---
格式版本: 2
标题: "Ten advances in mathematics and theoretical computer science"
原文链接: "https://openai.com/index/ten-advances-in-mathematics"
发布日期: "2026-08-01"
发布时间校准状态: "found"
发布时间需复核: "否"
发布时间来源: "llm:local:strict_original_body"
发布时间证据: "August 1, 2026"
发布时间校准原因: "该日期出现在标题附近（nearTitle=true），且为页面正文中明确标注的日期，符合文章发布时间特征。其他候选日期均为其他文章或链接的日期，不应作为本文章发布时间。"
发布时间校准置信度: "1"
发布时间候选数量: 16
发布时间严格候选数量: 4
发布时间原页读取状态: "source template page reused from URL open"
发布时间未找到原因: ""
发布时间校准时间: "2026-08-10T16:18:33+08:00"
发布时间仲裁状态: "confirmed"
发布时间仲裁尝试次数: 1
发布时间仲裁耗时毫秒: 5294
发现时间: "2026-08-10T16:14:13+08:00"
入库时间: "2026-08-10T08:18:38.908Z"
来源平台: "固定入口"
搜索渠道: "fixed_url"
搜索词: "https://openai.com/news/"
匹配关键词:
  - "deployment"
相关厂家:
  - "OpenAI"
相关专家:
  []
内容类型: "网页"
抓取工具: "Jina Reader"
清洗工具: "Jina Reader Markdown + Defuddle/Readability 正文提取"
原始附件:
  []
AI优质: "否"
AI打分: 0
AI分档: "非优质"
AI质检状态: "不通过"
AI打分理由: "内容为纯数学和理论计算机科学进展，与超节点/AI机柜级系统、硬件架构、供电散热互连等完全无关。"
AI质检模型: "deepseek-v4-flash"
AI质检时间: "2026-08-10T16:18:43+08:00"
AI主题相关性: 0
AI来源权威性: 0
AI新颖性: 0
AI技术细节: 0
AI商业部署信号: 0
AI完整性: 0
采集批次: "2026年8月10日15点46分00秒"
采集批次ID: "20260810-154600-0560986a"
去重键: "https://openai.com/index/ten-advances-in-mathematics"
---

Title: Ten advances in mathematics and theoretical computer science

URL Source: https://openai.com/index/ten-advances-in-mathematics

Markdown Content:
[Skip to main content](https://openai.com/index/ten-advances-in-mathematics#main)

[](https://openai.com/)

*   [Research](https://openai.com/research/index/)
*   Products
*   [Business](https://openai.com/business/)
*   [Developers](https://openai.com/api/)
*   [Company](https://openai.com/about/)
*   [Foundation(opens in a new window)](https://openaifoundation.org/)

Log in[Try ChatGPT(opens in a new window)](https://chatgpt.com/)

*   Research
*   Products
*   Business
*   Developers
*   Company
*   [Foundation(opens in a new window)](https://openaifoundation.org/)

[Try ChatGPT(opens in a new window)](https://chatgpt.com/)Login

OpenAI

August 1, 2026

[Publication](https://openai.com/research/index/publication/)

# Ten advances in mathematics and theoretical computer science

[Read the paper(opens in a new window)](https://cdn.openai.com/pdf/ten-proofs-oai.pdf)[Read the reasoning walkthroughs(opens in a new window)](https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf)

Loading…

[Audio 1](https://openai.com/index/ten-advances-in-mathematics)

Share

The results

*   [The results](https://openai.com/index/ten-advances-in-mathematics#the-results)
*   [Responsibility to the mathematical community](https://openai.com/index/ten-advances-in-mathematics#responsibility-to-the-mathematical-community)

*   [The results](https://openai.com/index/ten-advances-in-mathematics#the-results)
*   [Responsibility to the mathematical community](https://openai.com/index/ten-advances-in-mathematics#responsibility-to-the-mathematical-community)

We want to empower scientists and mathematicians with tools that accelerate discovery. That is why we recently announced [ChatGPT for Academic Researchers⁠](https://openai.com/index/chatgpt-for-academic-researchers/), an initiative providing 100,000 scientists and mathematicians with free access to our best ChatGPT models. We also continue to evaluate our models on open research problems during development.

