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

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

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Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

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

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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 $e_3,e_5,\dots$ onto the rational solution space $W$ 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: $W$ 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 $W$ 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.