{
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  "ref": "R1354",
  "content_hash": null,
  "slug": "planar-drums-finitely-different-spectra-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. 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.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. The MathOverflow page and every visible answer and comment were checked on 2026-07-27. No argument there decides whether a finite initial spectral discrepancy can occur for smooth planar domains. Planar isospectral constructions, including arXiv:1005.1839, were checked for examples with altered low eigenvalues. Their constructed pairs have identical full spectra and therefore do not answer the question. A 2026 preprint on eventual spectral agreement for locally symmetric manifolds was checked as nearby rigidity evidence. Its hypotheses do not cover bounded Euclidean planar domains with boundary. A local corpus search for eventual isospectrality, finite spectral difference, and planar drums found no duplicate.\n\nA complete resolution must satisfy: 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.",
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  "scope": null,
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    "citation": {
      "url": "https://mathoverflow.net/questions/474950/can-two-drums-almost-sound-the-same",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
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  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/474950/can-two-drums-almost-sound-the-same",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
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      "title": "Dated status and exact unresolved remainder",
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      "slug": "planar-drums-finitely-different-spectra",
      "title": "planar drums finitely different spectra",
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