与学术界合作建立科学理解
我们宣布在数学，计算机科学，和物理学方面关于口琴分析，复杂理论，和量子场理论的三个新发现。
根据相关学术界的反馈，我们的发布进程继续发展。 我们认为，这些手稿最好被理解为原始研究数据的一种形式。 为了将这些原始研究数据转化为科学理解，我们在本版中试验了一种新的方法。 我们联系了外部专家，要求他们充当牧羊人，组织小组了解新的结果。 他们建议采取三种不同的释放策略：
- 对于数学结果，释放AI输出，让研究人员自由地组织自己的努力来解释结果。
- 对于计算机科学的结果，组成一个外部合作小组来解释结果，同时将AI输出公开，供任何人检查和借鉴。
- 对于物理结果，组成一组外部合作者，并服从于他们释放材料。
我们还不知道这三种办法中哪一种最有用，我们欢迎就哪些办法可行和哪些办法可以改进提供反馈。
外部合作将遵循四项原则：
- 透明度。 我们对合作的存在是透明的。 将不要求与会者签署不披露协议，并可自由讨论工作、分享中间产品和发表其结论。
- 没有学术信用。 我们内部模型的发现并不寻求学术上的赞扬。 邀请研究人员协助管理协作也并不意味着他们或他们的工作组得出了最初的发现。
- 承认。 这一群体应因其形成的理解、撰写的文字以及他们制作的其他材料而获得承认。 该小组还将帮助确保这项工作适当评价研究者、技术和借鉴的理念。
-独立参与是自愿和无偿的。 工作组成员独立参与，他们的结论是自己的。 他们不代表OpenAI发言，OpenAI也不代表他们发言。
我们将把与这些成果有关的材料和文物交给协作。 对于计算机科学合作，小组决定立即公开自然语言证明和精益正规化，我们正直接向我们的GitHub寄存器发布。 对于物理合作，小组决定推迟释放人工智能制作的文物，而自行编写。
所有这些办法都涉及权衡。 一方面，这些协作使专家有能力推动整理成果，并在更广泛的学术界加以联系。 另一方面，选择最初的牧羊人必然会形成先参与并产生信息不对称的形状。 我们欢迎就如何扩大参与并使这一进程更加有用提供反馈。
数学的结果涉及口琴分析。 1991年，索格提出了当地的平滑猜想，该猜想涉及波浪成为焦点时的行为。 特别是猜想指出，最极端的焦点必须是短暂的。 在典型的随机时间集中关注的量总是比最坏情况下可能发生的情况要小得多。 这是整个数学中一个熟悉的模式：在最坏的情况下发生的事情比平均后观察到的更为极端。 由于这种猜想现象有多重要，它在整个数学分析中也具有广泛的影响。 例如，这意味着Kakeya猜想和Stein的限制猜想。
2019年，古斯，王，张(在新窗口打开)在两个空间维度上证明了当地的平滑猜想。 2025年，Wang和Zahl(在新窗口打开)在三个维度上证明了Kakeya猜想。 我们的模型证明了当地在每一个维度上的平滑猜想，因此也解决了卡基亚猜想和斯坦因在各个维度上的限制猜想。 新证据基于一个新颖的辅助部分微分方程。
2026年10月18日我们获得了AI产出，次日在利恩正式确定，之后我们与一位专家接触，商讨围绕这一结果形成合作。 根据他们的建议，我们决定公开发布AI产出，遵循我们2026年10月6日发布的同样战略实现这一结果。 发布既是我们模型制作的AI自然语言输出写入(在新窗口打开)，也是我们公共的GitHub寄存器中的Lean证明(在新窗口打开).
计算机科学合作将了解和解释关于“衡量时间假说”(ETH)的结果，这是对布尔公式的计算复杂性的长期猜测，这个问题与P对NP问题密切相关。 ETH被广泛认为是真实的，在上百篇论文中被用来在包括图论，计算生物学，数据库，计算几何等多种领域得出有条件的计算硬度结果。 令人惊讶的是，我们的系统最近证明了ETH是虚假的，这个结果随后在利恩被证实。 对ETH的反驳重新提出了关于这些任务可变成多快的问题，并可能有助于指导搜索更好的算法。 寻找这种算法，并将理论进展转化为实际改进，将需要进行大量的进一步研究。
这部作品将由罗素·英帕格利亚佐(英语：UC San Diego)，拉马莫汉·帕图里(英语：UC San Diego)，瑞安·威廉姆斯(英语：Ryan Williams (MIT))率领的团体进行核查和牧羊。 三者都是算法设计和复杂理论方面的专家。
- Impagliazzo和Paturi，与Francis Zane一起，是"指数时间假说"的最初提议者，在过去30年里，他们写了(在新窗口打开)关于这个主题的(在新窗口打开)有影响力(在新窗口打开)的(在新窗口打开)纸(在新窗口打开)数字(在新窗口打开).
