Mathematicians working with Shanghai AI Lab's Intern-S2 resolve Ghys's conjecture in complex dimension three; four main theorems checked in Lean by up to 100 agents in ~5 days
Shanghai AI Lab announced on 2026-10-15 that Disheng Xu (Great Bay University) and Jiesong Zhang (KTH), aided by an internal version of Intern-S2, proved the codimension-one case of Ghys's 1995 holomorphic Anosov conjecture, including complex dimension three (arXiv:2609.37695). Peking University's team formalised all four main theorems in Lean with the SHEAF framework in about five days with up to 100 concurrent agents and ~550k lines, sorry-free and accepted by Lean comparator, with Hsiang–Wall (1969) as an explicit hypothesis.
Key Takeaways
- Result: codimension-one Ghys conjecture proved — the manifold is a complex torus and the map is holomorphically conjugate to a hyperbolic torus automorphism; higher dimensions remain open (arXiv)
- Human–AI loop: problem graph grew 15 → 389 nodes over seven rounds; ~400-page first report, ~1,000 pages total (announcement)
- Formalisation: ~5 days, up to 100 agents, ~550k lines of Lean (team-reported); sorry-free, three standard axioms (repo)
- Tooling: SHEAF proves top-down over a DAG; in a prior run 62.5% (1,067/1,706) of non-Mathlib nodes were pruned as unnecessary
- Context: Intern-S1 scored 102 in the official AI evaluation at the 2025 Chinese Mathematical Olympiad finals vs a 78-point human gold threshold
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