{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-267",
  "title": "Erdős Problem #267",
  "canonical_statement": "If $n_1<n_2<\\cdots$ with $n_{k+1}/n_k\\ge c>1$, must $\\sum_k 1/F_{n_k}$ be irrational? The proposed proof closes the range $1<c<2$ left open by earlier criteria.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:erdos-267:event",
      "problem_id": "vibemathed:erdos-267",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-07-13T00:00:00.000Z",
      "title": "erdosproblems.com/267",
      "summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-267:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-267:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-267:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-267:attempt",
      "problem_id": "vibemathed:erdos-267",
      "model": "GPT-5.6 starships (Claude Fable 5 reviewer)",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-07-13T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer, according to the source.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "erdosproblems.com/267",
          "url": "https://www.erdosproblems.com/267",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #267",
          "url": "https://vibemathed.com/problem/erdos-267",
          "kind": "source-assertion",
          "license": "CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-267:verification",
      "problem_id": "vibemathed:erdos-267",
      "solution_event_id": "vibemathed:erdos-267:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The source reports a Lean check while the Erdős Problems community status remains pending; status therefore remains candidate.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #267",
          "url": "https://vibemathed.com/problem/erdos-267",
          "kind": "source-assertion",
          "license": "CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)"
        },
        {
          "label": "erdosproblems.com/267",
          "url": "https://www.erdosproblems.com/267",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/267",
      "url": "https://www.erdosproblems.com/267",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #267",
      "url": "https://vibemathed.com/problem/erdos-267",
      "kind": "source-assertion",
      "license": "CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-267:signal:verify_now",
      "problem_id": "vibemathed:erdos-267",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-08-17T15:54:39.560Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented."
  ],
  "generated_at": "2026-08-17T15:54:39.560Z"
}