Claimed AI-led

Sogge's local smoothing conjecture claimed in every dimension, with Bochner–Riesz and Kakeya as consequences

An OpenAI manuscript released with a Lean formalization claims Sogge's local smoothing conjecture for the wave equation in every spatial dimension, and from it the Bochner–Riesz conjecture, the Kakeya and Nikodym maximal conjectures, full Hausdorff dimension for every Euclidean Kakeya set, and Fourier restriction for the sphere, paraboloid and cone throughout the interior of the conjectured range.

Lab: OpenAI
Model: Unreleased internal OpenAI model, not named in the release
Field: Mathematics
Date: 2026-10-26
Problem posed: 1991 · open 35 yrs

Sources

Original work

GitHubOpenAI: Local smoothing for the Euclidean wave equation (PDF)↗GitHubopenai/math: scope of the Lean formalization for family 079↗GitHubopenai/math: Comparator challenge statement (LocalSmoothing.lean)↗

Announcement

GitHubopenai/math release history, 26 October 2026 entry↗

What was found

Local smoothing asks how much regularity solutions of the wave equation gain on average in time beyond what fixed-time estimates allow. Sogge conjectured the sharp form in 1991, and it was known to imply the Bochner–Riesz and Kakeya conjectures; Guth, Wang and Zhang proved it in two spatial dimensions in 2020. Family 079 of openai/math, added on 26 October 2026, pairs a manuscript dated 25 October with an earlier one claiming the critical $L^3$ estimate in three dimensions. OpenAI's scope page says the Lean formalization bounds, for every $n \ge 2$ and $2 \le p < \infty$ , the $L^p(\mathbb{R}^n \times [1, 2])$ norm of the half-wave by the $W^{s,p}$ norm of Schwartz initial data with $s = \max(0, (n-1)/2 - n/p) + \varepsilon$ , and also gives uniform Bochner–Riesz bounds, spherical extension estimates and Stein square-function bounds in their strict ranges, the Kakeya maximal bound with $\delta^{-\varepsilon}$ loss, and full dimension for every set containing a unit segment in each direction. The release note says the model output is published directly, as in the release of 6 October.

Novelty check

arXiv, 2026-10-29: abs:"local smoothing conjecture" returned 9 records and abs:"local smoothing" AND abs:"wave equation" returned 29; none claims the conjecture in all dimensions, the standing results being partial ranges and the two-dimensional case. Within this registry 2026-10-06-kakeya-3d-maximal-4d records OpenAI's earlier claims of the three-dimensional Kakeya maximal estimate and full dimension for four-dimensional Kakeya sets, both of which this result would contain; that entry stands, since its two manuscripts are separate arguments. 2026-10-26-eth-refuted-openai records the other result added the same day.

Caveats and known objections

Entered at claimed. A Lean formalization exists and its Comparator statements cover the headline estimates, which sets this apart from the earlier Kakeya entry, but it is OpenAI's own: no rebuild outside OpenAI, audit of the statements or Palomar registration was found, and an outside rebuild is what moved the one family from this release that this registry grades formal. Two scope limits are on OpenAI's own page: the estimates are proved for Schwartz initial data, and restriction is claimed throughout the interior of the conjectured range rather than at its endpoints, with Bochner–Riesz, restriction and square-function bounds stated in their strict ranges. Nobody outside OpenAI is on record as having read the manuscript, and nothing was found from the people whose results it would subsume. The reliability of the surrounding collection is part of the context: OpenAI withdrew three manuscripts on 7 October 2026 over a sign error. Autonomy is ai-led on the strictest defensible reading, matching every other family from this release: the model's output is released directly, while humans chose the problems and ran the formalization pipeline.

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Entry history (1 event)

2026-10-29AddedEntered the registry graded Claimed and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded claimed for verification and ai-led for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Sogge's local smoothing conjecture claimed in every dimension, with Bochner–Riesz and Kakeya as consequences. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-26-local-smoothing-all-dimensions

BibTeX

@misc{whataifound-openai-2026-dimensions,
  title        = {Sogge's local smoothing conjecture claimed in every dimension, with Bochner–Riesz and Kakeya as consequences},
  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/2026-10-26-local-smoothing-all-dimensions}
}

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