OpenAI Drops 722 Math Manuscripts Overnight: Kakeya Conjecture Pushed to Four Dimensions by AI

In July of this year, Hong Wang and Yu Deng received the Fields Medal at the International Congress of Mathematicians in Philadelphia. In the days when the news went viral, Anthropic employee Alek Dimitriev left a pointed remark on X: this year's Fields Medal would be the last one ever won by a human. Past recipient Timothy Gowers replied that he'd had similar thoughts, offering a relatively optimistic estimate — humanity would probably hold on until 2030.

Reality arrived faster than anyone expected. On October 7th, OpenAI released 722 mathematical manuscripts in one shot, uploading the papers and some Lean proofs to GitHub. Among them, a set numbered 074 contains just two papers, aimed squarely at the direction for which Wang won her medal — the Kakeya conjecture. One claims to prove a stronger version of the three-dimensional Kakeya maximal function conjecture; the other claims to prove the four-dimensional Kakeya conjecture. Together, the two manuscripts run 272 pages, with a single word in the author field: OpenAI.

What Is the Kakeya Conjecture, and Why Is It So Hard?

In 1917, Japanese mathematician Soichi Kakeya posed a question: how much area does it take to rotate a needle of length 1 a full 360 degrees in the plane? Besicovitch's answer was counterintuitive to an extreme — the area can be made arbitrarily small; in fact, a set can contain a unit line segment in every direction and yet have measure zero.

With area proving unreliable, mathematicians switched tools and used "dimension" as their measuring stick. The dimension here isn't the everyday notion of length, width, and height, but rather tools like Hausdorff dimension and Minkowski dimension, which gauge how irregular a set is across different scales. A set can have zero volume while still possessing a full integer dimension.

The Kakeya conjecture is therefore stated as follows: in n-dimensional space, any set that contains a unit line segment in every direction must have full dimension n. The difficulty lies in how these thin line segments interleave and overlap, and whether that overlap can compress the set's dimension.

The planar case was proved by Davies in 1971. The three-dimensional case stayed open for half a century, until Hong Wang and Joshua Zahl finally cracked it. Four dimensions and above had never been proved.

Hausdorff dimension is a type of fractal dimension introduced by mathematician Felix Hausdorff in the early 20th century, capable of assigning a meaningful dimension to "pathological" sets that fall between integers. Intuitively: a smooth curve has Hausdorff dimension 1, a planar region has dimension 2, and a fractal curve like the Koch snowflake has Hausdorff dimension approximately 1.26 — more complex than a line, yet unable to fill the plane. Minkowski dimension (also called box-counting dimension) is a similar tool that measures complexity by covering a set with small boxes of side length ε and tracking how the required number of boxes grows. The two agree for most "regular" sets but can diverge in extreme cases. Part of what made Wang and Zahl's 2025 proof so highly regarded was that it resolved both versions simultaneously, rather than just one.

What Each Manuscript Claims to Do

The first paper, 97 pages long, addresses the three-dimensional Kakeya maximal function conjecture. Compared to the "set version" proved by Wang and Zahl, the maximal function version is more refined: thicken each needle into a thin tube, and suppose that only a fraction λ of each tube is "filled"; can the total volume of all filled parts be guaranteed to be no smaller than on the order of λ³? The smaller λ is, the emptier the tubes and the harder the problem. The set version only requires a loose power of λ; the maximal function version demands the exponent be exactly 3.

OpenAI explicitly states in the introduction that Wang and Zahl's theorem yields a factor of λ^K(ε), while the demand for λ³ was raised as the natural next step after their theorem — and this paper claims to resolve it. The result matters because the Kakeya maximal function turns the question of "how thin tubes cluster across all directions" into a concrete analytic problem, tying it closely to the Fourier restriction conjecture and the Bochner-Riesz conjecture.

The second paper, 175 pages long, addresses the Hausdorff dimension of Kakeya sets in four dimensions. Prior human progress had been painfully slow: Wolff proved a lower bound of dimension 3 in 1995; Guth and Zahl pushed it to roughly 3.025; Katz and Zahl reached 3.059 in 2019, with subsequent progress inching along in the second or third decimal place. This paper claims to jump directly to 4.

What makes four dimensions so hard? Guth has noted that a key theorem in Wang and Zahl's three-dimensional proof simply fails in four dimensions, because in higher-dimensional space tubes can cluster near low-degree algebraic surfaces in ways that produce counterexample structures absent in three dimensions. OpenAI's central tool is using quadratic polynomials to locally approximate tube behavior at different scales, tracking lines that persistently cluster across scales — the word "polynomial" appears 213 times throughout the paper.

The two papers form a connected chain: one of the key inputs in the four-dimensional paper cites a lemma from the three-dimensional paper. But the scope of the results is clearly bounded — the four-dimensional paper proves only the Hausdorff dimension version, not the four-dimensional maximal function conjecture, and the five-dimensional-and-above cases remain unproved. Crucially, neither paper in set 074 has been Lean-formalized, and neither has undergone peer review. OpenAI states plainly in its README that some unformalized results "may be incorrect."

