王的猜想，被 AI 一夜通关
困扰人类几十年的世界级难题，AI 只用几天就交卷了。
就在刚刚，OpenAI 宣布其模型一次性带来三项突破：推翻了计算机科学界坚信近 30 年的指数时间假设；证明了所有维度的局部平滑化猜想；为量子色动力学和杨-米尔斯理论构造了严格的数学存在性证明。
OpenAI 宣告，顺着数学推导链条往回推，这意味着卡凯亚猜想（又名挂谷猜想）在所有维度宣告破解。
而 2025 年初，中国数学家王虹刚刚因为证明三维卡凯亚猜想（即三维挂谷猜想）震惊世界。仅仅一年半的时间，人类引以为傲的突破就成了 AI 成果的一个特例。
更离奇的是后续安排。OpenAI 请来了当年提出 ETH 假设的科学家本人，来审查 AI 对自己假设的反证；又请来诺贝尔物理学奖得主领衔的团队，来消化 AI 写出的千禧年难题证明。
科学家们花了半辈子没找到答案的问题，AI 几天就生成了解答。数学家们接下来的工作，不再是寻找证明，而是验证 AI 的证明。
数学史上最大的一次颠倒，正在发生。
从王虹的三维突破，到 AI 挑战所有维度
三项成果中，数学领域的突破尤其引人关注，因为它直接涉及去年中国数学家王虹取得的重大成果。
卡凯亚猜想源自一个看似简单的几何问题，如果一个集合能够容纳所有方向的单位线段，那么这个集合究竟需要具有怎样的维度？经过上百年的研究，它已经成为调和分析、几何测度论等领域的重要课题。
2025 年，中国数学家王虹与纽约大学数学家 Joshua Zahl 合作，成功证明了三维卡凯亚猜想，成为当年数学界最受关注的突破之一。
但三维问题的解决，并不能直接推广到四维、五维乃至任意更高维度。
OpenAI 此次公布的成果，恰恰跨越了这一限制。其 AI 模型证明了一个更具基础性的数学命题，即数学家 Christopher Sogge 于 1991 年提出的局部平滑化猜想。
该猜想研究波在传播过程中的聚焦现象。简单理解，波可以在某些瞬间形成极强的聚焦，但这种极端状态只能短暂存在。如果观察一段时间内的平均表现，波的聚焦程度就会明显低于最极端的时刻。
局部平滑化猜想之所以重要，是因为它能够推导出卡凯亚猜想、Stein 限制性猜想等其他重要结论。
2019 年，数学家 Larry Guth、王虹和张瑞勋成功证明了二维空间中的局部平滑化猜想。如今，OpenAI 的模型通过引入一种新颖的辅助偏微分方程，构造出了适用于所有维度的局部平滑化猜想证明。
按照数学上的推导关系，局部平滑化猜想在所有维度成立，也就意味着卡凯亚猜想和 Stein 限制性猜想在所有维度得到解决。
王虹与 Joshua Zahl 此前攻克了三维卡凯亚猜想，而 OpenAI 此次通过证明更强的局部平滑化猜想，将相关结果推广到了所有维度。
OpenAI 披露，其模型于 10 月 18 日生成自然语言证明，第二天便完成了 Lean 形式化。目前，两份证明材料均已上传至公开 GitHub 仓库，供全球数学家研究。
AI 推翻计算机科学近 30 年的基础假设
相比卡凯亚猜想，OpenAI 在计算机科学领域取得的成果，可能对整个算法研究产生更加广泛的影响。
此次被 AI 推翻的，是计算复杂性理论中著名的指数时间假设（Exponential Time Hypothesis，ETH）。
该假设由 Russell Impagliazzo、Ramamohan Paturi 和 Francis Zane 等研究者在 1999 年前后提出，核心问题是计算机解决某些复杂问题时，究竟需要付出多少计算成本。
其中一个典型例子，是布尔可满足性问题（SAT）。
假设有大量由逻辑变量组成的条件，计算机需要判断，是否存在一种变量赋值方式，让所有条件同时成立。随着变量数量增加，寻找答案可能变得异常困难。ETH 认为，对于一般的 3-SAT 问题，不存在时间复杂度为 (2^{o(n)}) 的算法，其中 (n) 代表变量数量。
换句话说，这类问题在最坏情况下的计算时间，无法被降低到关于变量数量的次指数级。
过去近 30 年，ETH 已经成为计算复杂性理论的重要基础假设。数百篇学术论文以 ETH 为前提，在图论、计算生物学、数据库等领域，推导出大量问题的条件性计算复杂性下界。
如今，OpenAI 宣布，其 AI 系统成功证明 ETH 为假，并且已经通过 Lean 完成形式化验证。
一个支撑了数百篇计算机科学论文的基础假设，如今被 AI 推翻了。
该成果意味着，部分长期被认为无法实现次指数时间求解的计算问题，可能存在此前尚未发现的更高效算法。建立在 ETH 基础上的大量条件性复杂性结论，也将迎来重新审视。
