The three-dimensional Bochner–Riesz conjecture

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on \(L^p(\mathbb R^3)\) for every \(1\le p\le\infty\) whenever \(\delta\gt \max\{3|1/p-1/2|-1/2,0\}\).

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1

In Bochner–Riesz multipliers in three dimensions, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

For every \delta>0, there is a finite constant C_\delta such that

\norm{T_\delta f}_{L^3(\RR^3)} \le C_\delta\norm f_{L^3(\RR^3)} \qquad(f\in\mathcal S(\RR^3)).

Consequently T_\delta has a unique bounded extension to L^3(\RR^3).

Reported proof verification: The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Bochner–Riesz multipliers in three dimensions

24 September 2026, current edition.

Read it here PDF

We prove the three-dimensional Bochner–Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner–Riesz multipliers on \(L^3(\mathbb R^3)\) for every positive order. Interpolation and duality then give the full conjectured strict range.

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

Elliptic capacity propagation and Fourier restriction to the sphereCurrent edition; no accompanying Lean statement Family Fourier restriction for positively curved surfacesCited by Bochner–Riesz multipliers in three dimensions.
Diagonal Fourier extension for positively curved surfaces in three dimensionsCurrent edition; no accompanying Lean statement Family Fourier restriction for positively curved surfacesCited by Bochner–Riesz multipliers in three dimensions.
Critical local smoothing for the three-dimensional wave equationCurrent edition; no accompanying Lean statement Family Local smoothing in three dimensionsCited by Bochner–Riesz multipliers in three dimensions.

Papers elsewhere in the release that cite this family’s papers

Critical local smoothing for the three-dimensional wave equationCurrent edition; no accompanying Lean statement Family Local smoothing in three dimensionsCites Bochner–Riesz multipliers in three dimensions.
Elliptic capacity propagation and Fourier restriction to the sphereCurrent edition; no accompanying Lean statement Family Fourier restriction for positively curved surfacesCites Bochner–Riesz multipliers in three dimensions.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

39 references in Bochner–Riesz multipliers in three dimensions

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanation Suggest a source attribution Suggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

Family 078 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

Family 078 The release has no Lean statement for this family. Open the form