The Deligne–Drinfeld conjecture
The Deligne–Drinfeld conjecture is the subject of a manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math). The conjecture predicts that the Grothendieck–Teichmüller Lie algebra is freely generated by one element in each odd weight 3, 5, 7, …. This Lie algebra consists of the elements of the free Lie algebra on two letters that satisfy an antisymmetry relation, a three-term relation and a pentagon relation, the infinitesimal compatibility conditions attached to Drinfeld's associators, with the Ihara bracket. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original result
- Claim. There are homogeneous solutions σ2k+1 of weight 2k + 1, one for each k ≥ 1, of the antisymmetry, three-term and pentagon equations such that the free Lie algebra on generators e3, e5, e7, … maps isomorphically onto the rational solution space with the Ihara bracket, and the induced map on completions by weight is an isomorphism. Source: The Deligne–Drinfeld conjecture (23 September 2026).
- Context, as summarized in the manuscript. Freeness of a Lie subalgebra of this form was known (Brown's theorem on mixed Tate motives over Z, and Willwacher's freeness criterion); the open part was that these elements exhaust all solutions. Computations of Naef and Willwacher had established the equality through weight 29. The manuscript's main new estimate is an upper bound on the dimension of the solution space in every weight, proved through a degeneration of the pentagon equation in characteristic two.
- Stated consequence. Combined with Willwacher's identification of the zeroth cohomology of Kontsevich's graph complex GC2 with the Grothendieck–Teichmüller Lie algebra, the manuscript states that H0(GC2) is isomorphic to the completed free Lie algebra on e3, e5, e7, ….
- Formalization, as described by the repository. The scope note lean/docs/008.md states that the Lean formalization establishes the statement for the rational solution space of antisymmetry, the three-term equation and the four-strand pentagon with the Ihara bracket, together with the continuous isomorphism after completion by weight; the regularized-transport proposition and the graph-complex consequences are not included.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 14 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | MathVet (GitHub; review of openai/math at commit adc7f12, tables at repository commit cd0063d, 13 October 2026) |
Review (fidelity of the formal statements) | Rates this family "full" and records a sandboxed Comparator pass for the challenge DeligneDrinfeld (8 October 2026). Its notes record that the formal statement concerns the graded solution space and that the formalization gives the Lie algebras discrete topologies, so the continuity clauses on the weight completions are essentially formal. |
The review states that "full" means the Lean states the headline and "does not say the mathematics is correct beyond what Lean checked"; it describes itself as a model-assisted reading without a referee yet. |
| 2026-10-11 | T. Willwacher, The Deligne–Drinfeld conjecture after OpenAI (arXiv:2610.15592, v1) | Exposition | The abstract describes "a simplified and more human-friendly version of the proof of the Deligne–Drinfeld conjecture recently given by OpenAI" and states that no originality is claimed. | arXiv v1. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.