What was already known
Ground-state condensation for the dilute hard-sphere gas was known; positive-temperature condensation in the exact interacting three-dimensional hard-sphere canonical state remained unproved.
AI involvement
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
Establishes macroscopic coherence at a fixed positive temperature in the exact dilute hard-sphere model.
Limitations
The linked Lean note checks zero-temperature ground-state condensation only; it gives no check of the positive-temperature Gibbs-state claim.
Technical verification
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 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
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision