新智元报道
困扰数学界30余年的世界级难题——三维Ghys猜想迎来了破解！
大湾区大学教授许地生团队借助中国科学多模态大模型，通过人机协同完成了三维Ghys猜想证明。北京大学董彬教授、刘若川院士团队利用其开发的SHEAF框架对本次证明完成形式化验证，验证了这一工作的正确性。这标志着中国科学家首次用中国大模型攻克世界级数学难题。
「Ghys猜想」由法国数学家艾蒂安·吉斯（Étienne Ghys）于1995年提出，猜测每一个紧致复流形上的全纯Anosov系统，都双全纯共轭于某个复infra-幂零流形上的全纯仿射自同构。该问题兼具理论难度与学科标志性价值，被动力系统领域数学专家公认为该领域最重要的长期公开问题之一，是检验数学推理能力的一块试金石。
相关论文已在预印本平台（arXiv）发布，其形式化代码也在Github平台公开。
论文链接：https://arxiv.org/pdf/2609.37695
形式化代码：https://github.com/frenzymath/HolomorphicAnosov
1995年，法国数学家艾蒂安·吉斯（Étienne Ghys）提出Ghys猜想，其核心是：混沌的动力系统行为的存在是否能决定几何上的分类。
这是复动力系统领域的一个重要难题。如果这一猜想成立，混乱的动力学轨迹的问题就能被「化归」成简洁、规整的标准代数模型。
如何理解Ghys猜想？可以想象在一个封闭有限、没有边界、每一小块都遵守复数运算规则的「世界」。
对Anosov系统，可借助揉面类比：按压、延展、折叠面团，部分方向被拉伸，部分方向被压缩。拉伸效应会让初始紧邻的点快速分离——初始条件微小差异，会带来截然不同的演化结果。
Ghys猜想说，只要存在这样一种「既像揉面又完全规则」的运动，那么这个「世界」就只能是一种由简单规则反复拼接、像晶体格子一样高度规整的形状。例如复环面：
换个视角看，这个运动本质上就是这个「世界」的简单平移、旋转或缩放。也就是说，运动方式太混沌，会反过来决定世界只能长成什么形状。
2026年国际数学家大会45分钟报告人，北京大学北京国际数学研究中心谢俊逸教授评价：「Ghys猜想的难点在于其对应的实（数）版本的类似断言仍是著名的完全公开的难题，复数的情形问题变得更加刚性，是复动力系统刚性现象研究中的重要问题，非常困难」。
大湾区大学教授许地生长期深耕该猜想的研究，曾与其导师、菲尔兹奖得主阿图尔·阿维拉（Avila）教授合作研究三维情形，但未形成有效的证明思路；此后他与其学生张洁松博士合作在三维情形下取得Ghys猜想的部分弱相关结果，相关成果刊发于数学顶级期刊《欧洲数学会杂志》（JEMS）。
「科学发现是对智能的终极考验。」
周伯文作出过一个判断：问题越古老、越基础，AI提效的空间越大——AI参与攻关的单克隆抗体、蛋白质折叠，以及至今悬而未决的千禧年难题纳维-斯托克斯方程，都符合这条规律。三维Ghys猜想正是这类基础性难题中的一个。
2026年3月，许地生教授与国内某AI实验室的科研团队组建联合团队，并同步搭建了面向数学研究场景的Harness系统。许地生教授结合数学科研经验，精准指出模型与系统的能力短板；依托真实前沿数学研究命题，驱动系统完成多轮迭代打磨。
针对三维Ghys猜想的首轮推理，模型产出400页推导报告。
在训练策略层面，科研团队坚持「通专融合」技术路线，在大规模科学知识预训练夯实模型底座基础上，精心构造研究级后训练数据，包括大规模来自数学论文的难题和可靠的校验方案，提升模型在研究级数学上的专业性、严谨性和规范性。
在训练算法层面，科研团队突破了异步长程推理强化学习训练方法，在可验证奖励下破解了误差指数级累积、长期状态维护、自我纠错等一系列根本性难题；团队还同时研发了将Harness带来的能力提升反向蒸馏回模型的自进化算法，在大量超出模型直推能力边界的高难度问题上持续获得有效监督信号，推进模型能力的自主提升。
据许地生教授团队介绍，基模能力已具备辅助科研团队攻关研究级数学问题的基本水平，但距离真正破解世界级难题，还需要更上一层楼，突破的关键在于更紧密的人机协同：
大模型及配套系统发挥长链条大规模推演的优势，铺开多条支线同步尝试、拆解复杂命题、猜想不同的思路，把海量繁琐的推导工作承接下来，每一轮迭代给出海量尝试结果；
研究人员则负责把握大方向，判断哪些思路值得深挖、哪些路径是死胡同、给出一些直觉建议，作为系统下一轮迭代的指令。二者并非单方独立推导，而是人机协同，通过多轮对话和反馈，在紧密互动中进行迭代。
系统的多轮交互过程及各轮交互中的递归求解结构。其中，边仅表示子问题的分解关系，不表示上下文依赖关系；每个节点均可访问时序上先于该节点的全部全局上下文。
在AI给出的首轮探索基础上，数学家许地生与张洁松敏锐抓住关键突破口，持续给模型反馈、校准探索方向，引导模型向有价值的思路深耕。
攻关过程中，大模型先后完成7批接续推演，问题图从15个节点扩展到389个，输出总量达上千页的严密推导，同时生成数十种全新论证思路，成功助力团队破解这一世界级数学难题，得到该方向国际前沿学者「思考深度远超预期」的评价。
有意思的是，在论文上传预印本网站arXiv之前，前文提及的动力系统专家谢俊逸教授也曾用GPT-6尝试挑战这道题，并没有找到证明方案。「这不仅印证了国内大模型在本次前沿数学探索中的独特价值，同时也传递出一个信号——中国的大模型同样有能力站上前沿数学的赛场，助力世界级难题的攻关。」
AI助力产出的成果是否经得起验证？
北京大学董彬教授、刘若川院士团队完成了本次论文全部四个主定理的形式化验证，证明已通过Lean检验。Lean作为交互式定理证明系统，能对形式化证明进行严格逻辑检查。
北大团队骨干成员姜杰东、吴彬介绍，整个过程仅约5天，峰值并发达100个智能体，产出约55万行Lean代码，展现出极大的并行规模与极快的验证速度。专家确认形式化陈述与论文主定理语义一致，Lean验证据此确保其严格正确性。
形式化的难度首先来自数学本身：自然语言证明常依赖读者的数学知识补全条件与推理，形式化则须将其落实为计算机可检查的完整证明。论文引用的结果若未进入基础定理库，还需补充形式化证明，或寻找利用已有结果的替代路线。
尤其动力系统领域原有形式化基础设施不足，此次工作补充了大量基础设施，覆盖论文依赖的众多文献结果。
对研究级数学而言，这些工作涉及定义的准确表达、证明结构的拆解和数学工具的选择；大量智能体并行时，还须确保各部分的定义、陈述和依赖相互衔接，局部证明才能组成完整证明链。
为组织这一过程，北大团队使用了自主开发的SHEAF框架：多个智能体将论文中的命题、定义及依赖关系组织成一张从目标定理逐层延伸到基础定理库的有向无环图；
模型在框架下参与命题分解与形式化证明构建，团队围绕该图编排工作，跟踪各节点的陈述、证明与剪枝状态，将大规模形式化任务分解为相互衔接、可检查的步骤。
早在 2021 年大模型尚未问世之时，北京大学北京国际数学研究中心董彬教授团队便敏锐洞察到人工智能将重塑数学研究范式。
北大数学团队是国际上最早布局AI for Math方向的研究团队之一，搭建起覆盖高难度评测集（FATE）、大规模数学知识检索（Matlas/LeanSearch）、自动推理与形式化验证（Rethlas/Archon/Danus/Iteris）的完整技术链条；
其开发的AI系统成功攻克安德森猜想这一公开数学难题，实现国内首例、同时也是国际上首个由高校团队主导，依托人工智能自主突破重大数学猜想的成果。近期，团队还独立完成了庞加莱猜想的形式化验证工作。
本项研究的意义还在于验证了一种新的科研范式。传统数学研究依赖个体天才的漫长摸索，而本次攻关将专家直觉系统性接入机器求解回路，形成「系统探索—专家判断—成果积累—反馈修正」的闭环。
这一范式可复制推广至更多「搜索空间巨大而专家稀缺」的基础研究领域，实现更多「从0到1」的原始创新突破。
参考资料：
https://arxiv.org/pdf/2609.37695
编辑：LRST
不喜欢
本文由 AI图鉴自动抓取并翻译，版权归原作者所有；如需查看原文请点下方链接。
New Ji-won.
The world-class puzzle that has plagued mathematics for more than 30 years -- three-dimensional Ghys guess has come up with solutions!
With the help of the Chinese science multi-modular LLM, the Shigeki team, a professor at the University of the Bay, completed a three-dimensional Ghys conjecture through a human machine. The formalization of this certificate was validated by the team of Professor Tung Bin of Beijing University and Master Liu Joagawa using the SHEAF framework developed. This marks the first time that Chinese scientists have used China's LLM to tackle world-class mathematical difficulties.
