The three-dimensional Kakeya conjecture has been proved: every three-dimensional Kakeya set has dimension three.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The three-dimensional Kakeya conjecture, which asserts that every Kakeya set in three-dimensional space (a set containing a unit line segment in every direction) has Minkowski and Hausdorff dimension three, was proved by Hong Wang and Joshua Zahl. Their proof appeared as an arXiv preprint in February 2025 and resolved a problem open in this form for roughly a century, closing the gap left by earlier partial results such as the 2.5-dimensional lower bounds of Wolff and of Katz and Zahl.
The proof has been examined and accepted by the mathematical community. Terence Tao analyzed the argument publicly within days of its release, Larry Guth studied and checked the proof and wrote two expository accounts of it, and no error has been found. The result was recognized by the 2026 Clay Research Award to Wang, Zahl, Orponen, and Shmerkin, and it was central to Wang's 2026 Fields Medal. No credible challenge to the proof's correctness has been raised.
Full reasoning: the evidence and decisions behind this verdict
The claim turns on two dependencies, both of which hold on direct examination of the record.
First, the proof exists: Wang and Zahl posted an arXiv preprint in February 2025 (arXiv:2502.17655, "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions"), an event documented by the arXiv record, Terence Tao's blog post of 25 February 2025 (terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/), and wide contemporaneous coverage (Quanta Magazine, Scientific American, NYU news).
Second, the proof is correct by the operative standard for mathematics, sustained expert scrutiny: the community accepts the Wang–Zahl proof as correct. Tao worked through and endorsed the argument's core mechanism. Larry Guth wrote two expository arXiv articles while digesting and checking the proof (arxiv.org/pdf/2505.07695 and arxiv.org/pdf/2508.05475), and in the latter states plainly that Wang and Zahl proved the 3-dimensional Kakeya conjecture. Quanta's July 2026 reporting on Wang's Fields Medal states that community review found no error and that the proof's logic was digested and streamlined. Institutional recognition converges on the same conclusion: the 2026 Clay Research Award citation (www.claymath.org/news/2026-clay-research-awards/) credits the laureates' work as leading to the proof of the Kakeya conjecture in three dimensions, and Wang received the 2026 Fields Medal substantially for this result.
All recorded source instances affirm the claim; searching turned up no expert dissent, no retraction, and no identified gap. The one residual caveat is the general one attaching to any recent proof: acceptance rests on expert vetting rather than on a formal machine verification, and mathematics has occasionally seen accepted proofs later revised. Given the depth and independence of the scrutiny here (Tao and Guth working through the argument, award committees vetting it further), that residue is small; credence is set at 0.97 rather than higher only for this reason. The conclusion would change only if a substantive gap were identified by experts, of which there is currently no sign.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- requiresa load-bearing premise: the parent is false without itsteward instructions →Hong Wang and Joshua Zahl posted an arXiv preprint in February 2025 claiming a proof of the three-dimensional Kakeya set conjecture. ↗︎
- requiresa load-bearing premise: the parent is false without itsteward instructions →The mathematical community accepts the Wang–Zahl proof of the three-dimensional Kakeya conjecture as correct. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The three-dimensional Kakeya conjecture says that a set's Minkowski dimension must be three.
Defining the content of the conjecture in terms of Minkowski dimension.
Hong Wang of New York University's Courant Institute and Joshua Zahl of the University of British Columbia have proved the three-dimensional Kakeya conjecture — they've established an absolute limit to how small such a pattern of movements can be.
Announcement of the new proof posted to arxiv.org.
Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set con…
Expert blog post by Terence Tao explaining the Wang–Zahl preprint and endorsing that it resolves the three-dimensional Kakeya set conjecture.
In the recent preprint [5], Hong Wang and Joshua Zahl proved the 3-dimensional Kakeya conjecture, building on previous papers [3] and [4].
Expository article by Larry Guth, who worked on digesting and checking the proof, stating in his own voice that Wang and Zahl proved the 3-dimensional Kakeya conjecture.
No error was found, the proof's logic was digested and streamlined, and everyone became convinced of its soundness. Wang and Zahl had proved the three-dimensional Kakeya conjecture.
Article on Wang's 2026 Fields Medal, reporting that community scrutiny of the Kakeya proof found no error and asserting the conjecture was proved.
A Clay Research Award is made to Tuomas Orponen (Jyväskylä), Pablo Shmerkin (UBC), Hong Wang (IHES and NYU), and Joshua Zahl (Nankai) in recognition of their remarkable work on geometric problems in harmonic analysis, leading to the proof of the Furstenberg set conjecture in the plane and the Kakeya conjecture in three dimensions.
Official award citation recognizing the work leading to the proof of the Kakeya conjecture in three dimensions.
The Kakeya Conjecture, a Decades-Old Math Problem, Is Solved in Three Dimensions
Popular science article asserting in its own voice that the three-dimensional Kakeya conjecture has been solved by Wang and Zahl.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.