{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-539",
  "title": "Erdős Problem #539",
  "canonical_statement": "For $|A| = n$, how small can the cofactor set $Q(A) = \\{a / \\gcd(a,b) : a, b \\in A\\}$ be? The answer is $h(n) = n^{1/2 + o(1)}$: a new upper bound $h(n) \\le n^{1/2} \\exp(O(\\sqrt{\\log n}))$ matches the classical lower bound.",
  "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:erdos-539:event",
      "problem_id": "vibemathed:erdos-539",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-06-10T00:00:00.000Z",
      "title": "erdosproblems.com/539",
      "summary": "main exponent determined; sharper subpolynomial factors remain open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-539:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-539:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-539:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-539:attempt",
      "problem_id": "vibemathed:erdos-539",
      "model": "ProofCouncil (GPT-5.5 Pro)",
      "provider": "OpenAI",
      "attempted_at": "2026-06-10T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The upper-bound construction was found by the ProofCouncil harness running GPT-5.5 Pro.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "erdosproblems.com/539",
          "url": "https://www.erdosproblems.com/539",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #539",
          "url": "https://vibemathed.com/problem/erdos-539",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-539:verification",
      "problem_id": "vibemathed:erdos-539",
      "solution_event_id": "vibemathed:erdos-539:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean record alongside the official Erdős problems update marking the exponent determined.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #539",
          "url": "https://vibemathed.com/problem/erdos-539",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/539",
          "url": "https://www.erdosproblems.com/539",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/539",
      "url": "https://www.erdosproblems.com/539",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #539",
      "url": "https://vibemathed.com/problem/erdos-539",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-539:signal:replay_ready",
      "problem_id": "vibemathed:erdos-539",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "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"
}