OpenAI Releases 722 Math Manuscripts, Some Machine-Verifiable

On local time October 6, OpenAI released a batch of mathematical results produced by an internal frontier model. This batch of results was organized into a GitHub repository, containing 722 manuscripts grouped into 372 result families, covering multiple mathematical branches from number theory to quantum field theory. The repository also released 10 model reasoning summaries, as well as compute consumption data estimated based on Pro usage.

The significance of this matter has two levels: one is mathematics itself—722 manuscripts, 372 result families, some of which have already been machine-verified; the other is the organizational mode of mathematical research. AI has begun to enter the stage of solving open problems, and mathematicians need to face a new question—how to incorporate AI-produced results into human understanding, and how to ensure that the direction of mathematical research is not entirely determined by the capability demonstrations of AI systems.

Regarding how these results came about, OpenAI stated in the repository that they used the same unreleased internal model to complete this work. After existing mathematics benchmarks reached saturation, they extended evaluation to open research problems. Throughout the process, the model was asked approximately 4,000 problems. Aggregating these outputs into result families and manuscripts, and then requiring a certain level of significance, produced the current catalog.

The average compute consumed per result is equivalent to about 3 hours of ChatGPT Pro thinking. This figure comes from the repository's compute consumption data estimated based on Pro usage, and is a self-report from OpenAI's side.

Some manuscripts in the repository come with Lean formalized proofs. Lean is a programming language in which mathematical proofs can be written as code, and then a computer checks whether each step of the derivation is valid. This is equivalent to moving part of the verification work from human eyes to machines.

But formalization is not full coverage. OpenAI also stated in the repository that unformalized results may have certain problems, and that they will correct them as soon as possible. At the same time, they are also exploring a community-hosted repository to store these materials. This means that the parts accompanied by Lean proofs and the unformalized parts are in different states of verifiability.

Among this batch of results there are two exceptions, both of which OpenAI also singled out. The first is the work on the zero-free region of the Riemann zeta function, and the second is the proof of the Hodge conjecture on CM abelian varieties. The Hodge conjecture is one of the seven Millennium Prize Problems of the Clay Mathematics Institute, and remains unproven to this day. What OpenAI refers to is the special case of CM abelian varieties.

The repository also states that the written text concerning the zero-free region of the zeta function for Re(s)>11/12 underwent manual editing, aimed at better readability. This note suggests that, at least for this text, human editing participated in the final presentation.

In the course of this matter, there is also a noteworthy episode. OpenAI initially contacted several mathematicians at the Institute for Advanced Study in Princeton, wanting to invite them to form an external advisory committee. However, the mathematicians did not accept this arrangement, and instead formed their own independent group, named the Advisory Group on Mathematics and AI.

Members include renowned mathematicians Timothy Gowers, Martin Hairer, Edward Witten, and others. They do not accept payment from AI companies, and independently issue recommendations regarding mathematical results released by AI companies. The group states on its own website that they do not hold decision-making power over any AI company, and that responsibility for AI company decisions still belongs to the AI companies themselves.

On October 6, this advisory group gave a response to OpenAI's released results. They said this is a very important event, whose impact on mathematics and the mathematical community far exceeds the impact of any single result. At the same time, they emphasized that publicly releasing results is only the first step. Making results public does not mean humans have truly understood these results, nor that these results have been integrated into the mathematical knowledge system.

They also said that future mathematical research cannot consist only of AI-produced results; mathematicians must be able to pose their own questions, develop their own methods, and explore directions that have not been selected by AI systems as capability demonstrations. To achieve this, fair access to powerful research tools and sufficient computational resources is a necessary condition.

OpenAI stated that they will fund a series of workshops, conferences, and special projects to discuss how to understand major results produced by AI. They also said they are responsibly preparing to release the model that produced these results, and will continue to update future standards for disclosing scientific progress based on community feedback.

But several key questions remain undecided: the correctness of this batch of mathematical results has not yet been verified by an independent third party; the specific scope and severity of problems in unformalized manuscripts are unclear; the exploration of a community-hosted repository has no clear result yet; how AI-produced results will be incorporated into human understanding and the mathematical knowledge system remains undecided; and how future standards for disclosing scientific progress will be updated with community feedback remains unclear.