---
title: "An L3 bound for the trilinear Hilbert transform"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 086; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "real-and-complex-analysis"
  - "unformalized"
seed_rank: 1943
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "An L3 bound for the trilinear Hilbert transform"
    url: "https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# An L3 bound for the trilinear Hilbert transform

## One-sentence takeaway

OpenAI's result family 086 (Real and complex analysis) claims: Proves that the principal-value trilinear Hilbert transform with shifts $x-t$, $x-2t$, $x-3t$ is bounded from $L^3(\mathbb R)^3$ to $L^1(\mathbb R)$.

## 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 L3 exponent case of the standard conjecture for slopes 1, 2, 3.
- *An L3 bound for the trilinear Hilbert transform*: We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps $L^3(\mathbb R)\times L^3(\mathbb R)\times L^3(\mathbb R)$ to $L^1(\mathbb R)$, resolving this case of the trilinear Hilbert transform 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: [An L3 bound for the trilinear Hilbert transform](https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/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