> ## Content Index
> Fetch the complete content index at: https://globalfeed.ai/llms.txt
> Use this file to discover other available public pages before exploring further.

# OpenAI publishes 722 math manuscripts produced by an internal model
- URL: https://globalfeed.ai/en/openai-publishes-722-math-manuscripts-produced-by-an-internal-model/
- Published: 2026-10-08T02:11:35.000Z
- Updated: 2026-10-08T02:11:35.000Z
- Description: OpenAI released 722 mathematics manuscripts on GitHub, produced by an internal model that has not been released yet. According to the company, they are grouped into 372 result families and include claimed solutions to long-standing open problems. Many proofs come with Lean formalizations that a comp
- Author: GlobalFeed Editor
- Tags: OpenAI, mathematics, formal verification, research, x-OpenAI, dil-en, elle

**OpenAI** shared a broad set of mathematical results on October 6, 2026, produced by an internal frontier model that it has not yet released. The results are in a GitHub repository called **openai/math**. According to the repository, the collection contains 722 manuscripts organized into 372 result families. A family groups related papers, such as a principal result, companion arguments, consequences or alternative proofs. The families are classified into 17 disciplines, including number theory, algebraic geometry, combinatorics, theoretical computer science and mathematical physics.

According to OpenAI's announcement, the work is part of how the company evaluates its models in mathematics. The repository says OpenAI expanded its evaluations to open research problems, questions mathematicians have not yet solved, after performance on its existing math evaluations saturated. Examples in the catalog include a paper on a "quasi-Riemann hypothesis" that claims Dirichlet L-functions have no zeros in the region Re(s) > 7/8, a proof related to Erdős's conjecture on short Egyptian fractions, and a counterexample to Kaplansky's direct-finiteness conjecture in characteristic two. These are the company's claims, and they await review by independent mathematicians.

## The results came from roughly 4,000 attempted problems

According to the repository, the vast majority of results were obtained with the same procedure and the same internal model. Over the course of the evaluation, the model was posed about 4,000 problems. The output was grouped into families and manuscripts, and only results meeting a set level of significance went into the catalog. OpenAI says the average result used compute equivalent to roughly three hours of ChatGPT Pro thinking. Some outputs build on earlier results produced by the model.

The company also lists the exceptions to that procedure: work on a zero-free region for the Riemann zeta function and the proof of the Hodge Conjecture for CM abelian varieties were produced outside the fixed process. The writeup for the Re(s) > 11/12 zero-free region was edited by people for readability. The repository also includes abridged summaries of the model's reasoning for ten results.

## Many proofs can be checked by computer in Lean

Lean is a programming language that lets a computer check a mathematical proof step by step. OpenAI added Lean formalizations for many of the manuscripts, along with a catalog that maps each paper to its formal proof. The repository also includes instructions for re-checking the proofs with Comparator, a tool from the Lean community. The company says it will add more formalizations as it obtains them.

## How the release is organized and what comes next

OpenAI says it consulted the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study on how to share the results and drew on its recommendations. Each manuscript comes with citation information; corrections will be recorded as new versions, and earlier versions will remain accessible. The company is also exploring community-hosted alternatives for the collection. The announcement says OpenAI will fund workshops, conferences and special programs on understanding major results produced by AI, and that it is working to release the model behind these results responsibly; no dates were given.

According to the repository itself, the collection includes results at different stages of verification. Not every manuscript has a Lean formalization, and OpenAI notes that some unformalized results could have issues, which it will try to fix quickly. A Lean proof confirms that the formal statement given to the computer is correct; whether that statement fully captures the paper's claim and the known open problem is left to mathematicians to review. Since the model that produced the papers is not publicly available, outsiders cannot yet repeat the process. For a fuller picture, follow evaluations from independent researchers as well.

Sources: [OpenAI: Sharing AI progress in mathematics](https://openai.com/index/sharing-ai-progress-in-mathematics/?ref=globalfeed.ai), [GitHub: openai/math](https://github.com/openai/math?ref=globalfeed.ai).