OpenAI 为这项研究邀请的专家阵容也颇有戏剧性。
加州大学圣迭戈分校的 Russell Impagliazzo、Ramamohan Paturi，以及麻省理工学院的 Ryan Williams，将共同领导外部研究团队。
其中，Impagliazzo 和 Paturi 正是 ETH 的原始提出者。
OpenAI 邀请了当年提出这一假设的科学家，亲自参与审查 AI 对该假设的反证。OpenAI 表示，AI 模型于 10 月 16 日生成自然语言反证，次日完成 Lean 形式化，两份材料现已公开。
外部研究团队还要求 OpenAI 公开全部原始数据，并资助举办学术会议，讨论这一成果对计算复杂性理论的影响，OpenAI 已经接受这些要求。
如果这一成果得到学界认可，人类可能需要重新思考，过去认定的计算难度极限，究竟还有多少改进空间。
诺贝尔物理学奖得主参与，AI 触及千禧年大奖难题
三项研究中，理论物理领域的成果可能最为复杂，也最具基础科学意义。
OpenAI 宣布，其 AI 模型已经为四维空间中包含规范场和费米子的某些相互作用量子场论，构造出严格的数学存在性证明。
研究涉及量子色动力学（QCD）、粒子物理标准模型的一种统一性完备化，以及纯杨-米尔斯理论的存在性问题。
现代粒子物理学主要依靠标准模型描述基本粒子及其相互作用，涉及电磁力、弱相互作用和强相互作用。量子色动力学则专门研究强相互作用，解释夸克和胶子如何相互作用。
虽然相关理论已经在大量实验中得到验证，但其严格的数学基础仍然存在一些悬而未决的问题。
杨-米尔斯理论就是其中最著名的例子。
2000 年，美国克雷数学研究所将杨-米尔斯理论的存在性与质量间隙问题列为七大千禧年大奖难题之一，要求研究人员严格证明四维量子杨-米尔斯理论的存在，并证明其具有正的质量间隙。
OpenAI 此次公布的成果涉及其中的存在性部分，并进一步将研究推进至量子色动力学及相关统一理论。
OpenAI 披露，模型于 9 月 9 日生成杨-米尔斯理论存在性的自然语言证明，9 月 18 日又生成量子色动力学的证明，并于 10 月 16 日完成杨-米尔斯理论存在性证明的 Lean 形式化。
为进一步研究这项成果，OpenAI 邀请了五位理论物理学家组成团队。
负责人 David Gross 是 2004 年诺贝尔物理学奖得主，他因参与发现强相互作用中的渐近自由现象而获奖，其研究为量子色动力学奠定了重要基础。
其他成员包括 Michael Douglas、Arthur Jaffe、Igor Klebanov 和 Nikita Nekrasov，均是量子场论和数学物理领域的重要研究者。
其中，Arthur Jaffe 曾参与制定杨-米尔斯理论千禧年大奖难题的正式表述，并参与创办克雷数学研究所。
考虑到证明内容十分庞大，专家团队决定暂不公开 AI 生成的原始材料，由研究人员进一步整理、审查并撰写正式研究成果。如果研究顺利推进，AI 有望帮助物理学家为现代量子场论建立更加严格的数学基础。
AI 开始批量证明难题，科学家可能看不过来了
值得一提的是，就在今年 10 月 6 日，OpenAI 曾一口气公开 722 份 AI 生成的数学研究手稿，涉及 372 个尚未解决的数学问题。
但随后部分证明被发现存在错误，OpenAI 不得不撤回 3 篇手稿，并修订另外 14 篇。
短短 20 天后，OpenAI 又宣布了三项重大成果，同时改变了发布策略，将 AI 生成的证明视为原始研究数据，并邀请外部科学家参与验证、解释和整理。
数学成果直接公开，计算机科学成果由 ETH 的原始提出者等专家组织审查，物理学成果则交由诺贝尔物理学奖得主领衔的团队研究。
OpenAI 还宣布不寻求这些发现的学术署名，并通过此前向西蒙斯基金会提供的 500 万美元资助，支持科学家开展相关研究。
从某种程度上说，AI 科学研究正在进入一个颇为荒诞的阶段。
过去，科学家可能需要花费几十年寻找一个证明。如今，AI 可以在几天内生成证明，甚至完成 Lean 形式化验证，但科学家仍然需要花费大量时间，才能真正理解这些成果究竟意味着什么。
当 AI 生成科学成果的速度开始超过人类理解成果的速度，科学研究的瓶颈，也可能从寻找答案转向验证和消化答案。曾经困扰人类几十年的科学难题，未来或许会越来越多地由 AI 给出答案。
AI 已经开始陆续交卷了，但人类可能连阅卷都快来不及了。
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。
The king's guess was cleared by AI overnight.
