{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:planar-berenstein-conjecture",
  "title": "The Planar Berenstein Conjecture",
  "canonical_statement": "The unrestricted planar Berenstein conjecture holds that overdetermined Dirichlet-Neumann data characterize the disc. Disproved: a bounded simply connected domain with real-analytic Jordan boundary that is not a disc, carrying a nonzero real eigenfunction with zero Dirichlet data and constant nonzero Neumann data.",
  "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:planar-berenstein-conjecture:event",
      "problem_id": "vibemathed:planar-berenstein-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-09T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The domain has dihedral symmetry of order 26 and is neither a disc nor centrally symmetric, and its eigenfunction changes sign - which is why an additional sign assumption rescues the statement.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:planar-berenstein-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:planar-berenstein-conjecture:method"
      ],
      "source_assertion_ids": [
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      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:planar-berenstein-conjecture:attempt",
      "problem_id": "vibemathed:planar-berenstein-conjecture",
      "model": "ChatGPT 5.6",
      "provider": "OpenAI",
      "attempted_at": "2026-08-09T00:00:00.000Z",
      "human_collaborators": [
        "Matthew J. Colbrook",
        "Siavash Sadeghi",
        "George Stepaniants"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The authors declare the role and are careful about its limits: \"We believe it is important to declare the use of AI in mathematical research, and in the present case its role is particularly noteworthy. ChatGPT 5.6 was given a substantial warm start consisting of an early draft and working code independently developed by MJC and GS\" for the earlier Pompeiu-Schiffer paper, material that \"already contained the central construction, conformal fixed-disc formulation, coefficient spaces, disk-polynomial algebra, tail strategy, and computational architecture on which the present paper rests. The system was used to explore the modification\" to the Dirichlet endpoint.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.08953",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Planar Berenstein Conjecture",
          "url": "https://vibemathed.com/problem/planar-berenstein-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
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  "verifications": [
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      "id": "vibemathed:planar-berenstein-conjecture:verification",
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      "solution_event_id": "vibemathed:planar-berenstein-conjecture: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. The existence claim is reduced by a Newton-Kantorovich argument to finitely many explicit inequalities certified in interval arithmetic, so the final step is machine-checkable, but nobody independent has rerun it.",
      "sources": [
        {
          "label": "VibeMathed: The Planar Berenstein Conjecture",
          "url": "https://vibemathed.com/problem/planar-berenstein-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.08953",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.08953",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Planar Berenstein Conjecture",
      "url": "https://vibemathed.com/problem/planar-berenstein-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:planar-berenstein-conjecture:signal:watch_only",
      "problem_id": "vibemathed:planar-berenstein-conjecture",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}