{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:steklov-isospectral-convex-plane-domains",
  "title": "The Planar Steklov Analogue of Kac's Question",
  "canonical_statement": "Can one hear the shape of a drum, in the Steklov setting and in the plane? No: there exist pairs of noncongruent bounded plane domains with identical Steklov spectra including multiplicities, simply connected, strictly convex, with real-analytic boundaries, and arbitrarily close to a disk in the C-infinity topology.",
  "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:steklov-isospectral-convex-plane-domains:event",
      "problem_id": "vibemathed:steklov-isospectral-convex-plane-domains",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-11T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Strict convexity and real-analytic boundaries are what make this sharp: the classical Gordon-Webb-Wolpert drums are non-convex polygons, so the obvious escape routes are closed off.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:steklov-isospectral-convex-plane-domains:attempt"
      ],
      "method_family_ids": [
        "vibemathed:steklov-isospectral-convex-plane-domains:method"
      ],
      "source_assertion_ids": [
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      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:steklov-isospectral-convex-plane-domains:attempt",
      "problem_id": "vibemathed:steklov-isospectral-convex-plane-domains",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-08-11T00:00:00.000Z",
      "human_collaborators": [
        "Tao Hu",
        "Jiachen Shi",
        "Quanyu Tang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The division of labour is set out in the AI statement: the idea of adapting the Sunada construction of Gordon, Webb and Wolpert to the planar Steklov problem \"was proposed by the authors\". From there, \"ChatGPT provided substantial assistance in developing the concrete counterexample construction, including the passage from the orbifold construction to weighted Steklov problems on the disk and their subsequent realization by Euclidean plane domains. It also assisted with several technical arguments.\" The authors checked, revised and rewrote the arguments.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10557",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Planar Steklov Analogue of Kac's Question",
          "url": "https://vibemathed.com/problem/steklov-isospectral-convex-plane-domains",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "cost_usd": null,
      "independence": "unknown"
    }
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      "solution_event_id": "vibemathed:steklov-isospectral-convex-plane-domains:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint days old, with no independent review.",
      "sources": [
        {
          "label": "VibeMathed: The Planar Steklov Analogue of Kac's Question",
          "url": "https://vibemathed.com/problem/steklov-isospectral-convex-plane-domains",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10557",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
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  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.10557",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Planar Steklov Analogue of Kac's Question",
      "url": "https://vibemathed.com/problem/steklov-isospectral-convex-plane-domains",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:steklov-isospectral-convex-plane-domains:signal:watch_only",
      "problem_id": "vibemathed:steklov-isospectral-convex-plane-domains",
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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"
}