{"schema":"postcutoff/event@1","as_of":"2026-10-15T23:34:00+02:00","url":"https://postcutoff.com/e/2026-10-15-ghys-conjecture-codimension-one-intern-s2-lean/","md":"https://postcutoff.com/e/2026-10-15-ghys-conjecture-codimension-one-intern-s2-lean/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-15-ghys-conjecture-codimension-one-intern-s2-lean","date":"2026-10-15","date_precision":"day","short_title":"Ghys's 1995 conjecture proved in the codimension-one case with Shanghai AI Lab's Intern-S2","deck":"Peking University formalizes it in Lean (~550k lines)","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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2609.37695: Codimension-one holomorphic Anosov diffeomorphisms (J. Zhang)","url":"https://arxiv.org/abs/2609.37695","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: frenzymath/HolomorphicAnosov (Lean formalization)","url":"https://github.com/frenzymath/HolomorphicAnosov","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"GitHub: frenzymath/SHEAF (formalization agent)","url":"https://github.com/frenzymath/SHEAF","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Shanghai AI Laboratory announcement (Chinese, 15 Oct)","url":"https://www.shlab.org.cn/news/5444307","type":"official","group":"primary","domain":"shlab.org.cn"},{"n":5,"title":"Global Times: Chinese scientists use homegrown LLM to solve world-class math problem","url":"https://www.globaltimes.cn/page/202610/1372364.shtml","type":"press","group":"press","domain":"globaltimes.cn"},{"n":6,"title":"Pandaily: Mathematicians use Intern-S2 to prove 3D Ghys conjecture, verified in Lean","url":"https://pandaily.com/shanghai-ai-lab-intern-s2-ghys-conjecture-proof-human-ai-collaboration-lean-sheaf","type":"press","group":"press","domain":"pandaily.com"},{"n":7,"title":"HyperAI: Chinese scientists use Shusheng-S2 to prove Ghys conjecture","url":"https://hyper.ai/en/stories/4cc34ccf7304036ca06b0eac04605283","type":"press","group":"press","domain":"hyper.ai"}],"official":4,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Shanghai AI Laboratory","Peking University"],"title":"Ghys's 1995 conjecture proved in the codimension-one case (incl. complex dimension 3) with Shanghai AI Lab's Intern-S2; Peking University formalizes it in Lean (~550k lines)","summary":"arXiv 2609.37695 (v1 29 Sept, v3 13 Oct 2026), \"Codimension-one holomorphic Anosov diffeomorphisms\" by Jiesong Zhang, proves that 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 Étienne Ghys's 1995 conjecture in the codimension-one case and so in complex dimension three. The v3 declares that an internal version of Shanghai AI Lab's Intern-S2 helped develop the ideas over many rounds of dialogue; Bin Dong and Ruochuan Liu's Peking University team formalized the main theorems in Lean with its open-source SHEAF agent. Shanghai AI Lab announced it on 15 Oct.","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"],"key_numbers":[],"tags":["math","dynamical-systems","complex-geometry","anosov","ghys-conjecture","intern-s2","lean","china"],"science":{"field":"mathematics","subfield":"dynamical systems / complex geometry","problem":"Ghys's conjecture (1995): every holomorphic Anosov diffeomorphism of a compact complex manifold is conjugate to a holomorphic affine automorphism of a complex infra-nilmanifold","result":"Proved in the codimension-one case: such maps are biholomorphically conjugate to hyperbolic automorphisms of a complex torus; this covers complex dimension 3.","open_since":"1995","ai_system":["Intern-S2 (internal version)","SHEAF"],"human_role":"AI-assisted: the author used Intern-S2 in many rounds of dialogue to develop ideas and arguments, then verified everything himself; the Lean formalization was agent-driven (SHEAF) with human checks of the final theorem","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean (Peking University, SHEAF; repo public); unrefereed arXiv preprint"},"status":"confirmed","shock":""},"body_md":"## What happened\n\nOn 15 Oct 2026 the Shanghai AI Laboratory said its Intern-S2 model (书生-S2) had helped prove the three-dimensional case of a\nconjecture Étienne Ghys posed in 1995 about holomorphic Anosov systems, and that a Peking University team had checked the\nproof in Lean. The paper behind it is Jiesong Zhang's arXiv preprint 2609.37695, first posted on 29 Sept. Its third version\n(13 Oct) adds a declaration of AI use and a formalization statement. It proves the whole codimension-one case. Complex\ndimension three follows, since dimension two was already settled by Ghys.\n\nThe paper credits Intern-S2 with helping \"in developing the mathematical ideas and arguments … through multiple rounds of\ndialogue\" and says the author verified all the arguments himself. The press accounts describe a team led by Disheng Xu,\nwhile the arXiv author line names only Zhang, who thanks Xu for discussions. The figures on derivation pages, graph nodes,\nagents and lines of Lean code come from the lab's announcement.\n\n## Why it matters\n\nMost AI-assisted proofs of named conjectures in this dataset used US models (GPT-6 Astra, Gemini, Claude). This one was\ndone 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\ncovers the paper's main theorems as well as the 1995 results it relies on. The general conjecture, beyond codimension one,\nis still open.","disputed":[],"related":[{"id":"2026-10-14-ai-assisted-results-mid-october-2026","url":"https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/","date":"2026-10-14","date_precision":"day","short_title":"Mid-October 2026: arXiv results with disclosed AI help","deck":null,"takeaway":"The 13 Oct arXiv listing alone brought a human write-up of GPT-6 Astra's rational exponents proof, Ulam's Lie group question answered with an Astra construction, a better-than-2 k-median algorithm, and about a dozen smaller results whose AI statements range from 'suggested by ChatGPT' to 'GPT 6 Pro was used … to obtain all other new results'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":20,"official":19,"filed":"2026-10-14","updated":"2026-10-15","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/","date":"2026-10-04","date_precision":"day","short_title":"Linear Hadwiger conjecture proved","deck":"K_t-minor-free graphs are O(t)-colourable; proof found by GPT-6 Astra (Norin & Steiner), Lean-verified by Codex","takeaway":"With the 3SUM/APSP refutation and the KLS proofs posted the same week, it marks a step change in AI-originated mathematics: famous problems now fall with proofs largely written by models and checked by machines.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-06","updated":"2026-10-13","orgs":["OpenAI","McGill University","ETH Zurich"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-15","type":"filed","text":"Created (Global Times; read arXiv v3 PDF and Shanghai AI Lab post)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-15","run":null,"sources_read":"Global Times; read arXiv v3 PDF and Shanghai AI Lab post","updated":"2026-10-15","human_review":null,"version":{"date":"2026-10-15"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":168,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":107,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":198,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":137,"in_training_data":false}],"short_url":null}