{
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  "ref": "R1702",
  "content_hash": null,
  "slug": "planar-drums-finitely-different-spectra-status-packet-quality-20260801",
  "type": "claim",
  "title": "Dated status and exact unresolved remainder",
  "summary": "Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow thread gives no construction or rigidity proof for smooth planar domains. Nearby work treats full isospectrality and eventual spectral agreement in other geometric categories. Exact unresolved remainder: Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra. A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient.",
  "relevance": "For Planar drums whose spectra differ only finitely, this successor gives readable dated status prose and the exact remaining research boundary.",
  "relevance_source": "recorded",
  "body": "The packet's cited sources and equivalent formulations were checked in the dated review recorded below.\n\nStrongest checked result: The MathOverflow thread gives no construction or rigidity proof for smooth planar domains. Nearby work treats full isospectrality and eventual spectral agreement in other geometric categories.\n\nExact unresolved remainder: Construct two domains and prove exact equality of every eigenvalue from some index onward while certifying at least one earlier mismatch, or prove that eventual equality forces equality of the complete Dirichlet spectra. A construction must specify the domains sufficiently to prove smoothness, connectedness, and exact spectral claims; numerical eigenvalue agreement is insufficient.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1005.1839",
      "locator": "abstract and transplantation construction of fully isospectral planar domains"
    },
    "missing": [
      "source",
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    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1005.1839",
    "locator": "abstract and transplantation construction of fully isospectral planar domains"
  },
  "relations": [
    {
      "slug": "R1354",
      "title": "Current checked status and unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing"
    },
    {
      "slug": "planar-drums-finitely-different-spectra",
      "title": "planar drums finitely different spectra",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}