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The three-dimensional Ball–Evans approximation problem

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Resolves the three-dimensional Ball–Evans approximation problem: every \(W^{1,p}\) homeomorphism between arbitrary bounded domains in ℝ3, for \(1\le p\lt \infty\), is a strong \(W^{1,p}\) limit of smooth diffeomorphisms onto the same target.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

What the papers state

Theorem 1.1

In Strong diffeomorphic approximation in three dimensions for p>2, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Let \Omega,\Lambda\subset\R^3 be bounded domains, let 2<p<\infty, and let f:\Omega\to\Lambda be a homeomorphism in W^{1,p}(\Omega;\R^3). There exist C^\infty diffeomorphisms f_j:\Omega\to\Lambda, each belonging to W^{1,p}(\Omega;\R^3), such that

\int_\Omega\bigl( |f_j-f|^p+|Df_j-Df|^p\bigr)\,dx\longrightarrow0.

Theorem 1.1

In Strong diffeomorphic approximation in three dimensions for 1≤p≤2, 24 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

Let \Omega,\Lambda\subset\R^3 be nonempty bounded connected open sets, let 1\le p\le2, and let f:\Omega\to\Lambda be a homeomorphism in W^{1,p}(\Omega;\R^3). There exist C^\infty diffeomorphisms f_j:\Omega\to\Lambda, each belonging to W^{1,p}(\Omega;\R^3), such that

\int_\Omega\bigl( |f_j-f|^p+|Df_j-Df|^p\bigr)\,dx\longrightarrow0.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

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Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

  • Family 368 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 368 The release has no Lean statement for this family. Open the form