{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1321",
  "content_hash": null,
  "slug": "harmonic-random-disk-cover-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The comments establish almost-sure coverage of each fixed point and zero area of the uncovered set. They do not establish complete coverage of the uncountable disk, and the exact critical sequence was not settled in the checked random-covering literature.",
  "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 question and every visible answer and comment were checked on 2026-07-27. Borel-Cantelli handles each fixed point, and Fubini gives zero uncovered area, while neither argument controls all points simultaneously. Penrose's arXiv:2101.06306 and related finite random geometric coverage results were checked. They use finite samples and threshold radii with boundary corrections, which do not directly decide this infinite nonidentical-radius process. Dvoretzky covering references were checked for the one-dimensional analogue. The planar boundary and two-dimensional holes require different capacity or net estimates. A local corpus search for harmonic random disks, n^{-1/2} covering, and complete random coverage found no duplicate.\n\nA complete resolution must satisfy: Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/458793/will-a-unit-disk-be-completely-covered-by-randomly-placed-disks-of-area-pi-fr",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/458793/will-a-unit-disk-be-completely-covered-by-randomly-placed-disks-of-area-pi-fr",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "relations": [
    {
      "slug": "R1320",
      "title": "Complete the stated acceptance conditions",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "R1590",
      "title": "Dated status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming"
    },
    {
      "slug": "harmonic-random-disk-cover",
      "title": "harmonic random disk cover",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}