{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:mckean-entropy-production-conjecture",
  "title": "McKean Entropy-Production Conjecture",
  "canonical_statement": "In 1966, McKean asked whether the entropy production of the Boltzmann equation must be monotone decreasing in time. We show that this is not the case even in the space-homogeneous setting for the Boltzmann collision operator with a constant angular cross section and the kinetic parameter $\\gamma\\in[0,1]$. This recovers the classical case of hard spheres and the simplest case of Maxwell molecules. Our examples are radially symmetric mixtures of Maxwellians.",
  "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:mckean-entropy-production-conjecture:event",
      "problem_id": "vibemathed:mckean-entropy-production-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-09-01T00:00:00.000Z",
      "title": "arXiv",
      "summary": "For the space-homogeneous Boltzmann equation in dimension $3$ with collision kernels\n$$B=\\frac{1}{4\\pi}|v-v_*|^\\gamma,\\qquad \\gamma\\in[0,1],$$\nthe paper constructs smooth, positive, radial mixtures $f_R=(1-p)M_1+pM_R$ for which $\\partial_tD(f_R)>0$ for sufficiently large $R$. Thus entropy production need not decrease even for Maxwell molecules ($\\gamma=0$) or hard spheres ($\\gamma=1$), negatively resolving McKean's monotonicity question for these physically standard kernels.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:mckean-entropy-production-conjecture:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:mckean-entropy-production-conjecture:attempt",
      "problem_id": "vibemathed:mckean-entropy-production-conjecture",
      "model": "GPT-5.6 Sol; Claude",
      "provider": "OpenAI; Anthropic",
      "attempted_at": "2026-09-01T00:00:00.000Z",
      "human_collaborators": [
        "Luis Silvestre"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A first version of the proof was obtained by GPT-5.6 Sol running in Codex Ultra with access to Silvestre's earlier paper and research notes. Claude Code then rewrote the initially difficult-to-read proof, after which Silvestre reinterpreted, restructured, and checked the argument and took responsibility for the final proof. The substantive counterexample proof therefore originated from GPT-5.6 Sol under access to author-supplied mathematical context.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.01753",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: McKean Entropy-Production Conjecture",
          "url": "https://vibemathed.com/problem/mckean-entropy-production-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:mckean-entropy-production-conjecture:verification",
      "problem_id": "vibemathed:mckean-entropy-production-conjecture",
      "solution_event_id": "vibemathed:mckean-entropy-production-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The arXiv preprint contains a complete analytic proof by a leading researcher in kinetic equations, who states that he reinterpreted, restructured, and assumes full responsibility for the final argument. No independent expert review, peer review, or formal proof-assistant verification is currently documented.",
      "sources": [
        {
          "label": "VibeMathed: McKean Entropy-Production Conjecture",
          "url": "https://vibemathed.com/problem/mckean-entropy-production-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2609.01753",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2609.01753",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: McKean Entropy-Production Conjecture",
      "url": "https://vibemathed.com/problem/mckean-entropy-production-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:mckean-entropy-production-conjecture:signal:watch_only",
      "problem_id": "vibemathed:mckean-entropy-production-conjecture",
      "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"
}