- 威廉姆斯驳斥了ETH(在新窗口打开)强(在新窗口打开)的版本(在新窗口打开)，将ETH用于(在新窗口打开)硬度(在新窗口打开)结果(在新窗口打开)，并连接(在新窗口打开)算法(在新窗口打开)和(在新窗口打开)更低的界限(在新窗口打开).
我们同意该小组的要求，即公布所有结果的原始数据，以便利社区独立核查，OpenAI承诺资助一次会议，以了解这项工作对整个实地的影响。 工作组请社区帮助组织对ETH权利主张的公开审查，并讨论AI产生的突破对核查、获取、信贷和研究的未来提出的更广泛问题。
我们的AI于2026年10月16日发现其自然语言对ETH的隔绝性，随后我们于2026年10月17日在利恩正式确定了结果，这使我们发展了这里描述的合作。 为此，我们有发布，既有我们模型制作的AI自然语言输出写入(在新窗口打开)，也有我们公共的GitHub寄存器中的Lean证明(在新窗口打开).
计算的复杂性有许多未解的问题和猜想，迄今为止，这些疑问和猜想的证明或反驳都未能使科学家做出最大努力。 我们希望拥有AI的人类能够在这一领域提供新的理解，并阐明高效计算固有的能力和局限性。
物理学协作将理解并解释一种结果，该结果在数学上为某些相互作用的量子场理论提供严格的存在证明，这些理论包含四个维度的量子场和热量，包括量子铬动力学(QCD)和粒子物理标准模型的统一完成。 我们的工作建立在Balaban在建设性量子场理论中(在新窗口打开)的方法之上。 作为次问题，这部作品还包括证明纯阳的存在—— 米尔斯理论，是量子杨–米尔斯理论克莱数学研究所千年奖问题(在新窗口打开)的一部分。
标准模型描述了除重力外4个基本力中3个基本亚原子粒子的相互作用。 存在证明涉及最简单的大统一理论，包括标准模型的测量场和热量(因为标准模型本身不能任意推导到短距离). QCD通过强力描述夸克和葡萄糖等粒子的相互作用。 纯阳-弥勒斯理论只描述胶体，力载体的自交，而不是夸克等物质。
大卫·格罗斯将领导一个团体，包括迈克尔·道格拉斯，亚瑟·贾夫，伊戈尔·克莱巴诺夫，以及尼基塔·涅克拉索夫，他们将主持这项工作。
- Gross1973年在Yang-Mills理论中的不对称自由工作帮助建立了QCD作为强相互作用的理论，并为他赢得了2004年诺贝尔物理学奖(在新窗口打开)的份额。 不对称自由也是这些理论作为数学严谨对象存在的关键机制。
- Jaffe与Witten共同制订了Quantum Yang-Mills理论千年奖问题(在新窗口打开)，并设想了千年奖问题，并共同创办了克莱数学研究所。
- 道格拉斯在过去20年中一直是这一问题的管理者，在2004年报告了该问题的现状(在新窗口打开)，然后在今年年初全面介绍了进展情况(在新窗口打开)。
- Klebanov帮助先行应用“双轨制”，这是QCD的重要财产，现在指导 Simons 协作进行“封闭式”和QCD Strings(在新窗口打开)
- 涅克拉索夫对杨的概括方法做了开创性的工作米尔斯理论，并于2023年获得丹妮·海尼曼数学物理学奖。
专家组要求将就这项工作与专家组任何成员和所有成员进行的所有通信都输入以下电子邮件账户：YANGMILLS4D@gmail.com。
我们的模特为杨提供了自然语言证明米尔斯于2026年9月9日存在，随后于2026年9月18日为QCD提供了证据。 鉴于工作时间长且复杂，我们推迟发布，直到精益正规化完成。 我们于2026年10月16日完成了对杨米尔斯存在的正式化
我们希望这些结果标志着一个过渡的开始，量子场理论，现代理论物理中最重要的框架，成为数学上严格的领域。 我们兴奋地为在学术上主导的实现这一目标的进展提供资源。
我们致力于支持更广泛的理解工作和在这些成果的基础上再接再厉。 这些协作补充了我们最近向西蒙斯基金会提供的500万美元捐款(在新窗口开放)。 该资金由西蒙斯基金会管理和授予，将用于支持讲习班、会议和特别方案，以帮助研究人员了解和借鉴大赦国际的成果。 我们承诺确保资助研究人员独立审查、解释和巩固上述成果。
我们继续就披露成果的最佳方式征求反馈意见，并与科学家接触以加快发现。
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。
Working with academic communities to build scientific understanding
We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
Our release process continues to evolve in response to feedback from the associated academic communities. In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach. We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies:
- For the math result, release the AI output, leaving researchers free to organize their own efforts to explain the result.