"Ghys Conjecture" was proposed by French mathematician Étienne Ghys in 1995 and speculates that every strictly pure Anosov system in a condensed retrending form is identical to a pure imitation of a non-flail retre. The problem, which is both theoretically difficult and discipline-based, is recognized by mathematical experts in the field of power systems as one of the most important long-term public problems in this field, and is a litmus test of mathematical reasoning.
The papers have been preprinted on the platform (arXiv) publish and their formalization code is available on the Github platform.
Paper link: https://arxiv.org/pdf/2609.37695
Formatted code: https://github.com/french/HolomopropicAnosov
In 1995, Étienne Ghys, a French mathematician, put forward the idea of Ghys, the core of which was whether the existence of a confused act of power system could determine geometric classification.
This is an important challenge in the field of regenerative power systems. If this assumption is established, the problem of a chaotic dynamic trajectory can be "regulated" into a simple, well-defined standard algebra model.
How do you understand Ghys' guess? It is possible to imagine a "world" in which there are limited closures, no boundaries, and every small block is subject to the plural operating rules.
Anosov systems can be applied by rubbing: by pressure, extension, folding, partial stretching and partial compression. The stretching effect allows for the rapid separation of points from the initial immediate vicinity — a slight difference in the initial conditions — resulting in a very different evolution.
Ghys assumed that as long as there was such a movement, which was both a face-to-face and a full rule, the world could only be a shape that was collided over and over again by simple rules, as well as a crystal grid. For example:
In other words, the movement is essentially a simple smoothing, rotation or scaling of the world. In other words, the way in which the movement is carried out is too confusing to determine what shape the world can only grow.
The 45-minute reporter of the International Mathematics Congress in 2026, Professor Xie Joon-woo of the Beijing International Centre for Research in Mathematics at Beijing University, commented: “The hard part of Ghys’ conjecture is that a similar assertion of the true (number) version of his response remains a well-known problem of complete openness, and the problem of complex situations becomes more rigid, an important issue in the study of the rigidity of the regenerative system, and very difficult.”
The University of the Bay Region, which taught Shigeki the idea of long-term deep cultivation, had worked with his mentor and winner of the Philtz Prize, Professor Atul Avila, to study the three-dimensional situation, but had not developed a valid line of proof; since then, he had worked with his student, Dr. Zhang Jeysson, to obtain some of the weak results associated with Ghys' guess in a three-dimensional context, the results of which were published in the top-level journal of mathematics, Journal of the European Digital Institute (JEMS).
"Scientific discoveries are the ultimate test of intelligence."
Zhou Boven has made a judgment that the older the problem, the more fundamental it is, the greater the space for AI to be effective — the monoclon antibodies that AI is involved in the attack, the folding of proteins, and the millennium dilemma Navi-Stox equation, which is still pending. Three-dimensional Ghys conjecture is one of these basic dilemmas.
In March 2026, Professor Xu Landson formed a joint team with the scientific team of an AI laboratory in the country and synchronized the establishment of the Harness system for mathematical research scenarios. In combination with the experience of mathematics science, Professor Quantino has precisely pointed out the ability of models and systems to be short boarded; depending on the real front-line mathematical research proposition, driving the system to do multi-temperature grinding.
The model produces 400 pages of extrapolation reports for the first round of reasoning on three-dimensional Ghys conjecture.
At the level of training strategies, the scientific team follows the technical path of "specialization" and elaborates post-research training data, based on large-scale scientific pre-training models, including large-scale problems from mathematical thesis and reliable validation programmes, and upgrades the specialization, rigour and regulation of models in research-level mathematics.
At the level of training algorithms, the scientific team has broken through the intensive learning and training methods of long-range distance reasoning, breaking a series of fundamental challenges such as the accumulation of error index levels, long-term state maintenance and self-correction under verifiable incentives; it has also developed a self-evolutional algorithm that brings Harness to the reverse distillation model, continuously receiving effective monitoring signals on a large number of high-probability issues that go beyond the boundaries of the model ' s ability to push straight forward, and boosting the autonomy of model capabilities.
According to the team of Professor Xuejien, the foundational modelling capability is already at a basic level to assist the scientific team in addressing mathematical problems at the level of research, but more needs to be done before the world-class problem can really be solved. The key to the breakthrough is closer human synergy:
LLM and its associated systems, with the advantage of a long chain of large-scale roll-out, spread multiple feeder synchronized attempts, decomplicated complex propositions, and conjecture different lines of thought, carry the heavy loads of the work ahead and give a big trial result for each of the rotations;
Researchers are responsible for taking the larger direction, determining which lines of thought are worth digging and which paths are dead end, giving some intuitive suggestions, as instructions for the next iterative cycle of the system. The two are not unilateral independent extrapolations, but rather human synergies, with multiple rounds of dialogue and feedback and iterative interactions.
