OpenAI's 272 Pages Take Aim at 4D Kakeya, Weeks After Wang Hong's 3D Proof
新智元 · wechat · 2026-10-10
Among OpenAI's 722 Oct 6 manuscripts, two target the Kakeya conjecture: a 97-page proof of the stronger 3D maximal function version (Wang Hong and Zahl proved the set version in Feb 2025), and a 175-page proof that every four-dimensional Kakeya set has full Hausdorff dimension — beyond the prior 3.059 bound, with only the 'sticky' case known in 4D.
The 3D manuscript demands a sharper lower bound controlling the fraction of each tube that is 'painted,' tracked across scales with geometric estimates and entropy analysis. The 4D proof treats quadratic-surface clustering (flagged by Guth) as a clue, describes local line relations with quadratic polynomials, and reuses a weighted estimate from the 3D manuscript — itself rooted in Guth-Wang-Zahl. Notably both proofs still lean on Wang and collaborators' prior estimates.
If verified, the 4D full-dimension result generalizes beyond sticky sets; the 4D maximal version and dimensions ≥5 remain open. Both manuscripts await independent verification.
Related event: OpenAI Drops 722 AI-Generated Math Papers in One Day, Sparking Backlash(17 posts)→
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