For three decades, a French mathematician's question about chaos and geometry stayed open. This autumn, the proof was not assembled by one mind working alone. It was built through a mathematician and a model trading drafts through the night, round after round, until a thousand pages of working material narrowed to a single clean argument.
The model was Intern-S2 (书生-S2), a scientific research model from Shanghai AI Lab (上海人工智能实验室).
The problem and the partnership
Ghys's conjecture was proposed in 1995 by Étienne Ghys. It asks a deceptively simple thing: if a space admits a certain kind of chaotic dynamics (a holomorphic Anosov system), does that force the space to have a rigid, classifiable algebraic structure? The real-version case is still unsolved and is a standard yardstick for high-level mathematical reasoning.
The resolution in complex dimension three came from a collaboration led by Disheng Xu, a professor at Great Bay University and visiting scholar at Shanghai AI Lab, with his student and collaborator Jiesong Zhang of KTH Royal Institute of Technology. Xu had studied the three-dimensional case for years, including with Fields Medal winner Artur Avila. What changed this time was a partner that could shoulder the brute-force exploration.
How the model and the mathematicians worked
Starting in August 2026, Intern-S2 and its supporting system explored the conjecture from many angles. Its first round produced a roughly 400-page report of candidate arguments. Xu and Zhang spotted a promising opening inside that material, then used repeated dialogue to test the logic, abandon dead ends and develop the useful threads.
Across seven successive rounds, the system's "problem graph" expanded from 15 nodes to 389, generating around a thousand pages of working material and dozens of possible proof lines. The mathematicians judged which ideas mattered, verified the steps, and fed constraints back in. The final proof grew out of that loop — not from the model alone, and not from the humans alone.
Intern-S2 combines broad scientific pretraining with specialized training on research mathematics. Shanghai AI Lab describes breakthroughs in asynchronous long-horizon reinforcement-learning training under verifiable rewards, plus a method that distills the system's own gains back into the model. On IMO-Proof and AdvancedMathBench, the lab says Intern-S2 leads existing open models and beats Gemini 3.1 Pro on some metrics, placing it among the strongest open-source long-horizon reasoning models.
Why this is a reasoning benchmark in itself
A math proof is the hardest kind of long-horizon task: one bad step invalidates the whole chain, and the path is unknown in advance. That is exactly what reasoning models are measured on, except most benchmarks score a final answer. Here the model had to sustain coherent argument across hundreds of branches and accept correction.
The result was independently checked. A Peking University team led by Prof. Bin Dong and academician Ruochuan Liu formalized the main theorems in Lean 4 using their SHEAF pipeline, producing about 550,000 lines of Lean code and finishing roughly five days after the paper appeared. At peak, 100 concurrent agents worked the sub-tasks. Formal verification turns "the mathematicians believe it" into "the proof checks."
What it says about open Chinese models
The preprint (arXiv:2609.37695) and the formalization code are public. Intern-S2's predecessor, Intern-S1, scored 102 in an official AI evaluation at the 2025 Chinese Mathematical Olympiad finals — above the human gold-medal threshold of 78 — which is what first drew researchers to test its depth. The lab frames the Ghys work as evidence that a Chinese large model (大模型) can stand on the frontier of research mathematics, not just imitate it.
Why a single proof matters beyond mathematics
A resolved conjecture is a clean result, yet the larger signal is how the work was produced. The loop of a thousand pages of working material, an expanding problem graph, and a human partner steering it is a template that travels. A chemistry group hunting a reaction pathway, a systems team tracing a compiler bug across hundreds of files, or an analyst mapping a regulatory edge case all face the same shape of problem: long, branchy, and unforgiving of a single wrong turn. The Ghys case suggests the value was not the model answering harder questions on its own, but the model absorbing the tedious exploration that used to consume the most patient hours of a researcher's week, leaving the human to judge rather than to grind. The formal Lean check then converts that human judgement into something a machine can vouch for, which is what makes the result usable outside the room where it was found.
Honest limitations
- The model did not produce the proof by itself. Mathematicians set direction, filtered reasoning, and took full responsibility for the manuscript. This is human-in-the-loop research, not autonomous mathematics.
- The conjecture remains open in higher dimensions. One resolved case is a milestone, not a general method.
- The "reasoning leap" is demonstrated on a specific, deeply studied problem class. It does not imply the model sweeps every benchmark.
- Formal verification covered the codimension-one (complex dimension three) case as presented; broader generalization is unproven.
- Reproducing the workflow needs the surrounding system (problem-graph tooling, human experts), not just the base model weights.
What readers can do now
- Read the open artifacts. The paper (arXiv:2609.37695) and the Lean formalization repository are public — a rare chance to study a machine-assisted proof end to end if you work in math or formal verification.
- Copy the workflow, not just the model. The 15-to-389-node "problem graph" loop is a reusable template for any research or engineering team building agents on hard, branch-heavy problems.
- Watch for the open release. If you build research-math tooling, track Intern-S2's public availability to test it on your own domain-specific reasoning tasks.