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The Kakeya set conjecture in four dimensions (Hausdorff dimension)

Theorem 1.1: if $K\subset\mathbb R^4$ contains, for every $e\in S^3$ , a segment $\{a_e+te:0\le t\le1\}$ , then $\dim_H K=4$ . No measurability, compactness, stickiness or packing-dimension hypothesis is imposed. The paper also derives packing and Minkowski-dimension consequences, a projection lower bound in higher dimensions, and Nikodym and curved-Kakeya applications via cited transfers. It does NOT prove the four-dimensional Kakeya maximal conjecture, and does not settle the conjecture in dimension five or higher.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Kakeya set conjecture in four dimensions (Hausdorff dimension) · Kakeya set conjecture in R^4

Confidence: Not scored

Registry verification: unreviewed · announcement · partial

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Unreleased internal OpenAI model
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VibeMathed record: The Kakeya set conjecture in four dimensions (Hausdorff dimension)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the introduction states it; it covers arbitrary (not necessarily measurable or compact) sets and arbitrary witnessing segments, which is the full Hausdorff form in R^4. The proof was not refereed. No Lean formalization exists. It uses a weighted plank lemma (Lemma 2.3) from the companion three-dimensional maximal paper, itself unreviewed, together with the published Guth-Wang-Zahl union estimate.

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The Kakeya set conjecture in four dimensions (Hausdorff dimension)

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Companion: The Kakeya maximal conjecture in three dimensions (input lemma)

paper · pending

paper link supplied by the VibeMathed registry.

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The Kakeya set conjecture in four dimensions (Hausdorff dimension) parent of this event

Companion: The Kakeya maximal conjecture in three dimensions (input lemma) evidence for this event

The Kakeya set conjecture in four dimensions (Hausdorff dimension) evidence for this event

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