{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-12",
  "title": "Erdős Problem #12",
  "canonical_statement": "Let $A \\subset \\mathbb{N}$ be infinite with no distinct $a, b, c \\in A$ such that $a \\mid (b + c)$ with $b, c > a$. Can $|A \\cap [1, N]|/\\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:erdos-12:event",
      "problem_id": "vibemathed:erdos-12",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-05-21T00:00:00.000Z",
      "title": "erdosproblems.com/12",
      "summary": "parts (i) and (ii) resolved - a near-linear-density construction exists, refuting the N^{1-c} decay; the reciprocal-sum part remains open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-12:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-12:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-12:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-12:attempt",
      "problem_id": "vibemathed:erdos-12",
      "model": "AlphaProof Nexus",
      "provider": "Google DeepMind",
      "attempted_at": "2026-05-21T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": null,
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "erdosproblems.com/12",
          "url": "https://www.erdosproblems.com/12",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #12",
          "url": "https://vibemathed.com/problem/erdos-12",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-12:verification",
      "problem_id": "vibemathed:erdos-12",
      "solution_event_id": "vibemathed:erdos-12: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-checked; formal proofs published with the AlphaProof Nexus report (arXiv:2605.22763).",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #12",
          "url": "https://vibemathed.com/problem/erdos-12",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/12",
          "url": "https://www.erdosproblems.com/12",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/12",
      "url": "https://www.erdosproblems.com/12",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #12",
      "url": "https://vibemathed.com/problem/erdos-12",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-12:signal:recent_partial_result",
      "problem_id": "vibemathed:erdos-12",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}