{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples",
  "title": "Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)",
  "canonical_statement": "For the integral $\\mathcal{I}(h)$ over the unit interval and the torus, the paper gives the three-term Laurent polynomial $f(x,z)=(1-z^{-1})((1-x)+xz)$ with $\\mathcal{I}(f^n)=0$ but $\\mathcal{I}(z^{-1}f^n)=(-1)^{n-1}/(n+1)\\neq0$. This disproves the $xz$-conjecture with one interval and one torus variable, shows $\\ker\\mathcal{I}$ is not a Mathieu–Zhao subspace, and by padding yields counterexamples for SU(2).",
  "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:xz-conjecture-mathieu-su2-counterexamples:event",
      "problem_id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-21T00:00:00.000Z",
      "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:xz-conjecture-mathieu-su2-counterexamples:attempt"
      ],
      "method_family_ids": [
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      "source_assertion_ids": [
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      ]
    }
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  "attempts": [
    {
      "id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples:attempt",
      "problem_id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples",
      "model": "ChatGPT 5",
      "provider": "OpenAI",
      "attempted_at": "2026-07-21T00:00:00.000Z",
      "human_collaborators": [
        "Christopher D. Long"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Under a heading on AI provenance and author responsibility, the paper states the counterexample and its lift were discovered by ChatGPT 5.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.19012",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)",
          "url": "https://vibemathed.com/problem/xz-conjecture-mathieu-su2-counterexamples",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples:verification",
      "problem_id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples",
      "solution_event_id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No independent review, but the counterexample is a three-term Laurent polynomial with closed-form moment identities, checkable by hand. Preprint, not refereed.",
      "sources": [
        {
          "label": "VibeMathed: Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)",
          "url": "https://vibemathed.com/problem/xz-conjecture-mathieu-su2-counterexamples",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
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          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.19012",
          "kind": "primary-mathematical-source",
          "license": null
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      ]
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  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2607.19012",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)",
      "url": "https://vibemathed.com/problem/xz-conjecture-mathieu-su2-counterexamples",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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  "recommended_exposure": "result_only",
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    {
      "id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples:signal:watch_only",
      "problem_id": "vibemathed:xz-conjecture-mathieu-su2-counterexamples",
      "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"
}