The Bochner-Riesz conjecture in three dimensions (strict-order range)

For $\delta\ge0$ the Bochner-Riesz means on $\mathbb R^n$ are $\widehat{T_\delta f}(\xi)=(1-|\xi|^2)_+^\delta\widehat f(\xi)$ . Fefferman (1971) showed the ball multiplier ( $\delta=0$ ) is unbounded on $L^p$ for $p\ne2$ when $n\ge2$ ; Carleson and Sjolin (1972) proved the conjectured range in the plane. The Bochner-Riesz conjecture asserts that $T_\delta$ is bounded on $L^p(\mathbb R^n)$ whenever $\delta>\max\{n|1/p-1/2|-1/2,0\}$ ; Tao (1999) showed it implies the restriction conjecture. In $\mathbb R^3$ the strict range was known only for $\max\{p,p'\}\ge 22/7$ (Lee, Wu, Gao-Wu-Xi). Is $T_\delta$ bounded on $L^p(\mathbb R^3)$ for every $1\le p\le\infty$ and every $\delta>\max\{3|1/p-1/2|-1/2,0\}$ ?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Harmonic analysis; Fourier multipliers and summability
Posed by: Bochner-Riesz summability (Riesz 1923; Bochner 1935-36); the manuscript cites no single source for the conjectured range
Year posed: —
Years open: —
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 62 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every $\delta>0$ , $\|T_\delta f\|_{L^3(\mathbb R^3)}\le C_\delta\|f\|_{L^3(\mathbb R^3)}$ . By complex interpolation and duality this gives boundedness on $L^p(\mathbb R^3)$ for all $1\le p\le\infty$ and $\delta>\max\{3|1/p-1/2|-1/2,0\}$ , the full strict-order range in three dimensions, hence $L^p$ convergence of Bochner-Riesz means for $3/2\le p\le3$ at every positive order. Via Tao's implication it yields the spherical extension estimate $\|Eg\|_{L^p(\mathbb R^3)}\le C_p\|g\|_{L^\infty(S^2)}$ for $p>3$ . Not shown: the endpoint order, the maximal Bochner-Riesz operator, or any dimension $n\ge4$ ; it is a partial result on the $n$ -dimensional conjecture.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (September 24, 2026) uses the sharp planar Furstenberg estimate of Ren and Wang as its geometric input and cites, as non-inputs, the release's sphere restriction, diagonal extension and critical local smoothing manuscripts.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 (L^3 boundedness for every delta > 0) and the interpolation statement of the full strict range were read against the conjecture as the introduction states it. The proof (entropy growth of positive masses on lines, wave-packet repayment, then the pseudoconformal multiplier step) was not refereed. No Lean formalization. Scope the paper itself states: strict order only; the endpoint order delta = 3|1/p - 1/2| - 1/2 is not claimed, nor dimensions above three. A reader should know that Zipeng Wang has proposed a proof in all dimensions (arXiv 2501.12742, cited by the manuscript), and that the release's family 079 (Sogge's local smoothing in 3D) claims the same strict Bochner-Riesz range by an independent route.

The Bochner-Riesz conjecture in three dimensions (strict-order range)

What was actually shown

What the AI did

Verification

Sources

Discussion