OpenAI Releases 722 Math Manuscripts, Claiming Kakeya Advances in Three and Four Dimensions

The first paper, 97 pages long, addresses the three-dimensional Kakeya maximal function problem. In this version, thin needles are treated as tubes, and each tube contains a solid portion of fraction λ; the question is whether the union of the solid parts has volume of order at least λ³. Wang and Zahl's theorem gave a bound of λ^{K(ε)}, and the OpenAI paper states in its introduction that a λ³ bound was identified as the next step after their work. The maximal function version is tied to Fourier analysis and to Fefferman's 1971 use of Kakeya sets to show that high-dimensional ball multipliers are unbounded except on L², the report said.

The second paper, 175 pages long, claims a Hausdorff dimension of 4 for the four-dimensional Kakeya set. Earlier progress, according to the report, included Wolff's 1995 proof of dimension at least 3; Guth and Zahl's polynomial method reaching 3+1/40, about 3.025; and Katz and Zahl's 2019 plane-brush result of 3.059. OpenAI's paper uses quadratic polynomials to fit local structures across scales and track lines that keep clustering, with the word polynomial appearing 213 times. It cites a lemma from OpenAI's three-dimensional paper, linking the two manuscripts. The four-dimensional paper claims only the Hausdorff dimension version, not the four-dimensional maximal function conjecture, and does not complete dimensions five and above, where it gives a projection lower bound. The 074 papers have not been formalized in Lean and have not been peer-reviewed; OpenAI's README says some unformalized results may be wrong, according to QbitAI.

Wang Hong's earlier work is central to the new manuscripts. She and Zahl proved the sticky Kakeya case in 2022, the Assouad dimension version in 2024, and released a 127-page complete proof in February 2025 that resolved the Hausdorff and Minkowski dimension versions of the three-dimensional Kakeya set conjecture. Nets Katz called it a once-in-a-century result. Wang and Ren Kang also solved the planar Furstenberg set conjecture. OpenAI's three-dimensional paper uses a simplified proof by Guth, Wang and Zahl—Guth was Wang's doctoral adviser at MIT—and a planar Furstenberg estimate by Ren and Wang; the four-dimensional paper uses a dot-product theorem by Wang and Zahl. If confirmed, the OpenAI papers would build additional layers on the framework Wang established. After winning the Fields Medal, Wang described AI as an initiative booster for mathematical research, saying that asking questions, creating concepts and building theories remain the core work of mathematicians, the report said.

Reaction has been split. Alek Dimitriev, an Anthropic employee, wrote on X after the Fields Medal announcement that this Fields Medal class would be the last humans receive; Timothy Gowers replied that he had similar thoughts but expected a lag, so humans might last until 2030. After the OpenAI release, Terence Tao led the AHM in issuing a joint statement opposing OpenAI. The statement said mathematicians had not asked for the work and that releasing more than 700 files at once displayed power rather than scholarship, and it urged mathematicians to stop cooperating with OpenAI and return to a scientific vision centered on human understanding. Tao, who has actively used AI in research, objects to treating mass solutions to famous problems as a product demonstration, the report said; once AI solves a problem, it cannot be returned to unsolved status, potentially cutting off new methods and understanding human mathematicians might have developed. Review has also become a bottleneck: few people can carefully read a 175-page four-dimensional Kakeya proof, and authorship and credit are becoming confused. Others are more enthusiastic. Daniel Litt, an assistant professor of mathematics at the University of Toronto, said mathematicians now have a great deal of exciting work to do. Yann LeCun said mathematics is entering a new era in which formal proofs will be largely automated and attention will shift to new concepts, abstractions, definitions and conjectures. Gowers said humans could last until 2030, when the next Fields Medal will be awarded; the report asked whether a human will still stand on the stage then.