- For the computer science result, form an external group of collaborators to explain the result, while making the AI output publicly available for anyone to examine and build on.
- For the physics result, form an external group of collaborators and defer to them on the release of materials.
We do not yet know which of these three approaches will be most useful, and we welcome feedback on what works and what we can improve.
The external collaborations will follow four principles:
- Transparency. We are transparent about the existence of our collaborations. Participants will not be asked to sign nondisclosure agreements and will be free to discuss the work, share intermediate products, and publish their conclusions.
- No Academic Credit. We do not seek academic credit for the discoveries produced by our internal models. Inviting a researcher to help steward a collaboration also does not imply that they or their working group made the original discovery.
- Recognition. This group should receive recognition for the understanding they develop, the writeups they author, and the other materials they create. The group will also help ensure that the work appropriately credits researchers, techniques, and ideas that it draws on.
- Independence. Participation is voluntary and unpaid. Working-group members participate independently, and their conclusions are their own. They do not speak on behalf of OpenAI, and OpenAI does not speak on their behalf.
We will turn over materials and artifacts related to these results to the collaborations. For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.
All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.
The result in mathematics concerns harmonic analysis. In 1991, Sogge proposed the local smoothing conjecture, which concerns the behavior of waves as they become focused. In particular, the conjecture states that the most extreme focusing must be short-lived. The amount of focusing at a typical random time is always much smaller than what can happen at the worst-case time. This is a familiar pattern throughout mathematics: what can happen in the worst case is more extreme than what is observed after averaging. Because of how fundamental this conjectured phenomenon is, it also has wide ranging implications throughout mathematical analysis. For instance, it implies the Kakeya conjecture and Stein’s restriction conjecture.
In 2019, Guth, Wang, and Zhang(opens in a new window) proved the local smoothing conjecture in two spatial dimensions. In 2025, Wang and Zahl(opens in a new window) proved the Kakeya conjecture in three dimensions. Our models produced a proof of the local smoothing conjecture in every dimension and therefore also resolve the Kakeya conjecture and Stein’s restriction conjecture in all dimensions. The new proof is based on a novel auxiliary partial differential equation.
We obtained our AI output on October 18th, 2026, which was formalized in Lean the following day, and then we reached out to an expert to discuss forming a collaboration around this result. Following their recommendation, we decided to release the AI output publicly, following the same strategy for this result as our October 6th, 2026 release. We have released both an AI natural language output writeup(opens in a new window) produced by our models and a Lean proof (opens in a new window)in our public GitHub repository.
The computer science collaboration will understand and explain a result concerning the Exponential Time Hypothesis (ETH), a longstanding conjecture about the computational complexity of Boolean formula satisfiability, a problem deeply related to the P vs. NP problem. The ETH has been widely believed to be true, and has been used in hundreds of papers to derive conditional computational hardness results across a wide variety of fields including graph theory, computational biology, databases, computational geometry, and more. Surprisingly, our system has recently proved that the ETH is false, a result that was subsequently verified in Lean. The refutation of the ETH reopens questions about how much faster these tasks could become, and may help guide the search for better algorithms. Finding such algorithms, and turning theoretical advances into practical improvements, will require significant further research.
