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 set conjecture, which states that every compact subset of three-dimensional space containing a unit line segment in every direction has Minkowski and Hausdorff dimension three, was proved by Hong Wang and Joshua Zahl in a preprint posted in February 2025. The result has since passed the kind of scrutiny that settles a proof in mathematics: Terence Tao worked through and endorsed the core argument within days; Larry Guth wrote two expository accounts while checking the proof and then a Séminaire Bourbaki survey of it; Guth, Wang, and Zahl jointly published a streamlined second proof in January 2026; and independent rigorous lecture notes have been prepared. Institutional recognition followed, with a 2026 Clay Research Award citing the proof and Wang receiving the 2026 Fields Medal substantially for this work.
No expert has identified a gap, no correction or retraction has appeared, and every credible source consulted asserts the result rather than questioning it. The only residual qualification is the general one attaching to any recent proof accepted through expert vetting rather than machine verification, and the existence of a second, simplified proof reduces even that. The claim rests on two facts, both established: that Wang and Zahl posted a preprint claiming the proof in February 2025 and that the mathematical community accepts the proof as correct.
Full reasoning: the evidence and decisions behind this verdict
This pass re-examined whether anything had changed since the previous assessment. It had not, except in the direction of further confirmation.
The proof exists: Wang and Zahl posted an arXiv preprint in February 2025 (arXiv:2502.17655), documented by the arXiv record, 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.
The proof is accepted: the community accepts the Wang–Zahl proof as correct. Beyond the evidence already on record (Tao's endorsement; Guth's two expository articles at arxiv.org/pdf/2505.07695 and arxiv.org/pdf/2508.05475; Quanta's July 2026 Fields Medal reporting that community review found no error; the 2026 Clay Research Award citation at www.claymath.org/news/2026-clay-research-awards/), this pass located three further items. Guth, Wang, and Zahl posted "A streamlined proof of the Kakeya set conjecture in R^3" in January 2026 (arxiv.org/abs/2601.14411), a second and simplified proof of the same theorem. Guth's Séminaire Bourbaki survey of April 2026 (arxiv.org/abs/2604.03416) states flatly that Wang and Zahl proved the Kakeya conjecture in dimension three; a Bourbaki exposé is itself a mark that the community regards a result as established. Alex Cohen at NYU has written rigorous lecture notes following the Guth–Wang–Zahl argument (cims.nyu.edu/~ac6074/KakeyaRigorous_4-7-26.pdf), a further independent working-through.
Searches aimed at finding a reported error, gap, or retraction returned nothing of the kind. All nine recorded source instances affirm the claim; none denies it. Credence stays at 0.97: the residue reflects only that acceptance rests on expert vetting rather than formal verification, and the existence of a second streamlined proof by the same authors with Guth narrows that residue further. The conclusion would change only if experts identified a substantive gap in both the original and the streamlined arguments, of which there is no sign. Another pass would add little unless such a report appeared.
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.
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.
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 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.
In [31], Wang and Zahl proved the Kakeya conjecture in dimension 3.
Séminaire Bourbaki survey of the proof, written for a broad mathematical audience, stating in its own voice that Wang and Zahl proved the three-dimensional Kakeya conjecture and sketching all the ideas in the proof.
We present a streamlined and simplified proof of the Kakeya set conjecture in R^3.
A second, simplified proof of the three-dimensional Kakeya set conjecture, co-authored by the original authors with Larry Guth, presented as an established result.
Assessment history
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Created by extractor · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.