus 4.8, Claude Fable 5",
      "provider": "OpenAI, Anthropic",
      "attempted_at": "2026-08-05T00:00:00.000Z",
      "human_collaborators": [
        "Gonzalo Cao-Labora",
        "Jaume de Dios Pont"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Models with coding harnesses were used in multiple parts of the research: numerically verifying the asymptotic estimates, producing first drafts of the proofs of the Bessel function estimates, and helping with exposition. \n\nThe Lean4 verification of the proof was written by GPT 5.6 from an early draft of the paper. The novel construction strategy is the authors' own.",
      "failed_routes": [],
      "artifacts": [
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          "url": "https://github.com/jaumededios/Schiffer",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Pompeiu problem Wikipedia article",
          "url": "https://en.wikipedia.org/wiki/Pompeiu_problem",
          "kind": "wikipedia",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.05114",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Schiffer's Conjecture and the Pompeiu Problem",
          "url": "https://vibemathed.com/problem/schiffer-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "cost_usd": null,
      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:schiffer-conjecture:verification",
      "problem_id": "vibemathed:schiffer-conjecture",
      "solution_event_id": "vibemathed:schiffer-conjecture:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A day-old preprint. The paper states that a Lean4 verification of the proof was written by GPT 5.6, available at https://github.com/jaumededios/Schiffer. It solves the Pompeiu Problem challenge provided by https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/PompeiuProblem.lean",
      "sources": [
        {
          "label": "VibeMathed: Schiffer's Conjecture and the Pompeiu Problem",
          "url": "https://vibemathed.com/problem/schiffer-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.05114",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.05114",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Schiffer's Conjecture and the Pompeiu Problem",
      "url": "https://vibemathed.com/problem/schiffer-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:schiffer-conjecture:signal:replay_ready",
      "problem_id": "vibemathed:schiffer-conjecture",
      "kind": "replay_ready",
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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"
}