The Fourier restriction conjecture and the Bochner-Riesz conjecture are two central open problems in harmonic analysis, deeply intertwined with the Kakeya conjecture at a technical level. The Fourier restriction conjecture asks: if you restrict the Fourier transform of a function to a curved surface (such as a sphere or paraboloid), can the L^p norm be controlled by the L^q norm of the original function? The Kakeya maximal function precisely captures how tube-shaped neighborhoods cluster across all directions, and this clustering behavior is fundamentally related to the rate at which Fourier integrals oscillate and decay near curved surfaces. Historically, each step forward in understanding Kakeya sets has tended to drive progress on these two conjectures as well, and vice versa — the three form a mutually reinforcing network of problems. This is why the maximal function version, rather than merely the set dimension version, is considered the more practically valuable result in the analysis community.

Wang's Work Is the Foundation of All This

Some have said Wang was lucky to claim the Fields Medal before AI got there. But reading through both papers, one of the names that appears most frequently is precisely Hong Wang — in the three-dimensional paper alone, "Wang" appears 23 times.

The mountain of three-dimensional Kakeya took many years to climb. Katz, Łaba, and Tao had long ago identified that near-extremal tube configurations exhibit three structural properties — stickiness, planiness, and graininess — but no one managed a complete proof for decades. Wang and Zahl approached it in three stages: proving the sticky case in 2022, handling the Assouad dimension version in 2024, and releasing a complete 127-page proof in February 2025 that resolved both the Hausdorff and Minkowski dimension versions of the three-dimensional Kakeya conjecture simultaneously. Nets Katz called it a "once-in-a-century result."

OpenAI's papers are built on this framework. The three-dimensional paper directly invokes the simplified proof by Guth, Wang, and Zahl (Guth was Wang's doctoral advisor at MIT), along with the planar Furstenberg estimate by Ren Kan and Wang; the four-dimensional paper uses a point-product theorem by Wang and Zahl. If both papers are ultimately confirmed correct, they will have added two more floors on top of a foundation Wang already built.

Wang herself is not hostile to AI. In a post-award interview, she described AI as an "active accelerator" for mathematical research, maintaining that posing questions, creating concepts, and building theory remain the core work of mathematicians.

A Divided Mathematical Community: Resistance and Excitement

What made this episode so explosive was its disruption of pace. In the past, a major mathematical result typically followed a recognizable lifecycle: a small team spends years on it, posts to arXiv, waits a year or two for peer review, and lets the field slowly absorb it. Wang and Zahl completed that journey — from the sticky case in 2022 to the Fields Medal in 2026 — in roughly four years. OpenAI released 722 manuscripts overnight, averaging about three hours of ChatGPT Pro compute per result, with Lean formalization for only about 42% of the papers.

Terence Tao, leading the Association for Human Mathematics (AHM), issued a joint statement in pointed terms: releasing over 700 documents at once is an exercise of power, not scholarship, and strongly urged mathematicians to stop collaborating with OpenAI. Tao himself has been an enthusiastic user of AI-assisted research; what he objects to is treating the mass conquest of famous open problems as a product demonstration. In his view, once a problem is "solved" by AI, it cannot be made unsolved again — and whatever new methods and insights humanity might have developed by working through it are simply lost.

Peer review has become the biggest bottleneck. For a 175-page four-dimensional proof, the number of people in the world capable of seriously reading it can be counted on one hand. Who will read it, how long it will take, and how the effort gets credited are all open questions. Attribution is becoming chaotic — researchers working on the same problems may wake up one morning to find their topic has been "claimed" by a model. Whether the proof is correct remains unknown, but the landscape for choosing research directions has already shifted significantly.

Others are excited. Daniel Litt, an assistant professor at the University of Toronto, believes mathematicians now have a wealth of thrilling work ahead; Yann LeCun argues mathematics is entering a new era in which formal proofs will be largely automated and the emphasis will shift to developing new concepts, new abstractions, new definitions, and new conjectures. His analogy: the invention of ships diminished the importance of swimming, yet allowed us to discover new continents.

Gowers said humanity would probably hold on until 2030 — and the next Fields Medal will be awarded in 2030. By then, will there still be a human standing on that stage? Right now, no one has the answer.

Lean is an interactive theorem prover developed by Microsoft Research and now maintained by the community, built on dependent type theory. When mathematicians write proofs in Lean, every step of the reasoning must conform to strict formal syntax, and the system automatically verifies whether the logical chain is sound — meaning that once Lean accepts a proof, its correctness is guaranteed at a machine level. Traditional peer review, by contrast, relies on human experts reading and judging, and can miss errors in extremely long or technically dense proofs (history is full of celebrated proofs that turned out to have gaps years after publication). The fact that about 42% of OpenAI's papers include Lean formalization has been seen as partial credibility backing; the unformalized portion — including the four-dimensional Kakeya conjecture — is one that OpenAI itself acknowledges in its README as potentially flawed, a key reason the release has caused such deep unease in the mathematical community.