The Ball–Evans approximation problem in three dimensions
From OpenAI Aftermath
The Ball–Evans approximation problem asks whether a Sobolev homeomorphism between domains can be approximated in the Sobolev norm by smooth diffeomorphisms onto the same target. The question comes from nonlinear elasticity, where injectivity expresses noninterpenetration of matter and convolution does not preserve it; Ball attributes it to Evans. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), both dated 24 September 2026, claim a positive answer in three dimensions for every finite exponent. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introductions of the manuscripts:
- In the plane, Iwaniec, Kovalev and Onninen proved strong diffeomorphic approximation for 1 < p < ∞, and Hencl and Pratelli established the endpoint p = 1, including approximation by locally finite piecewise affine homeomorphisms. Daneri and Pratelli and Pratelli treated related planar problems with control of the inverse.
- Higher dimensions have a genuine obstruction: Hencl and Vejnar initiated counterexamples at p = 1, and Campbell, Hencl and Tengvall corrected and extended them to 1 ≤ p < ⌊n/2⌋ in dimensions n ≥ 4. In dimension three, Hencl and Malý's orientation theorem gives a nonnegative Jacobian for sense-preserving Sobolev homeomorphisms, but rank-deficient derivatives still require separate constructions.
- Mora-Corral and Pratelli smoothed piecewise affine homeomorphisms in the plane, Campbell and Soudský gave C1 homeomorphic approximation of piecewise affine maps in every dimension, and Campbell, D'Onofrio and Vítek proved diffeomorphic smoothing of locally finite piecewise affine homeomorphisms in dimensions three and four; these results do not approximate an arbitrary Sobolev homeomorphism.
The original results
- Claim (1 ≤ p ≤ 2). The main theorem of Strong diffeomorphic approximation in three dimensions for 1 ≤ p ≤ 2 states that every W1,p homeomorphism between arbitrary bounded domains in ℝ3 is a strong W1,p limit of smooth diffeomorphisms onto the same target. No boundary regularity, inverse Sobolev regularity or nonvanishing Jacobian is assumed.
- Claim (p > 2). Strong diffeomorphic approximation in three dimensions for p > 2 states the same conclusion for every finite p > 2. Together the two manuscripts claim the three-dimensional problem for every 1 ≤ p < ∞.
- Method, as described: the proof first constructs a locally bi-Lipschitz map onto the target with small strong Sobolev error and then proves a smoothing theorem with arbitrarily fine pointwise control, using three-dimensional piecewise linear topology in Hamilton's relative formulation. The two exponent ranges use different constructions; the first manuscript states that smooth invertibility of each approximant is not a claim of Sobolev convergence of the inverses.
- Sources (openai/math repository, commit
7ef5c0c): Strong diffeomorphic approximation in three dimensions for 1 ≤ p ≤ 2; Strong diffeomorphic approximation in three dimensions for p > 2; both dated 24 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | Strong diffeomorphic approximation in three dimensions, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains topology and Sobolev convergence, the combined approximation claim, why rank-deficient derivatives cannot be discarded and the two constructions, with proof outlines on the subcritical cell estimate behind the exponent split, collapse onto lower-dimensional strata, routing, and sharp energy. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_174 (1 ≤ p ≤ 2) and rank_056 (p > 2) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap and no minor notes in either manuscript. Both reviews describe themselves as partial audits of long manuscripts, of about 4,500 and 10,000 lines of LaTeX, that checked the load-bearing analytic steps rather than every line. | Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The reviews are published under CC BY 4.0. |
| 2026-10-19 | AI Proved What?, episode 368 Smooth any 3-D deformation?, remade version of 24 October 2026 | Reading (exposition) | Video explainer of the claims. The episode states that convergence of the inverses or of elastic energies, matching boundary values and dimensions four and higher are not claimed, that neither paper states approximation by piecewise affine maps, as in Ball's original question, as a theorem, and that two trackers rate the result as major while a third calls it partial because Ball asked in every dimension. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, 62 + 130 pages, both dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that it found no specialist verdict and no competing claim. |
Notes
- The background summarizes the manuscripts' own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