The world-class puzzle that has plagued mankind for decades, and AI just handed it over in a few days.
Just recently, Open AI announced three breakthroughs in its model: overturning the computer science community's belief in nearly 30 years of index time assumptions; proving local smoothing assumptions of all dimensions; and establishing strict mathematical existence for quantum color dynamics and Yang Mills theory.
Open AI announces that it follows the mathematical chain of conduct, which means that Kakaya's guess (also known as the Hang Valley Conjecture) is declared to be broken at all dimensions.
At the beginning of 2025, Chinese mathematician Wang Rainbow just shocked the world by proving that three-dimensional Kakaia's conjecture (i.e., a three-dimensional hang-out). In just a year and a half, the breakthrough of human pride became an exception to the AI outcome.
What is more surprising is the successor arrangement. Open AI invited the scientists who presented the ETH hypothesis that year to examine AI's counter-assumption of their assumptions; and the Nobel Prize winner's team to digest the millennium dilemma certificate written by AI.
Scientists spent half their lives trying to find answers, and AI created answers in a few days. The next job for mathematicians is not to look for proof, but to verify the AI.
The biggest reverse in math history is happening.
From the three-dimensional breakthrough of the Wang Rainbow, to AI challenging all dimensions
Of the three outcomes, breakthroughs in mathematics are of particular concern, as they directly relate to the significant achievements of Chinese mathematician Wang Rainbow last year.
Cakaya's guess stems from a seemingly simple geometrical question, and if a cluster can accommodate a unit segment in all directions, what is the dimension of this cluster? After hundreds of years of research, it has become an important subject in the fields of analysis and geometry.
In 2025, Chinese mathematician Wang Rainbow worked with New York University mathematician Joshua Zahl to prove a three-dimensional Kakaia guess that was one of the most interesting breakthroughs in mathematics.
However, the solution to the three-dimensional problem cannot be extended directly to four-dimensional, five-dimensional or any higher dimension.
The results of Open AI's announcement just crossed this limit. Its AI model proves a more fundamental mathematical proposition, the local smoothing guess that mathematician Christopher Sogge made in 1991.
The idea is to study the focus of wave during transmission. Simplely understood, the wave can form a very strong focus in some moments, but the extreme is only temporary. If average performance is observed over time, the wave concentration is significantly lower than at the most extreme.
Local smoothing guesses are important because they can extrapolate other important conclusions, such as Kakaya's guess, Stein's restrictive guess.
In 2019, mathematicians Larry Guth, Wang Rainbow and Zhang Rue-hoon succeeded in proving the idea of local smoothing in two-dimensional space. Today, the Open AI model builds local smoothing proofs that apply to all dimensions by introducing an innovative auxiliary differential equation.
According to mathematical reasoning, local smoothing guesses are set up at all dimensions, which means that Kakaya's guess and Stein's restrictive guess are resolved at all dimensions.
Wang Rainbow and Joshua Zahl had previously conquered three-dimensional Kakaya's conjecture, and Open AI has extended the results to all dimensions by demonstrating a stronger local smoothing assumption.
Open AI discloses that its model generated a certificate of natural language on October 18th, and the next day the Lean formalization was completed. Currently, both supporting materials have been uploaded to the public GitHub repository for research by global mathematicians.
AI overturns the basic assumptions of nearly 30 years of computer science
Open AI ' s achievements in the field of computer science may have a wider impact on the entire algorithm than Kakaya ' s guess.
This time, AI overturned the famous index time assumption in the theory of complexity (Exponential Time Hypothesis, ETH).
This assumption was put forward around 1999 by researchers like Russell Impagliazzo, Ramamohan Paturi and Francis Zane, and the central question is how much computing costs are involved in solving certain complex problems.
A typical example of this is the question of Bul’s satisfaction (SAT).
Assuming that there are a large number of conditions consisting of logical variables, the computer needs to judge whether there is a variable-granting method that allows all conditions to be established simultaneously. As the number of variables increases, finding answers may become extremely difficult. ETH considers that for the general 3-SAT problem, there is no algorithm with time complexity (2{o(n)} in which (n) represents the number of variables.
In other words, the timing of the calculation of such problems in the worst-case scenario cannot be reduced to the sub-index level for the number of variables.
Over the past 30 years, ETH has become an important underlying assumption of the theory of computing complexity. Hundreds of academic papers, premised on ETH, have led to a number of problems that fall under the complexity of conditional calculations in the fields of graphics, computational biology and databases.
