{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:hyperkahler-period-index-conjecture",
  "title": "The Hyperkahler Period-Index Conjecture",
  "canonical_statement": "Huybrechts conjectured that for every Brauer class alpha on a hyperkahler variety X, the index divides the period raised to the power dim(X)/2, strengthening the usual period-index conjecture. Disproved on certain hyperkahler fourfolds, in both the K3^[2] and Kum^2 deformation types.",
  "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:hyperkahler-period-index-conjecture:event",
      "problem_id": "vibemathed:hyperkahler-period-index-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-10T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The counterexamples come with divisibility bounds on the Hodge-theoretic index, at 2-torsion and 5-torsion, on very general hyperkahler fourfolds.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:hyperkahler-period-index-conjecture:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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      ]
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  "attempts": [
    {
      "id": "vibemathed:hyperkahler-period-index-conjecture:attempt",
      "problem_id": "vibemathed:hyperkahler-period-index-conjecture",
      "model": "Unspecified",
      "provider": null,
      "attempted_at": "2026-08-10T00:00:00.000Z",
      "human_collaborators": [
        "Pieter Belmans",
        "James Hotchkiss"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The AI disclosure names the role precisely while leaving the model unnamed: \"The starting points for this paper were two different LLM-assisted constructions of counterexamples, obtained independently by the authors. The paper is the synthesis of these constructions, with LLMs used to help with the copyediting.\" Two authors reaching counterexamples independently with model help is a stronger signal than either would be alone.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.09436",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Hyperkahler Period-Index Conjecture",
          "url": "https://vibemathed.com/problem/hyperkahler-period-index-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
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  "verifications": [
    {
      "id": "vibemathed:hyperkahler-period-index-conjecture:verification",
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      "solution_event_id": "vibemathed:hyperkahler-period-index-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint days old, with no independent review.",
      "sources": [
        {
          "label": "VibeMathed: The Hyperkahler Period-Index Conjecture",
          "url": "https://vibemathed.com/problem/hyperkahler-period-index-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.09436",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.09436",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Hyperkahler Period-Index Conjecture",
      "url": "https://vibemathed.com/problem/hyperkahler-period-index-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:hyperkahler-period-index-conjecture:signal:watch_only",
      "problem_id": "vibemathed:hyperkahler-period-index-conjecture",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}