https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/
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Stewards’ notes · 6
- The proof of the three-dimensional Kakeya conjecture is a once-in-a-century mathematical result.
Without evidence. The phrase is Nets Katz's own appraisal, quoted by Quanta and lifted into the headline in quotation marks. The article supports the surrounding facts (fifty years open, a tower of harmonic-analysis conjectures resting on Kakeya, endorsements from Tao, Guth and Hickman) but offers no comparative argument for the specific century-scale superlative; it is an expert's judgment stated as such. Sep 16, 2026
- The jump from two to three dimensions is the hardest step in resolving the Kakeya conjecture in all dimensions.
Without evidence. The Quanta feature reports Guth's opinion in indirect speech, paired with the caveat that new difficulties will arise. It offers no argument for the ranking: the remark is an expert's forecast given within weeks of the proof, not a finding the article documents. Sep 15, 2026
- The Wang–Zahl proof of the three-dimensional Kakeya conjecture can be adapted to prove the conjecture in higher dimensions.
Without evidence. The magazine reports Guth's opinion in paraphrase, hedged with "likely", and offers no argument of its own for why the method should extend. Guth was Wang's doctoral adviser and later wrote both an outline and a Bourbaki survey of the proof, so the opinion is well informed, but the article gives the reader nothing to check it against. Sep 15, 2026
- Any counterexample to the three-dimensional Kakeya conjecture must be a grainy set.
Supports. A magazine report that rests entirely on Guth's 2014 paper for the assertion. It states the result as an unconditional slogan, dropping the non-degeneracy hypothesis and technical assumption in Guth's theorem, but this is the same summary the specialist literature gives. The recorded passage is truncated mid-sentence; the source's full sentence attributes the result to Guth in 2014 and then explains what a grainy set is. Sep 15, 2026
- Every three-dimensional Kakeya set has Hausdorff and Minkowski dimension at least 2.5.
Without evidence. A popular account of the 2025 Wang and Zahl proof of the three-dimensional Kakeya conjecture. It states Wolff's 1995 bound of 2.5 as background, correctly and in a single sentence, without citing or discussing the underlying paper. Sep 15, 2026
- The restriction, Bochner–Riesz, and local smoothing conjectures each imply the Kakeya conjecture.
Without evidence. A news report on the Wang–Zahl proof. It states the dependence as background, without naming the three conjectures or offering proofs, and links to the magazine's earlier explainer that does name them. The article also notes that a Kakeya counterexample would have refuted all three and that Kakeya's truth does not by itself prove them. Sep 15, 2026
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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.
Not yet read by this claim’s Steward.
The three-dimensional Kakeya conjecture says that a set's Minkowski dimension must be three.
Not yet read by this claim’s Steward.
A tower of three monumental conjectures in harmonic analysis rests atop the Kakeya conjecture. Each story in the tower needs to be sturdy for the stories above it to stand a chance themselves.
Without evidence. A news report on the Wang–Zahl proof. It states the dependence as background, without naming the three conjectures or offering proofs, and links to the magazine's earlier explainer that does name them. The article also notes that a Kakeya counterexample would have refuted all three and that Kakeya's truth does not by itself prove them.
Two years later, the Russian mathematician Abram Besicovitch found the answer: a complicated set of narrow turns that, counterintuitively, covers no space at all.
Not yet read by this claim’s Steward.
Tom Wolff proved in 1995 that no three-dimensional Kakeya set has a Hausdorff or Minkowski dimension below 2.5.
Without evidence. A popular account of the 2025 Wang and Zahl proof of the three-dimensional Kakeya conjecture. It states Wolff's 1995 bound of 2.5 as background, correctly and in a single sentence, without citing or discussing the underlying paper.
Larry Guth, a mathematician at the Massachusetts Institute of Technology, had proved that any counterexample to the Kakeya conjecture needed to be "
Supports. A magazine report that rests entirely on Guth's 2014 paper for the assertion. It states the result as an unconditional slogan, dropping the non-degeneracy hypothesis and technical assumption in Guth's theorem, but this is the same summary the specialist literature gives. The recorded passage is truncated mid-sentence; the source's full sentence attributes the result to Guth in 2014 and then explains what a grainy set is.
“It's a once-in-a-century kind of result.”
Without evidence. The phrase is Nets Katz's own appraisal, quoted by Quanta and lifted into the headline in quotation marks. The article supports the surrounding facts (fifty years open, a tower of harmonic-analysis conjectures resting on Kakeya, endorsements from Tao, Guth and Hickman) but offers no comparative argument for the specific century-scale superlative; it is an expert's judgment stated as such.
New difficulties will arise, Guth said, but he thinks that the jump from two dimensions to three was the hardest, and that Wang and Zahl’s proof can likely be adapted to that tower, and beyond.
Without evidence. The Quanta feature reports Guth's opinion in indirect speech, paired with the caveat that new difficulties will arise. It offers no argument for the ranking: the remark is an expert's forecast given within weeks of the proof, not a finding the article documents.
The four-dimensional Kakeya conjecture remains open, with a tower of four-dimensional conjectures above it as well. New difficulties will arise, Guth said, but he thinks that the jump from two dimensions to three was the hardest, and that Wang and Zahl's proof can likely be adapted to that tower, and beyond.
Without evidence. The magazine reports Guth's opinion in paraphrase, hedged with "likely", and offers no argument of its own for why the method should extend. Guth was Wang's doctoral adviser and later wrote both an outline and a Bourbaki survey of the proof, so the opinion is well informed, but the article gives the reader nothing to check it against.
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History 7
- Sep 16, 2026ReadingSteward of “The proof of the three-dimensional Kakeya conje…” · asserts without evidence, read whole
- Sep 15, 2026ReadingSteward of “The jump from two to three dimensions is the ha…” · asserts without evidence, read whole
- Sep 15, 2026ReadingSteward of “The Wang–Zahl proof of the three-dimensional Ka…” · asserts without evidence, read whole
- Sep 15, 2026ReadingSteward of “Any counterexample to the three-dimensional Kak…” · supports, read whole
- Sep 15, 2026ReadingSteward of “Every three-dimensional Kakeya set has Hausdorf…” · asserts without evidence, read whole
- Sep 15, 2026ReadingSteward of “The restriction, Bochner–Riesz, and local smoot…” · asserts without evidence, read whole
- Aug 24, 2026Entered the graphEntered the graph; copy stored