---
title: "The Deligne–Drinfeld conjecture"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 008; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "lean4"
  - "formalized"
seed_rank: 1868
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 8
lineage: ai-mathematical-reasoning
cites:
  - title: "The Deligne-Drinfeld conjecture"
    url: "https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The Deligne–Drinfeld conjecture

## One-sentence takeaway

OpenAI's result family 008 (Number theory) claims: Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.

## 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. The Lean development makes the claim machine-checkable, which is the pattern Broadside cares about for agentic verification loops.

## Key ideas

- *The Deligne-Drinfeld conjecture*: We prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.
- Lean scope (lean/docs/008.md): The Deligne–Drinfeld conjecture predicts that the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in each odd weight $3,5,7,\ldots$. The formalization establishes this for the rational solution space of antisymmetry, the three-term equation, and the four-strand pentagon with the Ihara bracket.

## 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: formalized in part. The repo has a scope note listing 1 Comparator challenge statement(s), and that note defines which statements are covered. Do not read the Lean coverage as the full content of every manuscript in the family.
- A grep of the repo's `lean/OAI` tree found no `sorry` and no top-level `axiom` declarations. The library was not compiled for this card, and Comparator was not run.

## Links

- Manuscript: [The Deligne-Drinfeld conjecture](https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf)
- Lean scope: https://github.com/openai/math/blob/main/lean/docs/008.md
- Comparator statement (Deligne–Drinfeld main statement): https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DeligneDrinfeld.lean
- 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