math README

The project's README from github.com/openai/math. As one markdown file: README.md.

Readme

This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.

As part of model development, we evaluate our models on open research problems. We expanded these evaluations after performance on our existing mathematical evaluations saturated. Some outputs build upon earlier results produced by the models.

This collection includes results at different stages of verification. Not all have accompanying Lean formalizations. We will continue to update this repository with Lean formalizations as we obtain them.

Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly. We are also exploring community-hosted repositories for these materials.

Navigating the collection

The current catalogue contains 719 manuscripts organized into 372 families. A family groups related papers, which may include a principal result, companion arguments, consequences, or alternative proofs. Each family is classified by mathematical discipline.

Reasoning summaries

We are also releasing abridged summaries of the model's reasoning, covering the following results:

Family Subject
007 Ordinary two-point correlations of multiplicative functions
017 The irrationality exponent of π
087 Symmetric and general Mahler conjectures
102 Ordinary NP-hardness at the basic semidefinite threshold
159 Quasipolynomial bounds for arithmetic progressions
197 Kaplansky's direct-finiteness conjecture in characteristic two
221 The Mézard–Parisi formula for diluted spin glasses
271 Spontaneous magnetization in the quantum Heisenberg ferromagnet
287 Isomorphism of free group factors
362 The three-dimensional relativistic Vlasov–Maxwell system

How the results were produced

The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model. Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above.

Exceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. Additionally, the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability.

Versions and citations

We will preserve the public release history of this collection. Corrections and revisions will be recorded as new versions, with previously released versions remaining accessible.

To cite the individual manuscript, use the BibTeX block in its directory.

math (openai/math)

Searching Database…

15K ·GitHub
Math
这是什么？一份公开目录，收录了 719 份数学手稿，分为 372 个系列，由 OpenAI 的一个内部模型在研究开放科研问题的评估过程中撰写。其中约 42% 的核心成果已经有了 Lean 形式化证明。
这是需要安装的工具，还是一套文档？是一套文档，不是软件产品：preprints/ 目录里存放着 PDF 和源文件，每篇论文都附有引用和构建说明，旁边还有一个与手稿关联的 Lean 形式化证明库。
我该怎么浏览这个合集？先打开总览查看各系列的介绍，再通过手稿地图找到具体的论文；形式化目录列出了已有的 Lean 证明及其验证配置。
这里面的每个证明都可靠吗？不是——各项成果处于不同的验证阶段，并不是全部都有 Lean 形式化；作者提醒，部分未经形式化的结果可能存在问题，他们会在发现后尽快修复。
这些成果究竟是怎么产生的？几乎所有成果都遵循同一个固定流程：向一个未公开发布的 OpenAI 内部模型提出约 4,000 个问题，每个成果平均耗费三个小时的 ChatGPT Pro 思考算力。输出结果随后被汇总成达到显著性门槛的系列和手稿。少数成果偏离了这一流程，例如黎曼 zeta 函数的无零点区域，以及 CM 阿贝尔簇上的 Hodge 猜想。
哪些成果附带了模型的推理摘要？共有十个系列发布了节选的推理摘要，其中包括 pi 的无理数指数、对称及一般 Mahler 猜想、等差数列的拟多项式上界、特征 2 情形下的 Kaplansky 直接有限猜想，以及三维相对论性 Vlasov-Maxwell 方程组。
它涵盖哪些数学领域？每个系列都按数学学科分类，总览中有全部领域的介绍。仅就已发布的摘要而言，内容横跨解析数论、统计物理、量子多体理论、算子代数、复杂性理论和相对论场方程。
这个合集以后还会更新吗？会。Lean 形式化证明会随着新成果不断增加；发布历史也会保留：更正和修订以新版本形式发布，早期版本仍然可以访问。团队还在探索由社区托管的仓库来存放这些材料。
What are similar projects?NavierStokesAndEuler, gpt-3, fermats-last-theorem, awesome-rsi, gpt-2, Engram, MSA, bdh
Which topics is it in?lean, theorem-proving, research-tools, science, math, reasoning, mathematics, catalog, papers, paper, dataset, gpt
Creator
openai
Size
792.5 MB
Stars
15K
Updated
CommitOct 12
Age
10d
Commits · 90 Days