{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-346",
  "title": "Erdős Problem #346",
  "canonical_statement": "Let $A=\\{1\\leq a_1< a_2<\\cdots\\}$ be a set of integers such that $A\\backslash B$ is complete for any finite subset $B$ and not complete for any infinite subset $B$. If $a_{n+1}/a_n \\geq 1+\\epsilon$ for all $n$, must $\\lim_n a_{n+1}/a_n=(1+\\sqrt{5})/2$? Under the reading where the ratio limit is assumed to exist, a Lean-verified argument forces the limit to be the golden ratio; a separate construction disproves the literal statement where convergence is not assumed.",
  "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-346:event",
      "problem_id": "vibemathed:erdos-346",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-06-21T00:00:00.000Z",
      "title": "erdosproblems.com/346",
      "summary": "The problem statement is ambiguous: the limit-exists reading is claimed proved (Lean), while the convergence-from-hypotheses reading was disproved by a Lean-checked construction of Price that the community classes as a variant",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-346:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-346:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-346:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-346:attempt",
      "problem_id": "vibemathed:erdos-346",
      "model": "ChatGPT, Codex",
      "provider": "OpenAI",
      "attempted_at": "2026-06-21T00:00:00.000Z",
      "human_collaborators": [
        "Kenta Kitamura"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Kitamura's affirmative Lean 4 formalization of the limit-exists reading was produced with ChatGPT and Codex; days earlier, GPT Pro with Codex had produced a Lean-checked disproof of the literal reading (Liam Price), which the forum classes as solving a variant with precursors in Burr-Erdős 1981.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Kitamura's Lean formalization",
          "url": "https://github.com/KitaKen1/erdos346-ratio-limit-lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Price's disproof of the literal reading",
          "url": "https://www.overleaf.com/read/tgrrgqpbjpht#c5a085",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/346",
          "url": "https://www.erdosproblems.com/346",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #346",
          "url": "https://vibemathed.com/problem/erdos-346",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-346:verification",
      "problem_id": "vibemathed:erdos-346",
      "solution_event_id": "vibemathed:erdos-346:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A community screening found the Lean of the variant disproof correct and corresponding to its paper (one typo); the affirmative limit-exists formalization reports standard axioms only. erdosproblems.com still lists the problem open.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #346",
          "url": "https://vibemathed.com/problem/erdos-346",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/346",
          "url": "https://www.erdosproblems.com/346",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/346",
      "url": "https://www.erdosproblems.com/346",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #346",
      "url": "https://vibemathed.com/problem/erdos-346",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-346:signal:verify_now",
      "problem_id": "vibemathed:erdos-346",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "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.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}