{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:simonovits-product-conjecture",
  "title": "The Simonovits Product Conjecture",
  "canonical_statement": "Simonovits conjectured that if a forbidden family $\\mathcal{F}$ with $p(\\mathcal{F}) > 1$ has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of $p$ graphs, each extremal for a family of chromatic number two. Disproved by a fixed finite family $\\mathcal{L}$ with $p(\\mathcal{L}) = 2$ and $\\mathrm{ex}(n,\\mathcal{L}) > t_2(n) + cn^{3/2}$ that nevertheless has, at every large order, an extremal graph with connected complement and hence no nontrivial join decomposition. The same construction disproves the Weak Product Conjecture of Furedi and Simonovits.",
  "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:simonovits-product-conjecture:event",
      "problem_id": "vibemathed:simonovits-product-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-03T00:00:00.000Z",
      "title": "arXiv:2608.02115 - A finite forbidden family with superlinear surplus and non-join extremal graphs",
      "summary": "one construction disproves both the product conjecture and its weak form",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:simonovits-product-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:simonovits-product-conjecture:method"
      ],
      "source_assertion_ids": [
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      ]
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  ],
  "attempts": [
    {
      "id": "vibemathed:simonovits-product-conjecture:attempt",
      "problem_id": "vibemathed:simonovits-product-conjecture",
      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-03T00:00:00.000Z",
      "human_collaborators": [
        "Chuandong Xu"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper's comment credits the counterexample to GPT-5.6 Sol, found during a Codex project devoted to the Product Conjecture. The exact extremal-number and equality-case analysis around it is the author's.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
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          "label": "arXiv:2608.02115 - A finite forbidden family with superlinear surplus and non-join extremal graphs",
          "url": "https://arxiv.org/abs/2608.02115",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Simonovits Product Conjecture",
          "url": "https://vibemathed.com/problem/simonovits-product-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "cost_usd": null,
      "independence": "unknown"
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  "verifications": [
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      "solution_event_id": "vibemathed:simonovits-product-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Single-author arXiv preprint; not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: The Simonovits Product Conjecture",
          "url": "https://vibemathed.com/problem/simonovits-product-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2608.02115 - A finite forbidden family with superlinear surplus and non-join extremal graphs",
          "url": "https://arxiv.org/abs/2608.02115",
          "kind": "primary-mathematical-source",
          "license": null
        }
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  "sources": [
    {
      "label": "arXiv:2608.02115 - A finite forbidden family with superlinear surplus and non-join extremal graphs",
      "url": "https://arxiv.org/abs/2608.02115",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Simonovits Product Conjecture",
      "url": "https://vibemathed.com/problem/simonovits-product-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:simonovits-product-conjecture:signal:watch_only",
      "problem_id": "vibemathed:simonovits-product-conjecture",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
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  "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"
}