Shanghai AI Lab's Shusheng-S2 Helps Prove 3D Ghys Conjecture; Peking University Team Verifies in Lean

The Ghys conjecture was proposed in 1995 by French mathematician Étienne Ghys. It posits that every holomorphic Anosov system on a compact complex manifold is biholomorphically conjugate to a holomorphic affine automorphism on some complex infra-nilmanifold. The problem is regarded by experts in dynamical systems as one of the field's most important long-standing open problems and as a test of mathematical reasoning.

According to the report, the conjecture asks whether the existence of chaotic dynamical behavior can determine geometric classification. For an Anosov system, the report uses a kneading analogy: pressing, stretching, and folding dough stretches some directions and compresses others, so nearby initial points separate rapidly and small differences lead to different outcomes. The conjecture says that if such a motion exists and is also completely regular, the underlying space can only be a highly regular shape assembled from simple rules, such as a complex torus, where the motion is essentially translation, rotation, or scaling.

Xie Junyi, a professor at Peking University's Beijing International Center for Mathematical Research and a 45-minute speaker at the 2026 International Congress of Mathematicians, said the difficulty of the Ghys conjecture lies in the fact that the analogous assertion in the real case remains a famous completely open problem, while the complex case is more rigid and an important problem in the study of rigidity phenomena in complex dynamical systems. Xu Disheng has long worked on the conjecture. He previously studied the three-dimensional case with his adviser, Fields Medalist Artur Avila, but did not form an effective proof. Later, with his student Zhang Jiesong, he obtained partial weak related results in the three-dimensional case, published in the Journal of the European Mathematical Society.

In March 2026, Xu became a visiting scholar at Shanghai AI Laboratory, and the collaboration deepened. The two sides formed a joint team and built a Harness system for mathematics research scenarios. Xu pointed out the model and system's capability shortcomings based on his mathematical research experience, and real frontier mathematical problems drove multiple rounds of iteration. In August, the internally iterated Shusheng-S2 and its supporting system had large-scale recursive reasoning and flexible human-machine interaction, a qualitative improvement in handling complex mathematical propositions. In the first round of reasoning on the three-dimensional Ghys conjecture, Shusheng-S2 produced a 400-page derivation report.

On training strategy, the team followed a general-special integration route. It pretrained on large-scale scientific knowledge to build the model base, then constructed research-level post-training data, including large numbers of difficult problems from mathematics papers and reliable verification schemes, to improve the model's professionalism, rigor, and standardization in research-level mathematics. On training algorithms, the team developed asynchronous long-horizon reasoning reinforcement learning, using verifiable rewards to address exponential error accumulation, long-term state maintenance, and self-correction. It also developed a self-evolution algorithm that distills Harness-driven capability gains back into the model, obtaining effective supervision signals on problems beyond the model's direct reasoning boundary.

On benchmarks such as IMO-Proof and AdvancedMathBench, which emphasize long-horizon rigorous reasoning in competition and advanced mathematics, Shusheng-S2 led existing open-source models overall and surpassed top closed-source models such as Gemini 3.1 Pro on some indicators, placing it among the strongest open-source models for long-horizon reasoning, according to the report. The team said that to crack a world-class problem, closer human-machine collaboration was key. Shusheng-S2 and its system handled long chains of large-scale deduction, explored multiple branches, decomposed complex propositions, proposed different approaches, and carried out massive derivation work, while researchers set the overall direction, judged which ideas deserved deeper exploration, identified dead ends, and offered intuitive suggestions as instructions for the next iteration.

Xu Disheng and Zhang Jiesong seized key breakthroughs from the AI's first round of exploration, continuously fed feedback to the model, calibrated the direction, and guided it toward valuable lines. During the effort, Shusheng-S2 completed seven successive batches of deduction. The problem graph expanded from 15 nodes to 389 nodes, with more than a thousand pages of rigorous derivation and dozens of new argument ideas, helping the team solve the problem. Frontline international scholars in the field said the depth of thought far exceeded expectations, the report said. Before the paper was uploaded to arXiv, Xie Junyi, the dynamical systems expert, had also tried the problem with GPT-6 and did not find a proof, according to the report, which said this underlined Shusheng-S2's unique value in this frontier mathematical exploration and signaled that Chinese large models can compete in frontier mathematics.

Peking University's Dong Bin and Liu Ruochuan teams completed formal verification of all four main theorems in the paper. The proof passed Lean, an interactive theorem proving system that strictly checks the logic of formal proofs. Team members Jiang Jiedong and Wu Bin said the process took about five days, with peak concurrency of 100 agents and about 550,000 lines of Lean code, showing large-scale parallelism and fast verification. Experts confirmed that the formal statements are semantically consistent with the paper's main theorems, and Lean verification thereby ensures strict correctness.

Formalization was difficult first because of the mathematics itself. Natural-language proofs often rely on readers' mathematical knowledge to fill in conditions and reasoning, while formalization must turn them into a complete proof checkable by computer. If cited results are not in the basic theorem library, formal proofs must be supplied or alternative routes using existing results found. Formal infrastructure in dynamical systems had been insufficient, and this work added substantial infrastructure covering many literature results the paper depends on. For research-level mathematics, this involves accurate expression of definitions, decomposition of proof structure, and selection of mathematical tools. With many agents working in parallel, definitions, statements, and dependencies must connect so that local proofs form a complete proof chain.

To organize this, the Peking University team used its self-developed SHEAF framework. Multiple agents organized the paper's propositions, definitions, and dependencies into a directed acyclic graph extending from the target theorem down to the basic theorem library. The model participated in proposition decomposition and formal proof construction, and the team coordinated around the graph, tracking statements, proofs, and pruning states at each node, breaking the large formalization task into connected, checkable steps. As early as 2021, before large models emerged, Dong Bin's team at Peking University's Beijing International Center for Mathematical Research saw that AI would reshape mathematical research. The Peking University mathematics team is one of the earliest international teams in AI for Math and has built a technical chain covering high-difficulty evaluation sets (FATE), large-scale mathematical knowledge retrieval (Matlas/LeanSearch), and automated reasoning and formal verification (Rethlas/Archon/Danus/Iteris). Its AI system solved the Anderson conjecture, described as China's first and the world's first major mathematical conjecture autonomously cracked by a university team using AI. Recently, the team also independently completed formal verification of the Poincaré conjecture.

The work's significance also lies in testing a new research paradigm. Traditional mathematical research relies on individual genius and long exploration, while this effort systematically connected expert intuition into the machine-solving loop, forming a closed cycle of system exploration, expert judgment, result accumulation, and feedback correction. This paradigm can be copied and extended to other basic research fields with huge search spaces and scarce experts, the report said. Next, based on the complete capability loop demonstrated by this research, the team will further package it into reusable mathematical verification modules for the Shusheng Duanyan Scientific Discovery Platform, to be called directly in later scientific research. Starting from the three-dimensional Ghys conjecture, Shanghai AI Laboratory will apply the accumulated reasoning capability to other disciplines with homologous mathematical structures. After new breakthroughs, it will package the new reasoning capability back into the Duanyan platform for reuse and seek the next round of breakthroughs with more disciplines, forming a self-evolving flywheel of breakthrough, packaging, reuse, and further breakthrough.