Key points
- The problem: The Cannon conjecture is a core problem in geometric group theory: if a word-hyperbolic group has a two-sphere as its boundary at infinity, must it act geometrically on three-dimensional hyperbolic space? For nearly 40 years no one has produced a complete proof.
- What the AI did: An internal OpenAI frontier model produced a 31-page proof titled "A Modulus Proof of Cannon's Conjecture" (dated September 23, 2026). Its core is the combinatorial modulus method, establishing a "uniform boundary modulus bound" that predecessors could not prove, and completing the argument with boundary geometry, continuous limit functions and two proofs by contradiction.
- Machine-verifiable: The paper comes with a Lean formalization whose theorem statements can be checked by machine, a key difference between AI mathematics results and traditional manuscripts.
Analysis
The proof of the Cannon conjecture is one of the most geometry-heavy single results among the 722 manuscripts released in bulk on October 6. It shows not just that AI is "fast at computing" but that it can construct, in a highly structured area of pure mathematics, key lemmas on which human experts had been stuck. Still, until independent mathematicians complete their review, this remains a "claimed proof," but the existence of the Lean version gives verification a machine-executable starting point.
Source: Cannon Conjecture: OpenAI's Breakthrough Proof Of 3D Geometry. This text is a summary for reference only.
