Partial differential equations·OpenAI Math·Result 368
The three-dimensional Ball–Evans approximation problem
One of the 372 results in OpenAI's public mathematics catalogue, filed under partial differential equations: 2 preprints, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the papers and abstracts, and embeds the animation.
In one line: OpenAI reports an answer to the three-dimensional Ball–Evans approximation problem (partial differential equations) in 2 preprints, the latest dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More, is on YouTube.
Chapter 53 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 18:09 · watch from there on YouTube →
What OpenAI claims
Resolves the three-dimensional Ball–Evans approximation problem: every W1,p homeomorphism between arbitrary bounded domains in ℝ3, for 1 ≤ p < ∞, is a strong W1,p limit of smooth diffeomorphisms onto the same target.
Papers (2)
Strong diffeomorphic approximation in three dimensions for 1≤p≤2
OpenAI's abstract We resolve the three-dimensional Ball–Evans approximation problem for 1 ≤ p ≤ 2. Every W1,p homeomorphism between arbitrary bounded domains in ℝ3 is a strong W1,p limit of smooth diffeomorphisms onto the same target.
Strong diffeomorphic approximation in three dimensions for p>2
OpenAI's abstract We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every W1,p homeomorphism between arbitrary bounded domains in ℝ3 can be approximated strongly in W1,p by smooth diffeomorphisms onto the same target.
Status
Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted 2 preprints, the latest dated 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Resolves the three-dimensional Ball–Evans approximation problem: every W1,p homeomorphism between arbitrary bounded domains in ℝ3, for 1 ≤ p < ∞, is a strong W1,p limit of smooth diffeomorphisms onto the same target.
- Where is the proof?
- In the paper, Strong diffeomorphic approximation in three dimensions for 1≤p≤2 (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 53).
Also in
- Chapter 4 of 9 in Boltzmann Equation, Hot Spots Conjecture and 7 More in Partial Differential Equations · starts at 1:04 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · Compilation · 29:20 · chapter 53 from 18:09 · description, sources and chapters →
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