---
title: "The three-dimensional Bochner–Riesz conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 078; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "real-and-complex-analysis"
  - "unformalized"
seed_rank: 1935
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Bochner–Riesz multipliers in three dimensions"
    url: "https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The three-dimensional Bochner–Riesz conjecture

## One-sentence takeaway

OpenAI's result family 078 (Real and complex analysis) claims: 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 $\delta\gt \max\{3|1/p-1/2|-1/2,0\}$.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- *Bochner–Riesz multipliers in three dimensions*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Bochner–Riesz multipliers in three dimensions](https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math