OpenAI claim summary for Math-2484
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
Sphere extension and packet capacity: The question combines restriction and Kakeya assertions in broad generality. The theorem covers the bounded-data three-dimensional sphere extension p>3, and its stated Kakeya maximal corollary covers dimension three. It does not settle restriction or Kakeya in all dimensions and general settings.
This page records partial support. Complete resolution of the catalogue question is not established by this record.
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Fourier Restriction and Kakeya Incidence Conjectures
OpenAI's claim
Sphere extension and packet capacity: For bounded data on the two-sphere, the announced extension bound has every output exponent larger than three. The threshold three is also asserted to be the sharp endpoint of this open range.Mathematical statement of the claimed result
Mathematical claim 1.
For surface area $\sigma$ on $S^2$ and $Eg(x)=\int_{S^2}e^{2\pi ix\cdot\omega}g(\omega)\,d\sigma(\omega)$, every bounded measurable complex $g$ satisfies \[\|Eg\|_{L^p(\mathbb R^3)}\le C_p\|g\|_{L^\infty(S^2,\sigma)}\quad(p>3).\] The constant depends only on $p$; the endpoint $p=3$ and other ambient dimensions are outside this theorem.Outline of the argument
Sphere extension and packet capacity: Joint tests of position and velocity use ellipse weights that penalize eccentricity. A geometric propagation estimate controls how this capacity evolves along lines, using overlap savings from differently directed tests. Exact packet analysis and synthesis retain the oscillatory information needed to convert the positive capacity estimate into Fourier extension bounds.Formal verification
Sphere extension and packet capacity: The pinned catalogue lists no Lean formalization for this result.This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.
Status caveat
OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.Sources
- OpenAI, Elliptic capacity propagation and Fourier restriction to the sphere, 2026-09-24; citation key
OAI:Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.