The Fourier restriction conjecture in three dimensions for positively curved surfaces: bounded-data and diagonal forms
For a compact smooth positively curved surface , such as the sphere, let . Stein's restriction conjecture in asks that map to whenever and . Its bounded-data form asks for for all (Guth reached ), and its diagonal form for for all (Wang and Wu reached ). The constant function on the sphere shows fails. Do the bounded-data and diagonal estimates hold for every ?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Harmonic analysis; Fourier restriction
- Posed by
- Elias M. Stein; the manuscripts date the problem to Stein's work and cite his Some problems in harmonic analysis (1979) and the Tomas-Stein theory
- Year posed
- 1979
- Years open
- 47y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 62 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Principal manuscript, Theorem 1.1: for every compact smooth positively curved , possibly with smooth boundary, and every , ; the proof in fact bounds data into . A corollary is free Schrodinger local smoothing in two space dimensions for , . Companion: for the sphere, for every , and through Bourgain factorization the strict mixed-norm range with , . Not shown: the scaling-line endpoint that the full conjecture includes, surfaces with vanishing or mixed-sign curvature, and higher dimensions.
What the AI did
The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 24, 2026; the diagonal-extension manuscript is taken as principal, since its theorem contains the sphere bounded-data case. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.
Verification
No independent mathematician has checked this yet. Checked here: both main theorems were read against the restriction conjecture as the manuscripts describe it. The ranges are the full open ranges, and the sphere paper says explicitly that it makes no assertion on the scaling line. The proofs (a packet propagation theorem through an elliptic capacity, epsilon removal by sparse balls, chart decomposition for general elliptic surfaces) were not refereed. No Lean formalization. Dependence: the diagonal theorem takes as inputs the companion's packet theorem and the three-dimensional Kakeya maximal theorem from family 074, both unreviewed. The sphere paper says it also gives an independent route to that Kakeya estimate, so families 074 and 077 stand or fall partly together.