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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Core Background & Industry Pain Points
In 1995 Étienne Ghys conjectured a classification of holomorphic Anosov systems; the complex-dimension-three case stayed open for 30+ years. On 2026-10-15 Shanghai AI Lab announced that Disheng Xu (Great Bay University) and Jiesong Zhang (KTH), working with an internal version of Intern-S2, settled the codimension-one case, which includes complex dimension three (arXiv:2609.37695).
Architecture Highlights & Internals
Starting August 2026, the first round produced a ~400-page report of candidate arguments; over seven rounds the problem graph grew from 15 to 389 nodes and ~1,000 pages of working material. The mathematicians judged, redirected, and assembled the proof; the paper's AI-use statement says Zhang verified the arguments independently and takes full responsibility. Peking University's team formalised the four main theorems with SHEAF, which organises a paper into a DAG of statements, proves top-down, prunes unused nodes, and lets many agents share one Lean library.
Authoritative Benchmarks & Measured Scores
No leaderboard applies. Team-reported figures: ~5 days, up to 100 concurrent agents, ~550k lines of Lean. The HolomorphicAnosov repo states the four theorems are proved with no sorry, using only propext, Classical.choice and Quot.sound, accepted by Lean comparator; Hsiang–Wall's 1969 homotopy-torus rigidity is an explicit hypothesis for Theorems A, 1.1 and 1.3 (not 1.2).
Developer Hands-on Guide
Verify with lake exe cache get, lake build, then lake env comparator Comparator/config.json (needs comparator, lean4export, landrun; Lean v4.35.0-rc1). To run your own formalisation, clone SHEAF and start Claude Code or Codex in tmux; the agent reads AGENTS.md. The conjecture remains open in higher dimensions in general.
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