The David–Semmes problem in higher codimension
From OpenAI Aftermath
The David–Semmes problem in higher codimension asks whether an n-dimensional Ahlfors–David regular measure in Rd whose n-dimensional Riesz transform is bounded on L2 must be uniformly rectifiable, in the intermediate dimensions 1 < n < d − 1 left open after the planar and codimension-one cases. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September 2026, claims a positive answer, with quantitative constants. The collection includes a Lean formalization of the theorem. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- David and Semmes developed uniform rectifiability as a quantitative form of rectifiability suited to singular integrals and asked whether boundedness of the Riesz transform alone forces it. Mattila, Melnikov and Verdera proved the planar case through the Cauchy integral and curvature, curvature methods also give the one-dimensional case in higher ambient dimension, and Nazarov, Tolsa and Volberg proved the codimension-one case, identifying the lack of a higher-codimension maximum principle as the main obstacle to extending their argument.
- Mas and Tolsa characterized uniform rectifiability in every integer dimension by an L2 bound for a variation of the Riesz truncations, a stronger hypothesis than the operator bound. Tolsa's 2026 ICM survey states the remaining range as an open question. Tolsa characterized ordinary rectifiability by principal values for sets of finite Hausdorff measure, and his theorem on uniform measures and the reflectionless-measure methods of Jaye and Nazarov are described as relevant to the proof.
The original result
- Claim. Theorem 1.3 of Riesz transforms and uniform rectifiability in higher codimension states that for integers d ≥ 4 and 2 ≤ n ≤ d − 2, every n-Ahlfors–David regular Radon measure on Rd whose hard truncations of the n-dimensional Riesz transform are bounded on L2 uniformly in the truncation parameter is uniformly n-rectifiable, with big pieces of Lipschitz images whose constants depend only on d, n, the regularity constant and the operator bound. A final section derives variation bounds and almost-everywhere principal values.
- Method, as described: flatness is propagated to smaller scales through a normalized-height excess. Testing the transform against normal heights gives a positive fractional energy and compactness; the limiting equation, which has an order-one Fourier symbol, forces affine limits, and a counting argument over dyadic cells supplies the packing condition of David and Semmes.
- Source. The manuscript Riesz transforms and uniform rectifiability in higher codimension at commit
1581d4c. - Formalization, as described by the repository. The scope note states that the formalization proves, for d ≥ 4 and 2 ≤ n ≤ d − 2, that an n-Ahlfors–David regular measure whose positive hard truncations have one uniform L2 operator bound has big pieces of Lipschitz images, with constants depending only on the dimensions and the stated bounds, at every support point and admissible radius, including unbounded support; the variation and principal-value corollary is not included. The Comparator statement is
RieszQuantitative.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 20 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statements) | Rates this family "full": the headline is stated, with uniform rectifiability encoded as big pieces of Lipschitz images, a standard equivalent form; the variation and principal-value corollary is not stated. It records a sandboxed Comparator pass for RieszQuantitative. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K, run 38019899278 | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the RieszQuantitative solution (2,082 s). The project reports two-kernel certification of 398 of 416 challenges of the collection. |
Repository. |
| 2026-10-10 | J. Hoffman and B. Jaye, On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group, arXiv:2510.26934v4 | Remark | The paper extends the theorem of David and Semmes to parabolic space and to all Heisenberg groups. Version 4 states that the intermediate higher-codimension cases form the longstanding David–Semmes problem and that a recent preprint, the OpenAI manuscript, claims a positive resolution; it adds that a version of its cylindrical blow-up technique also appears in that claimed solution. | arXiv preprint; version 1 of 30 October 2025, version 4 of 10 October 2026. |
| 2026-10-10 to 2026-10-14 | con-leche development journal (task #348) | Verification (second kernel) | In its run over all Comparator challenges of the collection at commit fd4aeeb, the journal's table of exceptions (time-outs, memory, errors) does not list RieszQuantitative, so the challenge was accepted by con-leche and by Lean's official kernel. |
Development journal. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
