Ghys’s 1995 conjecture proved in the codimension-one case with Shanghai AI Lab’s Intern-S2
Peking University formalizes it in Lean (~550k lines)
Result confirmed
The takeaway
Jiesong Zhang’s preprint proves that every codimension-one holomorphic Anosov diffeomorphism of a compact complex manifold is conjugate to a hyperbolic torus automorphism, settling Ghys’s conjecture in that case and in complex dimension three. It declares that an internal Intern-S2 helped develop the ideas; a Peking University team formalized the main theorems in Lean with its SHEAF agent.
Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Lean-verified
- Importance
- 3 of 5
- Last verified
- 15 October 2026
Your AI and this story
- GPT-6 Astra168 days after its cutoff
- Claude Opus 5.5107 days after its cutoff
- Gemini 4 Argon198 days after its cutoff
- Grok 4.7137 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 107 days before it.
Key facts
- Paper: arXiv 2609.37695 [math.DS], single author Jiesong Zhang; v3 (13 Oct) ‘Added a declaration of AI use, a statement on proof formalisation, and acknowledgements. 19 pages’
- Abstract: ‘every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This verifies a conjecture of Ghys in the codimension-one case and, in particular, in complex dimension three.’ (Complex dimension 2 already follows from Ghys 1995, Theorem A)
- Declaration of AI use (v3): ‘The author used an internal version of Intern-S2 to assist in developing the mathematical ideas and arguments presented in this paper through multiple rounds of dialogue … The author has independently verified all mathematical arguments and takes full responsibility for the manuscript.’
- Formalisation (v3): Theorem A and Theorems 1.1–1.3, plus the Ghys 1995 results used, ‘have been formalised in Lean by the Bin Dong–Ruochuan Liu team in Peking University, using the open-source formalisation agent SHEAF’; carried out by Jiedong Jiang and Bin Wu; code at github.com/frenzymath/HolomorphicAnosov (repo created 13 Oct 2026)
- Shanghai AI Lab (15 Oct): Intern-S2 (书生-S2) produced a ~400-page derivation report in its first round; over 7 rounds the problem graph grew from 15 to 389 nodes, with 1,000+ pages of derivations; the Lean check took about 5 days with up to 100 agents in parallel and ~550,000 lines of Lean (lab’s figures, not checked by us)
- Who led: the lab and Global Times credit a team led by Disheng Xu (Great Bay University, visiting scholar at the lab) with Jiesong Zhang (KTH, per the lab); the arXiv paper lists only Zhang as author and thanks Xu for discussions and for hosting his stay at the lab
- Lab’s account: Junyi Xie called the complex case ‘very difficult’ and reportedly tried GPT-6 on it without success (lab claim, not independently verified)
- Section 5 builds a holomorphic Anosov automorphism of a nilmanifold whose invariant distributions are smooth but not holomorphic, so the paper’s regularity theorem fails in higher codimension; the full conjecture remains open
What happened
On 15 Oct 2026 the Shanghai AI Laboratory said its Intern-S2 model (书生-S2) had helped prove the three-dimensional case of a conjecture Étienne Ghys posed in 1995 about holomorphic Anosov systems, and that a Peking University team had checked the proof in Lean. The paper behind it is Jiesong Zhang’s arXiv preprint 2609.37695, first posted on 29 Sept. Its third version (13 Oct) adds a declaration of AI use and a formalization statement. It proves the whole codimension-one case. Complex dimension three follows, since dimension two was already settled by Ghys.
The paper credits Intern-S2 with helping “in developing the mathematical ideas and arguments … through multiple rounds of dialogue” and says the author verified all the arguments himself. The press accounts describe a team led by Disheng Xu, while the arXiv author line names only Zhang, who thanks Xu for discussions. The figures on derivation pages, graph nodes, agents and lines of Lean code come from the lab’s announcement.
Why it matters
Most AI-assisted proofs of named conjectures in this dataset used US models (GPT-6 Astra, Gemini, Claude). This one was done with a Chinese lab’s model and presented by Chinese state media as a national milestone. The Lean formalization, done with a separate agent system (SHEAF), is public, and it covers the paper’s main theorems as well as the 1995 results it relies on. The general conjecture, beyond codimension one, is still open.
Sources
7 sources from 6 sites. Numbers match the chips in the text.
7 sources: 4 primary, 3 press
Primary
- arXiv 2609.37695: Codimension-one holomorphic Anosov diffeomorphisms (J. Zhang)arxiv.org, paper
- GitHub: frenzymath/HolomorphicAnosov (Lean formalization)github.com, code
- GitHub: frenzymath/SHEAF (formalization agent)github.com, code
- Shanghai AI Laboratory announcement (Chinese, 15 Oct)shlab.org.cn, official
Press
Changes
- Filed (Global Times; read arXiv v3 PDF and Shanghai AI Lab post)