{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:nathanson-product-intersection-sets",
  "title": "Nathanson's Problems on Product Intersection Sets",
  "canonical_statement": "Nathanson asked which subsets of $\\mathbb{N}$ can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.",
  "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:nathanson-product-intersection-sets:event",
      "problem_id": "vibemathed:nathanson-product-intersection-sets",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-04-20T00:00:00.000Z",
      "title": "arXiv",
      "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_discovered",
      "attempt_ids": [
        "vibemathed:nathanson-product-intersection-sets:attempt"
      ],
      "method_family_ids": [
        "vibemathed:nathanson-product-intersection-sets:method"
      ],
      "source_assertion_ids": [
        "vibemathed:nathanson-product-intersection-sets:assertion"
      ]
    }
  ],
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    {
      "id": "vibemathed:nathanson-product-intersection-sets:attempt",
      "problem_id": "vibemathed:nathanson-product-intersection-sets",
      "model": "Harmonic Aristotle",
      "provider": "Harmonic",
      "attempted_at": "2026-04-20T00:00:00.000Z",
      "human_collaborators": [
        "Wouter van Doorn",
        "Pietro Monticone",
        "Quanyu Tang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "\"Both classifications were autonomously discovered and formally verified in Lean by Aristotle.\" The appendix documents the prompts: Aristotle was asked to characterize the two cases separately, then combine them; it even adopted a stronger definition than the source paper's and the Lean code verifies the equivalence explicitly.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2604.18869",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Nathanson's Problems on Product Intersection Sets",
          "url": "https://vibemathed.com/problem/nathanson-product-intersection-sets",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
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      "id": "vibemathed:nathanson-product-intersection-sets:verification",
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      "solution_event_id": "vibemathed:nathanson-product-intersection-sets:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Discovered and kernel-checked in Lean by the same system; the human authors audited the informal-to-formal correspondence. Tier: Aristotle wrote both proof and formal statements (and at one point adopted a stronger definition than the source paper's, caught by the authors) - exactly the failure mode an independent statement audit exists for, and none has happened.",
      "sources": [
        {
          "label": "VibeMathed: Nathanson's Problems on Product Intersection Sets",
          "url": "https://vibemathed.com/problem/nathanson-product-intersection-sets",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2604.18869",
          "kind": "primary-mathematical-source",
          "license": null
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      "label": "arXiv",
      "url": "https://arxiv.org/abs/2604.18869",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Nathanson's Problems on Product Intersection Sets",
      "url": "https://vibemathed.com/problem/nathanson-product-intersection-sets",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:nathanson-product-intersection-sets:signal:replay_ready",
      "problem_id": "vibemathed:nathanson-product-intersection-sets",
      "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"
}