This work will be verified and shepherded by a group led by Russell Impagliazzo (UC San Diego), Ramamohan Paturi (UC San Diego), and Ryan Williams (MIT). All three are experts in algorithm design and complexity theory.
- Impagliazzo and Paturi, along with Francis Zane, are the original proposers of the Exponential Time Hypothesis, and have authored a (opens in a new window)number(opens in a new window) of (opens in a new window)influential(opens in a new window) (opens in a new window)papers(opens in a new window) on the topic(opens in a new window) over the last 30 years.
- Williams has refuted (opens in a new window)strong(opens in a new window) (opens in a new window)versions(opens in a new window) of the ETH, used ETH for (opens in a new window)hardness(opens in a new window) (opens in a new window)results(opens in a new window), and connected (opens in a new window)algorithms(opens in a new window) and (opens in a new window)lower bounds(opens in a new window).
We have agreed to the group’s requests that all raw data of the results be made public to facilitate independent verification by the community, and that OpenAI commit to funding a conference to understand the implications of this work on the field as a whole. The group invites the community to help organize an open examination of the ETH claim and to discuss the broader questions that AI-generated breakthroughs raise about verification, access, credit, and the future of research.
Our AI found its natural language disproof for the ETH on October 16th, 2026 and then subsequently we formalized the result in Lean on October 17th, 2026, which led us to develop the collaboration described here. To that end, we have released both an AI natural language output writeup(opens in a new window) produced by our models and a Lean proof (opens in a new window)in our public GitHub repository.
Computational complexity has many open questions and conjectures, whose proofs or refutations have so far eluded the best efforts of scientists. We hope that humans with AI can provide new understanding in this field, and shed light on the inherent capabilities and limits of efficient computation.
The physics collaboration will understand and explain a result that gives a mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics. Our work builds on Balaban’s approach(opens in a new window) in constructive quantum field theory. As a subproblem, this work also includes a proof of the existence of pure Yang–Mills theory, which is part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem(opens in a new window).
The Standard Model describes the interaction of fundamental subatomic particles for three of the four fundamental forces, excepting gravity. The existence proof pertains to the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model (as the Standard Model itself cannot be extrapolated to arbitrarily short distances). QCD describes the interaction of particles such as quarks and gluons via the strong force. Pure Yang–Mills theory only describes the self-interactions of the gluons, the force carriers, rather than matter such as quarks.
David Gross will lead a group including Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov that will shepherd this work.
- Gross’s 1973 work on asymptotic freedom in Yang-Mills theories helped establish QCD as the theory of the strong interactions and earned him a share of the 2004 Nobel Prize in Physics(opens in a new window). Asymptotic freedom is also the key mechanism underlying the existence of these theories as mathematically rigorous objects.
- Jaffe formulated the Quantum Yang-Mills Theory Millenium Prize problem(opens in a new window) with Witten and also envisioned the idea of Millennium Prize problems and co-founded the Clay Mathematics Institute.
- Douglas has already been a steward of this problem over the last 20 years, reporting on its status in 2004(opens in a new window) and then providing a comprehensive update on the progress(opens in a new window) at the beginning of this year.
- Klebanov helped pioneer the application of gauge–string duality to confinement, an important property of QCD, and now directs the Simons Collaboration on Confinement and QCD Strings(opens in a new window),
- Nekrasov has done groundbreaking work on exact methods for generalizations of Yang–Mills theory and won the Dannie Heineman Prize for Mathematical Physics in 2023.
The group requests that all communications with any and all members of the group regarding this effort be directed to the following email account: YANGMILLS4D@gmail.com.
Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026, which led us to develop the collaboration described here.
We hope that these results mark the beginning of a transition in which quantum field theory, the most important framework in modern theoretical physics, becomes a mathematically rigorous field. We are excited to provide resources for academic-led progress towards that goal.
We are committed to supporting the broader work of understanding and building on these results. These collaborations complement our recently announced $5 million contribution(opens in a new window) to the Simons Foundation. Managed and awarded by the Simons Foundation, that funding will support workshops, conferences, and special programs to help researchers understand and build on results produced by AI. We are committed to ensuring funding for researchers to independently examine, explain, and build on the results above.
We continue to invite feedback on the best way to disclose results and engage with scientists to accelerate discovery.
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