In May, we shared an [AI-generated disproof⁠](https://openai.com/index/model-disproves-discrete-geometry-conjecture/) of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. This work has already inspired further developments in mathematics and theoretical computer science[1](https://openai.com/index/ten-advances-in-mathematics#citation-bottom-1). Today, we are sharing a selection of ten results, each of which resolves or makes substantial progress on a long-standing open problem. These problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. All of these problems are of substantial interest to their respective mathematical communities, and several are of broad interest across mathematics as a whole.

## The results

We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a [Lean certificate⁠(opens in a new window)](https://github.com/openai/ten-proofs). We are also releasing for each solution a model’s narration of its thinking process.

1.   **High-dimensional sphere packing.** New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
2.   **Binary and spherical codes.**Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
3.   **Non-sofic groups.** A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.
4.   **Connes’s rigidity conjecture.** Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras.
5.   **Arithmetic circuit complexity.** New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n 4/log n.
6.   **Quantum parallel repetition.** An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
7.   **Closest vector problem.** Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
8.   **Ehrhart’s volume conjecture.** Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point.
9.   **Multicolor Ramsey numbers.** A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
10.   **Extremal number conjectures.** Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.

## Responsibility to the mathematical community

The emergence of systems capable of contributing to mathematical research raises questions that cannot be answered by a technology company alone. There are many views as to the role of AI in mathematics, and we have deep respect and understanding for those concerned with its impact, including the signers of the [Leiden declaration on AI and Mathematics⁠(opens in a new window)](https://leidendeclaration.ai/). We believe attribution should honestly reflect how a result was produced: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and the nature of genuine human intellectual work. We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system. We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.

As AI systems evolve into more sophisticated research collaborators, ensuring widespread access is fundamental to supporting scientists and mathematicians as they navigate and define the future of their disciplines during this transformative era.

*   [2026](https://openai.com/research/index/?tags=2026)

## Footnote

1.   1 Subsequent research includes Bloom, Sawin, Schildkraut, and Zhelezov, “[The sum-product conjecture is false for real numbers⁠(opens in a new window)](https://arxiv.org/abs/2605.28781)”; Pohoata, “[Split primes and the Elekes-Rónyai problem⁠(opens in a new window)](https://arxiv.org/abs/2606.13619v2)”; Saha, Xu, and Ye, “[Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH⁠(opens in a new window)](https://arxiv.org/abs/2606.25887)”; Goh and Hatami, “[Communication complexity of point-line incidences over the reals⁠(opens in a new window)](https://arxiv.org/abs/2606.25192)”; and Lee, Pohoata, and Zhu, “[The Minkowski grid has robustly many repeated distances⁠(opens in a new window)](https://arxiv.org/abs/2607.05374).”[](https://openai.com/index/ten-advances-in-mathematics#citation-top-1) 

## Keep reading

[View all](https://openai.com/news/)

![Image 1: ARC-AGI-3 art-card 1x1](https://images.ctfassets.net/kftzwdyauwt9/71xyJRWnTkM6EW1JKDUc59/91d2690ff9e7abaa975b4a5f64089056/ARC-AGI-3_art-card_1x1.png?w=3840&q=90&fm=webp)

[How enabling two settings tripled our scores on the ARC-AGI-3 benchmark Research Jul 29, 2026](https://openai.com/index/how-two-settings-tripled-our-arc-agi-3-scores/)

![Image 2: Scientific computing agentic AI card image (1x1)](https://images.ctfassets.net/kftzwdyauwt9/5opqp3rNWM7eax6GUc6MAl/1c710a4aba8c5c1e0b7d1c1c7c8b32db/1x1__1_.png?w=3840&q=90&fm=webp)

