The McKean entropy-production conjecture asks whether entropy production for the spatially homogeneous Boltzmann equation must decrease with time. Writing 
for the Boltzmann entropy functional and 
, the conjectured inequality is 
. The usual entropy inequality gives 
and does not imply this stronger assertion. McKean's
question and its relation to stronger alternating-sign assertions are discussed by
Silvestre (2026).
Silvestre (2026) constructed counterexamples for constant angular cross section and collision kernels with kinetic factor 
throughout 
. This includes Maxwell molecules at 
and hard spheres at 
. The examples are radially symmetric mixtures of Maxwellian
densities. They extend the failure of monotonicity to these classical kernels, rather
than merely to a specially chosen collision law.
GPT-5.6 Sol supplied the initial argument, Claude helped rewrite it, and Silvestre reorganized and checked the proof. Independent external verification had not been reported as of Sep. 7, 2026.
