{
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  "source": "vibemath",
  "problem_id": "vibemathed:batyrev-stringy-hodge-numbers",
  "title": "Batyrev's Stringy Hodge Number Conjecture",
  "canonical_statement": "For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)",
  "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",
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    {
      "id": "vibemathed:batyrev-stringy-hodge-numbers:event",
      "problem_id": "vibemathed:batyrev-stringy-hodge-numbers",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-21T00:00:00.000Z",
      "title": "arXiv:2607.19184",
      "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_co_developed",
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        "vibemathed:batyrev-stringy-hodge-numbers:attempt"
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      "id": "vibemathed:batyrev-stringy-hodge-numbers:attempt",
      "problem_id": "vibemathed:batyrev-stringy-hodge-numbers",
      "model": "GPT",
      "provider": "OpenAI",
      "attempted_at": "2026-07-21T00:00:00.000Z",
      "human_collaborators": [
        "Matthew Satriano",
        "Jeremy Usatine"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Satriano and Usatine found the counterexample with the assistance of GPT: $X = M_0 \\times \\mathbb{P}^1$, where $M_0$ is the coarse moduli space of rank-2 semistable bundles with trivial determinant over a genus-3 curve. $X$ is a 7-dimensional projective variety with Gorenstein terminal singularities whose stringy $E$-function is a polynomial, yet its stringy Hodge number $h^{2,5}_{\\mathrm{st}}(X) = -1$ is negative.",
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          "label": "arXiv:2607.19184",
          "url": "https://arxiv.org/abs/2607.19184",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Batyrev's Stringy Hodge Number Conjecture",
          "url": "https://vibemathed.com/problem/batyrev-stringy-hodge-numbers",
          "kind": "source-assertion",
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      "id": "vibemathed:batyrev-stringy-hodge-numbers:verification",
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      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint 2607.19184 (21 Jul 2026) by Matthew Satriano and Jeremy Usatine. The proof is short and fully explicit: the stringy $E$-function is written out and its $u^2 v^5$ coefficient gives $h^{2,5}_{\\mathrm{st}} = -1$, so it is hand-verifiable. A domain-expert preprint, not yet peer-reviewed. Distinct from the unrelated Batyrev-Manin conjecture on rational points.",
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          "label": "VibeMathed: Batyrev's Stringy Hodge Number Conjecture",
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      "url": "https://arxiv.org/abs/2607.19184",
      "kind": "primary-mathematical-source",
      "license": null
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    {
      "label": "VibeMathed: Batyrev's Stringy Hodge Number Conjecture",
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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.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}