OpenAI reports Boltzmann nonuniqueness with exact local conservation

The three-dimensional periodic hard-sphere Boltzmann equation has one nonnegative initial density with bounded velocity support and finite mass, energy and absolute entropy that yields two distinct global entropy solutions, both satisfying exact local conservation of mass, momentum and kinetic energy and strong L¹ continuity.

Sources for this record

What was already known

The companion paper had constructed two periodic renormalized solutions with the same data, local mass conservation and global energy control. General renormalized solution theory can carry local momentum and energy defects, leaving nonuniqueness within the exact locally conservative class unresolved.

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Major

Shows that enforcing every local collision-invariant balance does not restore uniqueness.

Limitations

The linked Lean note checks the earlier periodic nonuniqueness construction without exact local momentum and kinetic-energy conservation; it does not check this refinement.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The OpenAI-authored manuscript reports and argues the result, but the reviewed sources do not document a completed check of the full assessed claim.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.

Read this published revision