Result 008, Number theory

The Deligne–Drinfeld conjecture

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.

Lean formalization Proof

The bigger picture

Why it matters

A complicated algebra in number theory may be built from a remarkably sparse set of ingredients. The manuscript claims that one ingredient in each odd weight suffices, with no unexpected algebraic relations.

What changes?

The rational Grothendieck-Teichmüller Lie algebra is a vector space over the rational numbers with an operation called the Ihara bracket. Weights organize its elements by degree. The manuscript reports one free generator in every odd weight starting at three. These generators build the algebra through brackets and linear combinations, with no relations beyond the universal Lie algebra identities. The corresponding isomorphism with the free Lie algebra holds after weight completion, which allows infinite sums across increasing weights.

What does that help mathematicians do?

The claim would replace uncertainty about hidden generators and relations with a precise structural description. Researchers could deduce the number of independent elements in any fixed weight using free Lie algebra rules. Odd generator weights do not mean that even-weight elements vanish: brackets combine weights. The result would therefore explain how more complicated parts arise, while ruling out additional relations not forced by the Lie algebra identities.

Are there practical applications?

The immediate value is foundational, not a demonstrated practical technology. A free presentation provides a framework for calculations and comparisons involving this particular number-theoretic algebra: researchers can work with generators and universal bracket rules rather than an unknown collection of constraints. The supplied abstract does not establish an applied algorithm or deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Deligne-Drinfeld conjecture

September 23, 2026 41 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Deligne-Drinfeld-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{The Deligne--Drinfeld conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf}{OAI:The-Deligne-Drinfeld-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/008.md.

The Deligne–Drinfeld conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

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,…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. It also gives the corresponding continuous isomorphism after completion by weight.

The regularized-transport proposition and graph-complex consequences are not included.

Comparator links

Result Comparator statement
Deligne–Drinfeld main statement DeligneDrinfeld.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.