{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:odd-area-unit-disks",
  "title": "The Odd Area Conjecture for Unit Disks",
  "canonical_statement": "For a family $F$ of an odd number $n$ of unit disks in the plane, let $\\mathrm{OA}(F)$ be the area covered by an odd number of disks. It was conjectured that $\\mathrm{OA}(F) \\ge \\pi$, the area of a single disk. False: configurations exist with smaller odd area.",
  "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:odd-area-unit-disks:event",
      "problem_id": "vibemathed:odd-area-unit-disks",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-06-06T00:00:00.000Z",
      "title": "arXiv:2606.08337 - A Remark on the Odd Area of Unit Disks",
      "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.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:odd-area-unit-disks:attempt"
      ],
      "method_family_ids": [
        "vibemathed:odd-area-unit-disks:method"
      ],
      "source_assertion_ids": [
        "vibemathed:odd-area-unit-disks:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:odd-area-unit-disks:attempt",
      "problem_id": "vibemathed:odd-area-unit-disks",
      "model": "AlphaEvolve",
      "provider": "Google DeepMind",
      "attempted_at": "2026-06-06T00:00:00.000Z",
      "human_collaborators": [
        "Stefan Steinerberger"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The counterexample configurations, at 51 and 151 disks, were investigated using AlphaEvolve. The author notes the behaviour differs between small and large numbers of disks, which is what the search surfaced.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2606.08337 - A Remark on the Odd Area of Unit Disks",
          "url": "https://arxiv.org/abs/2606.08337",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Odd Area Conjecture for Unit Disks",
          "url": "https://vibemathed.com/problem/odd-area-unit-disks",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:odd-area-unit-disks:verification",
      "problem_id": "vibemathed:odd-area-unit-disks",
      "solution_event_id": "vibemathed:odd-area-unit-disks:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The refutation is explicit finite disk configurations, so the odd area is a direct computation. Single-author arXiv note, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: The Odd Area Conjecture for Unit Disks",
          "url": "https://vibemathed.com/problem/odd-area-unit-disks",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2606.08337 - A Remark on the Odd Area of Unit Disks",
          "url": "https://arxiv.org/abs/2606.08337",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2606.08337 - A Remark on the Odd Area of Unit Disks",
      "url": "https://arxiv.org/abs/2606.08337",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Odd Area Conjecture for Unit Disks",
      "url": "https://vibemathed.com/problem/odd-area-unit-disks",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:odd-area-unit-disks:signal:watch_only",
      "problem_id": "vibemathed:odd-area-unit-disks",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "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"
}