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The Kakeya maximal conjecture in three dimensions

For ω∈S2\omega\in S^2 and 0<δ<10<\delta<1 let Kδf(ω)K_\delta f(\omega) be the supremum, over unit tubes of radius δ\delta in direction ω\omega, of the average of ∣f∣|f| on the tube. The Kakeya maximal conjecture in Rn\mathbb R^n asks that ∥Kδf∥Ln(Sn−1)≤Cεδ−ε∥f∥Ln(Rn)\|K_\delta f\|_{L^n(S^{n-1})}\le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^n(\mathbb R^n)} for every ε>0\varepsilon>0; it implies the Kakeya set conjecture. In R3\mathbb R^3 Wang and Zahl proved the set conjecture in 2025, but their union estimate carries a density power λK(ε)\lambda^{K(\varepsilon)} where the maximal bound needs λ3\lambda^3, a gap stated after their theorem. Does the Kakeya maximal estimate hold in R3\mathbb R^3 for every ε>0\varepsilon>0?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Harmonic analysis; Kakeya problems
Posed by
Maximal form of the Kakeya conjecture (Cordoba, Bourgain); the manuscripts cite the formulations of Katz and Tao (2002) and Zahl's 2025 survey, Conjecture 1.3''
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every ε>0\varepsilon>0 there is CεC_\varepsilon such that ∥Kδf∥L3(S2)≤Cεδ−ε∥f∥L3(R3)\|K_\delta f\|_{L^3(S^2)}\le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^3(\mathbb R^3)} for all 0<δ<10<\delta<1 and f∈L3(R3)f\in L^3(\mathbb R^3). This is the n=3n=3 case of the maximal conjecture; it strengthens the Wang-Zahl set theorem by getting the cubic dependence on shading density. The paper also derives Nikodym maximal and local curved Kakeya estimates by cited transfers. It does not treat dimension four or higher for the maximal operator; the four-dimensional Hausdorff case is a separate entry.

What the AI did

The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 23 and 24, 2026; this entry draws on the September 23 one. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the maximal conjecture as stated in the paper's introduction and, independently, in the companion four-dimensional paper (which quotes the Katz-Tao formulation). It is the full three-dimensional case, with exactly the epsilon loss the conjecture allows. The proof (critical exponents for a density-sensitive multiplicity inequality, equality configurations, plate structure) was not refereed. No Lean formalization exists for this family. Inputs are published theorems: the Guth-Wang-Zahl set estimate, multilinear Kakeya with Guth's endpoint, and the Ren-Wang planar Furstenberg theorem. The four-dimensional paper and the family 077 restriction papers use this theorem, so an error would propagate; the sphere-restriction companion in family 077 also claims an independent route to it.

Sources

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