Claimed AI-led

Kakeya maximal conjecture in three dimensions and the Kakeya set conjecture in four

Two manuscripts claim that every set in four-dimensional space containing a unit segment in every direction has full Hausdorff dimension, and that the Kakeya maximal function estimate holds in three dimensions, extending Wang and Zahl's 2025 resolution of the three-dimensional set conjecture.

Lab: OpenAI
Model: Unreleased internal OpenAI model, not named in the release
Field: Mathematics
Date: 2026-10-06

Sources

What was found

The four-dimensional paper proves that every $K \subset \mathbb{R}^4$ containing a unit segment in every direction has $\dim_H K = 4$ , with no measurability assumed on the set or on the choice of segments. Its introduction gives the best earlier lower bound as $3.059$ , from Katz and Zahl's planebrush argument, against Wolff's $3$ from 1995; Rai Choudhuri's $13/4$ covers only sticky sets. The three-dimensional paper proves the stronger maximal form: for every $\varepsilon > 0$ the radius- $\delta$ tube maximal operator maps $L^3(\mathbb{R}^3)$ to $L^3(S^2)$ with norm $O_\varepsilon(\delta^{-\varepsilon})$ . It builds on the streamlined Guth, Wang and Zahl proof of the set estimate and addresses the gap Wang and Zahl state after their own theorem, replacing a density loss $\lambda^{K(\varepsilon)}$ by $\lambda^3$ . Scientific American's report names only the four-dimensional result.

Novelty check

Read both manuscripts' introductions at commit adc7f1241b42e322a6451854ab7e4b4c146bf78a on 2026-10-07. Wang and Zahl's 2025 preprint resolved the Hausdorff and Minkowski dimension Kakeya set conjecture in three dimensions; the maximal conjecture in three dimensions and the set conjecture in four were open, and both papers say so with citations. No earlier claim of either result was found in a web search or in the registry; 2025-11-03-alphaevolve-at-scale mentions Kakeya only for an explicit construction that turned out to match prior work, not for a dimension bound.

Caveats and known objections

No Lean formalization accompanies either paper, unlike much of the release, so the only check available is reading them, and both are long multiscale harmonic-analysis arguments of the kind that take specialists months: the source of the four-dimensional paper runs to about 640,000 characters and the three-dimensional one to about 340,000. Nobody outside OpenAI is on record as having read either. Autonomy is graded ai-led because the README places neither among the exceptions to its single-agent procedure, but that account rests on OpenAI alone and the model is unreleased; it is not graded autonomous because humans chose the problems and filtered the output. The three-dimensional result depends directly on the Guth, Wang and Zahl set estimate, which is itself recent.

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Entry history (1 event)

2026-10-07AddedEntered the registry graded Claimed and AI-led.

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Cite this entry

Plain text

whataifound.org. (2026). Kakeya maximal conjecture in three dimensions and the Kakeya set conjecture in four. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-06-kakeya-3d-maximal-4d

BibTeX

@misc{whataifound-openai-2026-4d,
  title        = {Kakeya maximal conjecture in three dimensions and the Kakeya set conjecture in four},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by OpenAI. Verification: Claimed. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-10-06-kakeya-3d-maximal-4d}
}

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