{
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  "ref": "R1701",
  "content_hash": null,
  "slug": "planar-convex-hot-spots-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring result: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension. Exact unresolved remainder: No convex planar counterexample or general proof was located.",
  "relevance": "This is the dated publication status for the canonical target Hot spots conjecture for convex planar domains.",
  "relevance_source": "recorded",
  "body": "The problem was checked as open on 2026-08-01.\n\nThe strongest neighboring result found in the cited sources is: Many planar classes are proved; convex counterexamples now exist in sufficiently high dimension.\n\nThe exact unresolved remainder is: No convex planar counterexample or general proof was located.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove boundary attainment for every bounded convex planar domain, with a stated boundary regularity convention.\n- Or give a convex planar domain whose first Neumann eigenfunction has an interior global extremum.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2405.19508",
      "locator": "Lawford Hatcher, “The hot spots conjecture for some non-convex polygons,” arXiv:2405.19508v3. introduction and main theorems"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2405.19508",
    "locator": "Lawford Hatcher, “The hot spots conjecture for some non-convex polygons,” arXiv:2405.19508v3. introduction and main theorems"
  },
  "relations": [
    {
      "slug": "R1700",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1698",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1699",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "planar-convex-hot-spots",
      "title": "planar convex hot spots",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}