What was already known
Silvestre had earlier disproved the broader monotonicity claim using a singular collision kernel. That construction did not cover these physical kernels.
AI involvement
Silvestre credits ChatGPT 5.6 Sol in Codex Ultra with the first proof of the physical-kernel result, working from his earlier paper and notes. Claude Code made an initial rewrite; Silvestre reinterpreted and restructured the final argument.
The first proof of the new physical-kernel claim is expressly model-origin, within a human-defined program and with substantial later human reconstruction; independent model autonomy is not supported.
Importance
Provides counterexamples in the standard Maxwell-molecule and hard-sphere settings.
Limitations
The claim concerns the specified homogeneous Boltzmann kernels and two-Maxwellian data; it does not address the Landau equation or arbitrary kernels. The paper gives an analytic proof and a human-reworked explanation, but no explicit completed proof check or machine verification of this theorem is documented.
Technical verification
The paper contains an analytic proof and Silvestre reworked it, but neither an explicit complete human check nor a machine check of this exact result is documented. Author responsibility and registry labels alone do not establish Formal.
Human understanding
Silvestre says he reinterpreted and restructured the initial model proof, and his exposition explains the asymptotic balance yielding a positive derivative.
Correction and publication history
The paper attributes the physical-kernel proof to ChatGPT and presents a human-reworked argument.
Read this published revision