{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:period-index-conjecture",
  "title": "The Period-Index Conjecture",
  "canonical_statement": "For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \\geq 3$ there is a $d$-dimensional variety over $k$ carrying a Brauer class that violates it, for Hodge-theoretic reasons. For $d = 3$ the construction needs no uncountability, so the conjecture fails already over $\\overline{\\mathbf{Q}}$.",
  "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:period-index-conjecture:event",
      "problem_id": "vibemathed:period-index-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-04T00:00:00.000Z",
      "title": "arXiv:2608.03684 - The period-index conjecture is false",
      "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",
      "attempt_ids": [
        "vibemathed:period-index-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:period-index-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:period-index-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:period-index-conjecture:attempt",
      "problem_id": "vibemathed:period-index-conjecture",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-08-04T00:00:00.000Z",
      "human_collaborators": [
        "Alexander Perry"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper's AI disclosure is unusually specific about a partial success. Prompted to find a counterexample by the Hodge-theoretic strategy, ChatGPT produced an example that was flawed, but whose shape survived into the final solution: a quotient of the same form, with an abelian surface, a genus two curve and the group Z/4. The author refined that into the working construction.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2608.03684 - The period-index conjecture is false",
          "url": "https://arxiv.org/abs/2608.03684",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Period-Index Conjecture",
          "url": "https://vibemathed.com/problem/period-index-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:period-index-conjecture:verification",
      "problem_id": "vibemathed:period-index-conjecture",
      "solution_event_id": "vibemathed:period-index-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint, not yet peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: The Period-Index Conjecture",
          "url": "https://vibemathed.com/problem/period-index-conjecture",
          "kind": "source-assertion",
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        },
        {
          "label": "arXiv:2608.03684 - The period-index conjecture is false",
          "url": "https://arxiv.org/abs/2608.03684",
          "kind": "primary-mathematical-source",
          "license": null
        }
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  "sources": [
    {
      "label": "arXiv:2608.03684 - The period-index conjecture is false",
      "url": "https://arxiv.org/abs/2608.03684",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Period-Index Conjecture",
      "url": "https://vibemathed.com/problem/period-index-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:period-index-conjecture:signal:watch_only",
      "problem_id": "vibemathed: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"
}