[Scientific computing in the age of agentic AI Publication Jul 28, 2026](https://openai.com/index/scientific-computing-agentic-ai/)

![Image 3: GPT-Red art card](https://images.ctfassets.net/kftzwdyauwt9/6q32m87ClqE8Ovn6vD424h/05ced72e47bbe42711fbac6a082cbff2/Art_Card.png?w=3840&q=90&fm=webp)

[GPT-Red: Unlocking Self-Improvement for Robustness Safety Jul 15, 2026](https://openai.com/index/unlocking-self-improvement-gpt-red/)

Research
*   [Research Index](https://openai.com/research/index/)
*   [Research Overview](https://openai.com/research/)
*   [Economic Research](https://openai.com/signals/)

Latest Advancements
*   [GPT-5.6](https://openai.com/index/gpt-5-6/)
*   [GPT-5.5](https://openai.com/index/introducing-gpt-5-5/)
*   [GPT-5.4](https://openai.com/index/introducing-gpt-5-4/)

Safety
*   [Safety Approach](https://openai.com/safety/)
*   [Deployment Safety(opens in a new window)](https://deploymentsafety.openai.com/)
*   [Security & Privacy](https://openai.com/security-and-privacy/)
*   [Trust & Transparency](https://openai.com/trust-and-transparency/)

Products
*   [ChatGPT(opens in a new window)](https://chatgpt.com/)
*   [ChatGPT Business(opens in a new window)](https://chatgpt.com/business/)
*   [ChatGPT Enterprise(opens in a new window)](https://chatgpt.com/business/enterprise/)
*   [ChatGPT for Education(opens in a new window)](https://chatgpt.com/business/education/)
*   [Codex](https://openai.com/codex/)
*   [Release Notes](https://openai.com/products/release-notes/)

API Platform
*   [Overview](https://openai.com/api/)
*   [API Log In(opens in a new window)](https://platform.openai.com/login)
*   [Docs(opens in a new window)](https://developers.openai.com/api/docs)

Business
*   [Overview](https://openai.com/business/)
*   [Solutions](https://openai.com/solutions/)
*   [Resources](https://openai.com/business/learn/)
*   [Customer Stories](https://openai.com/business/customer-stories/)
*   [Partner Network](https://openai.com/business/partners/)
*   [Contact Sales](https://openai.com/contact-sales/)

Developers
*   [Apps SDK(opens in a new window)](https://developers.openai.com/apps-sdk)
*   [Open Models](https://openai.com/open-models/)
*   [Docs(opens in a new window)](https://developers.openai.com/)
*   [Resources(opens in a new window)](https://developers.openai.com/learn)
*   [Developer Forum(opens in a new window)](https://community.openai.com/)

Company
*   [About Us](https://openai.com/about/)
*   [Our Charter](https://openai.com/charter/)
*   [Careers](https://openai.com/careers/)
*   [News](https://openai.com/news/)

Support
*   [Help Center(opens in a new window)](https://help.openai.com/)

More
*   [Stories](https://openai.com/stories/)
*   [Academy](https://openai.com/academy/)
*   [Supply Co.](https://openai.com/supply/)
*   [Livestreams](https://openai.com/live/)
*   [Podcast](https://openai.com/podcast/)
*   [RSS](https://openai.com/news/rss.xml)

Terms & Policies
*   [Terms of Use](https://openai.com/policies/terms-of-use/)
*   [Privacy Policy](https://openai.com/policies/privacy-policy/)
*   [Other Policies](https://openai.com/policies/)

[(opens in a new window)](https://x.com/OpenAI)[(opens in a new window)](https://www.youtube.com/OpenAI)[(opens in a new window)](https://www.linkedin.com/company/openai)[(opens in a new window)](https://github.com/openai)[(opens in a new window)](https://www.instagram.com/openai/)[(opens in a new window)](https://www.tiktok.com/@openai)[(opens in a new window)](https://discord.gg/openai)

OpenAI © 2015–2026 Your privacy choices

English United States