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The Deligne-Drinfeld conjecture: the Grothendieck-Teichmuller Lie algebra is free on odd generators

Let grt1\mathfrak{grt}_1 be the graded space of Lie polynomials ψ(x,y)\psi(x,y) in two letters satisfying antisymmetry, the hexagon (three-term) relation and the pentagon relation in the infinitesimal pure braid Lie algebra t4\mathfrak t_4, with the Ihara bracket. Ihara and Drinfeld produced nonzero elements σ2k+1\sigma_{2k+1} in each odd weight 2k+1≥32k+1\ge3; Brown's mixed Tate motives theorem (2012) shows they can be chosen to generate a free Lie subalgebra, and Naef-Willwacher checked equality through weight 29. The Deligne-Drinfeld conjecture asserts that grt1\mathfrak{grt}_1 is exactly a free Lie algebra with one generator in each odd weight 3,5,7,…3,5,7,\dots. Is grt1⊗Q\mathfrak{grt}_1\otimes\mathbb Q freely generated by one element in each odd weight at least three?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Grothendieck-Teichmuller theory; multiple zeta values
Posed by
Pierre Deligne and Vladimir Drinfeld
Year posed
1990
Years open
36y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there are homogeneous σ2k+1∈W2k+1\sigma_{2k+1}\in W_{2k+1} such that e2k+1↦σ2k+1e_{2k+1}\mapsto\sigma_{2k+1} is a graded Lie isomorphism LieQ⟨e3,e5,… ⟩≅(W,{ ,})\mathrm{Lie}_{\mathbb Q}\langle e_3,e_5,\dots\rangle\cong(W,\{\,,\}), with WW closed under the Ihara bracket, and the induced map on weight completions is a continuous isomorphism. Via Willwacher's theorem this gives H0(GC2)≅Lie⟨e3,e5,… ⟩H^0(\mathrm{GC}_2)\cong\mathrm{Lie}\langle e_3,e_5,\dots\rangle. With Brown's theorem it identifies grt1\mathfrak{grt}_1 with the motivic Lie algebra. Generators are not canonical, and no other graph-complex degrees are computed.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 23, 2026.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it asserts a graded Lie isomorphism from the free Lie algebra on e3,e5,…e_3,e_5,\dots onto the rational solution space WW of (1.2)-(1.4) with the Ihara bracket, and the completed isomorphism. The key new input is the all-weight dimension upper bound (Corollary 5.4), proved by a characteristic-two degeneration of the pentagon. lean/formalization.yaml lists a main result (comparator DeligneDrinfeld, declaration OAI.DeligneDrinfeld.main). The comparator statement was read here: WW is defined as the rational solutions of the three equations, and the statement gives a weight-graded linear equivalence from the free Lie algebra on odd weights to WW intertwining the bracket with the Ihara bracket, plus continuous inverse maps on completions. This states the headline. lean/docs/008.md says the graph-complex consequence is not formalized. Not rebuilt here.

Sources

Changelog1 change

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