Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps \(L^3(\mathbb R^3)\) to \(L^3(S^2)\) with norm \(O_\varepsilon(\delta^{-\varepsilon})\) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
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In The Kakeya maximal conjecture in three dimensions, 23 September 2026
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For every \varepsilon>0 there exists a finite constant C_\varepsilon such that, for every 0<\delta<1 and every f\in L^3(\R^3),
In Every four-dimensional Kakeya set has full Hausdorff dimension, 24 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let K\subset\R^4. Suppose that for every e\in S^3 there is a point a_e\in\R^4 such that
Then \dim_H K=4.
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The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.
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Scientific American (byline not recorded), The most exciting claims from OpenAI’s heap of new proofs, 8 October 2026. Source
Describes the four-dimensional Kakeya result as extending the three-dimensional case of Wang and Zahl.
mo271, Wikipedia/Kakeya: the Kakeya set conjecture holds in dimension 4 (issue 6910), 7 October 2026. Source
The release manuscript proves every four-dimensional Kakeya set has full Hausdorff dimension.
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An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Would win a Fields Medal”. The page says that nothing on it is verified.
Author not stated. Source
Robert Hart, ‘Pure insanity’: Mathematicians will need years to make sense of OpenAI’s latest drop, 9 October 2026. Source
Jared Duker Lichtman (Stanford) puts the four-dimensional Kakeya result among tens of results they would rate at the level of major breakthroughs, alongside progress on the Riemann hypothesis and a special case of the Hodge conjecture.
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Why this passage is shown
Theorem 1.1
In The Kakeya maximal conjecture in three dimensions, 23 September 2026
Why this passage is shown
Theorem 1.1
In Every four-dimensional Kakeya set has full Hausdorff dimension, 24 September 2026