The multi-wheel interactive process of the system and the retrogressive solvency structure of the cycle. Of these, only the decomposition of the sub-issue does not mean the dependency of context; each node can access the entire context system ahead of that node.
Based on the first round of exploration given by AI, Mathematicians Xue and Zhang Jie-soon sensibly seized critical breakthroughs, continuously giving feedback to models, calibrating the directions of exploration, and guiding models deep into valuable thinking.
In the course of the attack, LLM completed seven series of rollovers, the problem map was expanded from 15 nodes to 389, with thousands of pages of output closely extrapolated, and produced dozens of completely new lines of reasoning that helped the team solve this world-class mathematical dilemma, which was evaluated by the international front-runners in the direction of "thinking depths far beyond expectations."
Interestingly, before the paper was uploaded to the paper’s arXiv site, Professor Xie Joon-soo, the above-mentioned power system expert, also tried to challenge the subject with GPT-6, without finding a proven solution. “Not only does LLM’s unique value in this cutting-edge mathematical exploration have been demonstrated, but it has also sent a signal that – - China's LLM also has the ability to stand on the cutting edge of mathematics to help the world-class challenge.”
Can AI-assisted outputs be validated?
Prof. Tung Bin of Beijing University and the team of Masters of Liu Yookawa completed the formalization of all four principals of the present paper, which proved to have passed the Lean test. As an interactive theorem certification system, Lean is able to conduct a rigorous logical examination of formalization certificates.
Kang Jie-dong and Oh Bin, members of the northern team, described the process as having lasted only about five days, peaking and developing 100 AI agents, producing about 550,000 lines of Lean code, showing a huge parallel scale and a very fast pace of validation. Experts confirmed that the formalized statements were consistent with the theorem of the paper, and Lean certified them as strictly correct.
The difficulty of formalization arises first and foremost from the mathematics itself: the natural language proves that it is often dependent on the reader ' s mathematical knowledge completion conditions and reasoning, while formalization must be translated into a complete computer-readable proof. If the results cited in the paper do not enter the base theorem, they need to be supplemented by formalized certificates or alternative routes to take advantage of the results already achieved.
In particular, there is a lack of formal infrastructure in the field of power systems, which has been supplemented by a large amount of infrastructure, covering the numerous literature on which the papers rely.
For research-level mathematics, these involve the precise expression of the definition, the decomposition of the structure and the selection of mathematical tools; while a large number of AI agents go hand in hand, it is also necessary to ensure that parts are defined, presented and relied upon in order to form a complete chain of proof.
To organize this process, the Great Northern Group used the self-developed SHEAF framework: several AI agents organized the titles, definitions and dependencies of the papers into a free-ring map extending from the goal theorem to the base theorem;
Within the framework of this project, the team is involved in the tasking and formalization of the project, which is structured around this map, following up on the statements, certificates and cuttings of the various nodes and decomposing large-scale formalization tasks into interconnective, detectable steps.
As early as 2021, when LLM was not introduced, the team of Professor Tung Bin, the Beijing International Centre for Research in Mathematics at Beijing University, was keenly aware that artificial intelligence would reshape the mathematical research paradigm.
The Northern Grand Mathematics Team is one of the first international teams to organize the AI for Math orientation, setting up a complete technology chain to cover the high-cost assessment (FATE), large-scale mathematical knowledge retrieval (Matlas/LeanSearch), automatic reasoning and formalization (Rethlas/Archon/Danus/Iteris);
The AI system that it developed successfully addressed the open mathematical dilemma of Anderson, achieving the first time in the country and the first ever in the world to be led by a college team, relying on artificial intelligence autonomy to break through major mathematical conjectures. Recently, the team has also independently completed the formalization of Pongarai ' s guess.
The significance of the study is also to validate a new scientific paradigm. Traditional mathematics research relies on a long search by individual geniuses, and this attack has led to the intuitive and systematic access of specialists to the machine, leading to a "system-exploration-expert judgement-accumulation-result correction."
This paradigm can be replicated in more basic research areas where "search space is vast and expertise is scarce" and where more "0-1" original innovation breakthroughs can be achieved.
References:
https://arxiv.org/pdf/2609.37695
Edit: LRST
I don't like it.
Auto-fetched and translated by AI Atlas. All rights belong to the original author.