---
title: "The $L\\log L$ Fourier-convergence conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 075; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "real-and-complex-analysis"
  - "unformalized"
seed_rank: 1932
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Almost-everywhere Fourier convergence in L log L"
    url: "https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The $L\log L$ Fourier-convergence conjecture

## One-sentence takeaway

OpenAI's result family 075 (Real and complex analysis) claims: Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L(\mathbb T)$ converge almost everywhere along the full sequence.

## 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

- Family summary (repo): This resolves the classical sufficiency conjecture at the $L\log L$ scale.
- *Almost-everywhere Fourier convergence in L log L*: We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L$ on the circle converge almost everywhere to the function along the full sequence. This establishes the classical $L\log L$ sufficiency conjecture.

## 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: [Almost-everywhere Fourier convergence in L log L](https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.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