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Precise 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.
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fidelity, correctness, priority, or novelty.
What AI did
GPT-5.6 Sol; Claude
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.
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.
For the space-homogeneous Boltzmann equation in dimension $3$ with collision kernels
$$B=\frac{1}{4\pi}|v-v_*|^\gamma,\qquad \gamma\in[0,1],$$
the 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.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
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.