OpenAI claims local smoothing and Kakeya in all dimensions

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Key facts

• Repository changelog (history.md, 26 Oct, verbatim): ‘Local smoothing for the Euclidean wave equation. The proof uses an auxiliary partial differential equation to establish Sogge’s local smoothing for waves in every spatial dimension at least two. It also yields the Kakeya maximal and dimension conjectures and Stein’s spherical restriction estimates throughout the interior of the conjectured range. We are releasing the model output directly, as we did in our October 6 release.’
• Theorem 1.1 (paper): for n ≥ 2, 2 ≤ p < ∞, ε > 0, ‖e^{-iT|D|}f‖{Lᵖ(ℝⁿ×[1,2])} ≤ C‖J^{s₊(p)+ε}f‖{Lᵖ}, with s₊(p) = max(0, (n−1)/2 − n/p); a focusing wave shows the threshold is sharp (Prop. 8.1)
• Abstract (verbatim): ‘We deduce the Bochner–Riesz conjecture, maximal estimates for these means, and sharp bounds for Stein’s square function. We obtain Fourier restriction estimates for the sphere, the compact paraboloid, and the truncated cone throughout the interior of the conjectured range, as well as local smoothing for the Schrödinger equation. We also prove the Kakeya and Nikodym maximal conjectures. In particular, every Euclidean Kakeya set has full Hausdorff dimension.’
• Lean scope note (lean/docs/079.md): covers the half-wave local smoothing bound for every n ≥ 2 and 2 ≤ p < ∞; uniform Bochner–Riesz bounds, spherical extension and Stein square-function bounds in their strict ranges; an Lⁿ Kakeya maximal bound with δ^(−ε) loss; and ‘Every subset of ℝⁿ containing a unit segment in each direction has Hausdorff dimension n’. Comparator file: ComparatorChallenges/LocalSmoothing.lean. We have not run the check
• Before this, family 079 held only ‘Critical local smoothing for the three-dimensional wave equation’ (165 pp., 24 Sept), with no Lean link as of 19 Oct; family 074 claimed the 3D Kakeya maximal and 4D Kakeya dimension conjectures (manuscript only)
• Human state of the art: the Kakeya set conjecture is proved in ℝ² (Davies 1971) and ℝ³ (Hong Wang and Joshua Zahl, 2025); it is open in ℝⁿ for n ≥ 4. Local smoothing was known in 2+1 dimensions (Guth–Wang–Zhang, 2020) and in partial ranges elsewhere. Local smoothing is known to imply Bochner–Riesz, which implies restriction, which implies Kakeya
• Other results in the same OpenAI announcement: ‘The Exponential Time Hypothesis is False’ (separate entry) and, an existence claim for 4D quantum field theories including QCD and pure Yang–Mills, reviewed by a group led by David Gross with the AI artifacts withheld (entry OpenAI claims 4D Yang–Mills and QCD existence)
• Unlike the ETH paper, the changelog names no review group for this paper. No expert reaction found on the evening of 26 Oct