OpenAI releases 175 pages of proof of the four-dimensional Kakeya conjecture

On October 12, 2026, Beijing time, Qubit and New Wisdom focused on reporting on a four-dimensional Kakeya problem manuscript published by OpenAI in the past 24 hours. The official PDF has a total of 175 pages, titled "Every Four-Dimensional Kakeya Set Has Full Hausdorff Dimension", and the author column only reads OpenAI. The manuscript claims to prove the four-dimensional Hausdorff dimension version of Kakeya's conjecture, but it is still a public preprint; OpenAI does not provide journal peer review acceptance information, and the public Lean catalog does not list this result. Therefore, the paper's claims cannot be written as a theorem that has been confirmed by the mathematics community.

Open warehouse and four-dimensional proposition

On October 6, OpenAI announced the release of a batch of mathematical results generated by internal cutting-edge models, and put the manuscript, revision records, and citation rules into a public GitHub repository. The cover date of the Shidi-Kakeya manuscript is September 24, and the GitHub interface shows that the corresponding PDF went online with the initial submission to the warehouse at 21:58:50 UTC on October 6. The official catalog lists it in Group 074 together with a three-dimensional Kakeya maximum function manuscript. The former discusses the Hausdorff dimension of four-dimensional sets, and the latter deals with different three-dimensional maximum function problems.

The Kakeya set can be understood as a set that accommodates a unit line segment in each direction in space; line segments in different directions can overlap a lot, making the volume of the set very small, but the dimensionality problem requires determining how thin it can be. Theorem 1.1 of the manuscript claims that in a four-dimensional Euclidean space, as long as a set contains one unit line segment in each direction, its Hausdorff dimension must be equal to four. The paper also makes it clear that there is no requirement that this set or the family of line segments used to witness each direction satisfy compactness, measurability, or regularity conditions.

Conclusion Scope and Research Background

The paper deals with the conclusion of the four-dimensional Hausdorff dimension, not the stronger four-dimensional Kakeya maximum function conjecture. The introduction to the manuscript separates the two: the maximal function version requires the establishment of an estimate of the line tube averaging operator of a certain radius, while the ensemble version focuses on the dimensionality at the coverage scale. The official catalog only describes the four-dimensional results as the Hausdorff dimension conjecture, and places the maximum function conclusion in the same group of three-dimensional manuscripts. Therefore, the public materials cannot support that "all four-dimensional Kakeya problems have been solved", nor can this conclusion be extended to a full-dimensional theorem in any dimension.

The research background reviewed in the manuscript shows that the previous lower bound of the Hausdorff dimension of the general four-dimensional Kakeya set was 3.059. Other recent works have achieved full-dimensional union estimates with additional conditions such as viscosity, approximately canonical line tube families, convex Wolff conditions, and dense coloring, but these conditions do not cover arbitrary Kakeya sets. In terms of three dimensions, Wang Hong and Joshua Zahl have proved the Kakeya set conjecture, and Larry Guth, Wang Hong and Joshua Zahl have given a simplified proof. The above results are existing research cited in the new manuscript, and external authors are not listed as authors of this four-dimensional paper.

Proof Routes and Claimed Inferences

According to the manuscript's own proof summary, the argument first transforms the hypothesized Hausdorff dimension gap into a weighted line problem, and then uses a quadratic local fit that is preserved across scales to track line segment aggregation. The paper sets up a quantity to measure the length of time that local fitting can be maintained, and handles it according to the optimized narrowing rate of zero, finite positive number or infinite. Each case attempts to construct longer or more elaborate partial graphs, ultimately conflicting with the positive time cost that the hypothetical counterexample must satisfy. This route comes from the paper's self-report, and there is no public independent review conclusion.

The proof chain also invokes the multilinear Kakeya estimate, the restricted triple product theorem, the multiplicative convolution estimate, and the weighted estimate derived from the three-dimensional Kakeya result. The paper claims that the main conclusion also deduces that the packing dimension, lower Minkowski dimension and upper Minkowski dimension of the four-dimensional Kaketani set are all four; for four-dimensional and higher-dimensional spaces, it at least gives the conclusion that the Hausdorff dimension is not less than four. At the end of the article, the inferences of the Nikodym set on a four-dimensional manifold with constant curvature and several specific curved Kakeya problems are also listed, and the existing transfer theorems relied on are noted one by one.

Validation boundary conflicts with quantity

Verification status and release size must be viewed separately. The OpenAI repository states that currently about 42% of top-level results are formalized, but this proportion covers the entire directory; the Lean list at the time of collection does not have Group 074 or Kakeya entries. The warehouse README also reminds that some results that have not yet been formalized may have problems, and says that the public version history will be retained and the corrections will be recorded as new versions. Disclosing a long proof, listing citations, and saving a revision track provides a material basis for external inspection, but it does not mean that the proof has been formalized or reviewed by experts.

There is also a discrepancy in the size of the catalog: Xinzhiyuan's information flow summary stated that OpenAI published the first batch of 722 mathematical manuscripts, while the official README and CONTENTS documents at the time of collection stated 719 manuscripts and 372 result groups. The existing materials did not explain whether the three differences were due to deletion, merging, or revision. OpenAI also stated that the internal evaluation asked about 4,000 questions to the model, and each result used an average of about three hours of calculations equivalent to ChatGPT Pro thinking; these are official self-reported generation process data. Signals that can be checked in the next step include whether there are new Lean entries in Group 074, whether a revised version appears in the PDF, and whether independent mathematicians or journals make public review opinions.