Today, Open AI announces that its AI system has successfully proven ETH to be false and has completed formalisation through Lean.
A fundamental assumption that supported hundreds of computer science papers is now overturned by AI.
The result means that some of the computational problems that have long been considered unattainable for sub-index time may be more efficient algorithms that have not been previously discovered. A large number of conditional complex conclusions based on ETH will also be re-examined.
Open AI has also been very dramatic in inviting experts for this study.
Russell Impagliazzo, Ramamohan Paturi, University of California, San Diego, and Ryan Williams, Massachusetts Institute of Technology, will co-lead the external research team.
Among them, Impagliazzo and Paturi were original authors of ETH.
Open AI invited scientists who had presented this assumption in the course of the year to personally participate in the review of AI's rebuttal of the assumption. Open AI states that on October 16, the AI model generated a natural language counter-proof, that the next day the Lean format was completed and that two materials are now available.
The external research team has also requested Open AI to make available all raw data and has funded academic meetings to discuss the impact of this outcome on the theory of computational complexity, which Open AI has accepted.
If this result is accepted by the academic community, human beings may need to rethink how much room there is for improvement in what has been determined to be difficult to calculate.
Nobel Prize winner in physics, AI, touching on the Jubilee Challenge.
Of the three studies, the results in the field of theoretical physics are likely to be the most complex and fundamental scientific.
Open AI announces that its AI model has built a strict mathematical proof of existence for some of the interactions in the four-dimensional space that contain the normative field and the Phemium.
The study dealt with quantum color dynamics (QCD), a uniform completeness of particle physics standard models, and the existence of pure Yang-Mirs theory.
Modern particle physics relies mainly on standard models to describe basic particles and their interactions, involving electromagnetic forces, weak interactions and strong interactions. Quantum colour dynamics specialize in strong interactions and explain how quarks and glue interact.
Although the theory has been tested in a number of experiments, there are still some outstanding questions about its strict mathematical base.
The Yang-Mills theory is one of the most famous examples.
In 2000, the Clay Mathematics Institute of the United States listed the existence and quality gap between the Yang-Mills theory as one of the seven millennium prize challenges, requiring researchers to prove strictly the existence of the four-dimensional quantum Yang-Mirs theory and to prove that it had a positive quality gap.
The results published by Open AI relate to the existence of these components and will further study the theory of advancement to quantum colour dynamics and related harmonization.
Open AI disclosed that the model produced natural language proof of the existence of the Yang-Mirs theory on September 9, and a certificate of quantum colour dynamics on September 18, and completed the formalization of the Lean certificate of the existence of the Yang-Mirs theory on October 16.
To further study this result, Open AI invited five theoretical physicists to form a team.
The leader, David Gross, was the 2004 Nobel Prize winner in physics, who was awarded the award for his involvement in the discovery of the phenomenon of progressive freedom in powerful interactions, and whose research provided an important basis for quantum color dynamics.
Other members, including Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, are important researchers in the field of quantum field theory and mathematics physics.
Among them, Arthur Jaffe was involved in the formal presentation of the Jubilee Award for Yang Mills Theory and in the creation of the Clay Math Institute.
Given the magnitude of the certification, the team of experts decided not to disclose the original material generated by AI for the time being, and to further collate, review and write the official research results by researchers. If research progresses well, AI is expected to help physicists build a more rigorous mathematical base for modern quantum field theory.
AI, start bulk proof of difficulty. Scientists may not see it.
It is worth mentioning that on October 6th of this year, Open AI published 722 AI-generated manuscripts on mathematics, covering 372 outstanding mathematical issues.
Later, however, part of the error was found and Open AI had to withdraw three manuscripts and revise another 14.
Just 20 days later, Open AI announced three major results while changing the publish strategy, treating the AI-generated certificates as raw research data and inviting outside scientists to participate in validation, interpretation and collation.
Mathematics are directly open, computer science results are reviewed by experts such as the original authors of ETH, and physics results are studied by teams of Nobel Prize laureates.
Open AI also announced that it would not seek academic signatures for these discoveries and supported scientists in conducting related research through a previous $5 million grant to the Simons Foundation.
To some extent, AI scientific research is entering a rather ridiculous phase.
In the past, scientists may have spent decades looking for a certificate. Today, AI can produce proof in a few days, or even complete the Lean formalization, but scientists still need a lot of time to really understand what these results mean.
When AI begins to generate scientific results faster than humans understand, the bottlenecks in scientific research can also shift from finding answers to validating and digesting them. The scientific difficulties that have plagued humanity for decades may be answered increasingly by AI.
AI has begun to file, but humanity may be too late to read it.
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