OpenAI claims 3D restriction and Bochner–Riesz conjectures

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Key facts

• Family 077 (catalogue): ‘Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions … bounded from Lᵖ(Σ) to Lᵖ(ℝ³) for every p > 3.’ Two papers dated 24 Sept 2026: ‘Elliptic capacity propagation and Fourier restriction to the sphere’ (103 pp.; ‘the Fourier extension operator maps L∞(S²) boundedly into Lᵖ(ℝ³) for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp’) and ‘Diagonal Fourier extension for positively curved surfaces in three dimensions’ (25 pp.)
• Prior state of the art, as the sphere paper itself summarises it: Guth’s polynomial partitioning gave the bounded-data estimate for p > 13/4, Hong Wang’s broom method p > 42/13, and Wang–Wu the diagonal estimate for p > 22/7. The claim closes the remaining gap to p > 3
• Dependencies: the diagonal paper ‘uses two theorem inputs’: the packet propagation theorem of the sphere paper and ‘The Kakeya maximal theorem [12, Theorem 1.1]’, i.e. OpenAI’s family 074 manuscript ‘The Kakeya maximal conjecture in three dimensions’ (23 Sept). The restriction claim therefore stands or falls with the Kakeya maximal claim, which Levent Alpöge linked to a scooping dispute
• Family 078: ‘Resolves the three-dimensional Bochner–Riesz conjecture in its strict range’ (paper ‘Bochner–Riesz multipliers in three dimensions’, 114 pp.; main estimate: Tδ bounded on L³(ℝ³) for every δ > 0, then interpolation and duality)
• Family 079: ‘Resolves Sogge’s local smoothing conjecture for the Euclidean wave equation in three spatial dimensions’ (paper ‘Critical local smoothing for the three-dimensional wave equation’, 165 pp.; critical L³ estimate with every positive Sobolev loss)
• Family 080: ‘Proves almost-everywhere convergence e^{itΔ}f → f … for every f ∈ H^{n/(2(n+1))}(ℝⁿ) and every dimension n ≥ 2’, the sharp Sobolev endpoint of Carleson’s problem (planar paper ‘Endpoint convergence for the planar Schrödinger equation’, 83 pp.)
• Verification: the catalogue entries for 074, 077, 078, 079 and 080 carry no Lean link (073 Falconer and 075 L log L Fourier convergence do), lean/docs/077.md returns 404, and neither 077 PDF mentions Lean. Korobenko–Sawyer’s remark that OpenAI proved the diagonal case ‘with L∃∀N certification’ is not supported by the repository as checked on 19 Oct
• First follow-up: Lyudmila Korobenko (Reed College) and Eric T. Sawyer (McMaster), ‘Probabilistic Fourier extension and the Schrödinger equation’, arXiv 2610.22634 (submitted 15 Oct, 8 pp.): ‘Just recently, OpenAI has claimed to prove the diagonal inequality p = q in the case of d = 3 dimensions’; their averaged Sobolev regularity result for the Schrödinger equation is the probabilistic counterpart of ‘the deterministic analogue in three dimensions … obtained recently by OpenAI’. The paper has no AI-use statement