Fourier extension for positively curved surfaces in three dimensions
From OpenAI Aftermath
Fourier extension for positively curved surfaces in three dimensions concerns the Fourier restriction problem for curved surfaces in R3: for which exponents the Fourier extension operator of a compact positively curved surface, such as the sphere, is bounded into Lp(R3). Two manuscripts dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim the bounded-data estimate from L∞(S2) to Lp(R3) for every p > 3, the full conjectured open range for the sphere, and the diagonal estimate from Lp(Σ) to Lp(R3) for every p > 3 and every compact smooth positively curved surface Σ, which contains the sphere case. The collection has no Lean formalization of these results. This page summarizes the published claims 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 introductions of the manuscripts:
- The restriction problem originates in Stein's work, and the Tomas–Stein theory gives the L2 to L4 extension estimate. The work of Carleson and Sjölin and of Hörmander, Fefferman's ball-multiplier obstruction and Córdoba's multiplier estimates form the analytic and geometric background, and Bourgain connected restriction with the geometry of tubes. The bilinear framework of Tao, Vargas and Vega, Wolff's cone theorem, Tao's paraboloid theorem, the multilinear estimates of Bennett, Carbery and Tao and the approach of Bourgain and Guth developed the role of wave packets and transverse interactions.
- Guth's polynomial partitioning gave the bounded-data estimate for p > 13/4, Shayya and Kim obtained diagonal estimates in that range, Wang's broom method improved bounded-data restriction for the paraboloid, and Wang and Wu proved the diagonal estimate for p > 22/7 on compact positively curved surfaces.
The original results
- Claim. Theorem 1.1 of Elliptic capacity propagation and Fourier restriction to the sphere states that for every p > 3 there is a constant Cp such that the extension of every bounded measurable function g on S2 satisfies ‖Eg‖Lp(R3) ≤ Cp‖g‖L∞; the manuscript notes that the constant function shows the range p > 3 to be sharp, and it makes no endpoint assertion. Theorem 1.1 of Diagonal Fourier extension for positively curved surfaces in three dimensions states that for every compact smooth positively curved surface Σ ⊂ R3, possibly with smooth boundary, and every 3 < p < ∞, the extension operator is bounded from Lp(Σ) to Lp(R3). Its Corollary 1.2 gives local smoothing for the free Schrödinger evolution in two space dimensions, from Ws,p(R2) to Lp(R2 × [0, 1]) for 3 < p < ∞ and s > 2 − 6/p.
- Method, as described: the sphere manuscript introduces an elliptic capacity that limits how much mass can concentrate simultaneously in differently oriented tests, proves a propagation theorem for this quantity along conditional laws of wave-packet arrays, and then removes a small spatial power loss. The diagonal manuscript continues the phase of each elliptic chart into the class allowed by the packet theorem, sums the elliptic-capacity factor with a weighted tube estimate derived from the Kakeya maximal theorem, and passes to L3-type data through a joint estimate on sparse families of balls.
- Sources. The manuscripts Elliptic capacity propagation and Fourier restriction to the sphere and Diagonal Fourier extension for positively curved surfaces in three dimensions at commit
1581d4c. - Formalization. The repository has no Lean scope note for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 20 October 2026) |
|---|---|---|---|---|
| 2026-10-13 | M. Colbrook and contributors, AIM — Open Applied Problems (GitHub), problem 208 | Catalogue | The collection records its problem 208, sharp Fourier extension from the sphere in three dimensions, as solved by the sphere manuscript, which it states covers the same operator, measure, inputs and every p > 3; it adds that the diagonal manuscript also implies the sphere case. | The status review is described as "an AI statement-level review of a published manuscript, not a fresh independent audit of its complete proof or a Lean kernel replay". The collection is not affiliated with the American Institute of Mathematics. |
| 2026-10-15 | L. Korobenko, E. T. Sawyer, Probabilistic Fourier extension and the Schrödinger equation, arXiv:2610.22634v1 | Remark (citation) | Applies a probabilistic Fourier extension theorem for the paraboloid in d dimensions to the global initial value problem for the Schrödinger equation, with sharp averaged Sobolev regularity over martingale transforms of the data. The abstract states that the deterministic analogue in three dimensions has been obtained recently by OpenAI; the paper cites Corollary 1.2 of the diagonal manuscript for that case and describes the diagonal inequality in three dimensions as claimed by OpenAI. | arXiv preprint, version 1 of 15 October 2026. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews of the sphere manuscript and the diagonal manuscript in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Both reviews report no gap. The review of the sphere manuscript has no minor notes. The review of the diagonal manuscript, which states that the paper was checked line by line, has one note on the Schrödinger corollary: it cites a theorem of Lee, Rogers and Seeger with a Besov summation exponent and an admissible exponent range that the reviewer could not confirm offline, and it states that the corollary still follows from the case of equal exponents, the remaining question being the exponent range. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
