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  "problem_id": "vibemathed:homogeneous-polynomial-lyapunov-conjecture",
  "title": "The Homogeneous Polynomial Lyapunov Converse Conjecture",
  "canonical_statement": "Does every globally asymptotically stable homogeneous polynomial vector field admit a homogeneous polynomial Lyapunov function? No. A planar homogeneous cubic vector field with integer coefficients is globally asymptotically stable yet admits no positive definite homogeneous polynomial with nonpositive Lie derivative, and no real-analytic Lyapunov function even locally, though it does have exponential and rational strict Lyapunov functions.",
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      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-17T00:00:00.000Z",
      "title": "arXiv:2607.16171 - A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field Without a Homogeneous Polynomial Lyapunov Function",
      "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.",
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      "attempted_at": "2026-07-17T00:00:00.000Z",
      "human_collaborators": [
        "Jun Liu",
        "Maxwell Fitzsimmons"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The acknowledgement is direct: the authors used GPT-5.6-Sol Pro to discover the counterexample and to produce initial versions of the mathematical arguments in Sections II to IV. Codex assisted with drafting and with the Lean 4 formalization. The authors independently verified all AI-generated content.",
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    "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"
}