OpenAI reports strong diffeomorphic approximation in three dimensions

Every W^{1,p} homeomorphism between nonempty bounded connected open domains in R³, for 1≤p≤2, is a strong W^{1,p} limit of smooth diffeomorphisms onto the same target domain.

Sources for this record

What was already known

Strong target-preserving diffeomorphic approximation was known in the plane for all p≥1. The three-dimensional Ball–Evans problem remained open in the low-exponent range addressed here.

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

Resolves the three-dimensional Ball–Evans approximation problem for 1≤p≤2.

Limitations

No smooth boundary, boundary extension, inverse Sobolev regularity or nonvanishing Jacobian is assumed; the original target domain is preserved.

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, and prior-work review for non-notifying local inclusion; technical verification remains Reported.

Read this published revision