OpenAI publishes 722 math papers written by an unreleased AI model

The repository holds a lot more than a pile of PDFs. According to its README and Unite.AI's walkthrough of the release, here's what's inside:

• Manuscripts: 722 papers, grouped into 372 result families.
• Families: each one bundles related papers, such as a main result, companion arguments, consequences or alternative proofs. Every family is labeled by area of mathematics.
• Preprints folder: PDFs, source files, and build and citation instructions for each paper.
• Lean library: formal proofs plus a catalogue showing which paper each proof belongs to and how it was checked.
• Reasoning summaries: 10 abridged write-ups of how the model reached certain results.
• License: Apache-2.0, so anyone can read and reuse the material.

Each manuscript folder also carries a BibTeX block, the standard snippet researchers paste in when they cite a paper.

OpenAI says they started as tests. As part of building its models, it evaluates them on open research problems, the kind nobody has solved yet. It expanded those tests after its models maxed out its existing math benchmarks.

By the numbers: the model was posed about 4,000 problems. OpenAI says the average result used the equivalent of about three hours of ChatGPT Pro thinking. It then kept only results that met its bar for significance and grouped them into families and papers.

The repository also says some outputs build on earlier results the models produced. So part of the batch is the model building on its own earlier work.

Mostly no, OpenAI says. The vast majority of results came from the same fixed procedure with the same internal model.

There are two named exceptions:

• The Riemann zeta function: work on a zero-free region, a stretch where the function provably has no zeros. The writeup for the region where Re(s) is greater than 11/12 was human edited for readability.
• The Hodge Conjecture for CM abelian varieties: a proof covering one special family of shapes in geometry.

Both topics sit close to famous unsolved problems. The Riemann Hypothesis is about where the zeta function's zeros lie, and the full Hodge Conjecture is one of the seven Millennium Prize Problems. These results tackle narrower pieces, not the full prizes.

This is where Lean comes in. Lean is a programming language for writing proofs so a computer can verify every single step. If a proof passes, it doesn't depend on a tired reviewer spotting a slip on page 40.

Think of it like a spell checker for logic, except it doesn't let anything slide, not even one "obviously true" step.

The catch: not every paper has a Lean proof yet. OpenAI's own README warns that some of the unformalized results could have issues. The company says it will fix problems quickly and add formal proofs as it gets them.

It also promises a paper trail. Corrections and revisions will be saved as new versions, and older versions will stay public.

In real life If you're a student or a math hobbyist, you can open the repository today, pick a family in your favorite area, and see whether it has a Lean proof attached. The ones that do are the ones a computer has already checked.

The 10 reasoning summaries give a sense of the range. They cover:

• Number theory: two-point correlations of multiplicative functions, the irrationality exponent of π, and quasipolynomial bounds for arithmetic progressions.
• Geometry: the symmetric and general Mahler conjectures.
• Computer science: NP-hardness at the basic semidefinite threshold.
• Algebra: Kaplansky's direct-finiteness conjecture in characteristic two.
• Physics-flavored math: the Mézard–Parisi formula for diluted spin glasses, spontaneous magnetization in the quantum Heisenberg ferromagnet, and the three-dimensional relativistic Vlasov–Maxwell system.
• Operator algebras: the isomorphism of free group factors.

You don't need to know what any of these mean. The point is the spread: the batch runs from prime-number puzzles to models of magnets.

Because mathematicians asked. The Verge reports that in September, OpenAI said its model had "resolved more than 100 long-standing open problems across most areas of mathematics," without saying which ones. Mathematicians had been waiting weeks for the details.

On September 29, the Advisory Group on Mathematics and Artificial Intelligence (AGMAI), an independent group at the Institute for Advanced Study in Princeton, published recommendations for AI labs. Unite.AI reports the group drew on more than 600 replies from the math community. Its main asks:

• Release where no lab is in control: scholarly repositories with permanent, citable identifiers.
• Show the work: the model name, the prompts, a summarized chain of thought, the time taken and the estimated compute cost for each result.
• Formalize: check proofs by computer as far as possible.
• Count the misses: for big batches, say how many problems of similar difficulty the model tried and failed, and how problems were chosen.
• Support people: fund workshops, summer schools and books that help humans understand AI-made math, with nonprofits deciding where the money goes.

The group also asked labs not to treat math results as marketing for their models. And it went further: "we do not endorse this practice," it wrote about testing hard problems on proprietary models, "and we ask them to stop."

OpenAI says it consulted the group and drew on its advice for this release. The compute estimates, the 4,000 attempted problems and the reasoning summaries line up with several of the asks. The model's name and a home outside GitHub don't, yet.

OpenAI described its plan in a statement quoted by The Verge. It chose GitHub for now, "with protocols for paper revisions and citations," and is "continuing to explore other community-hosted alternatives" that meet the committee's guidelines. For future releases, it says it will improve the citations, the explanations and the presentation of results.

The big picture: The Verge notes the field is still digesting a fast-growing pile of AI math results from OpenAI and rivals such as Anthropic, including results connected to a Millennium Prize problem. It also reports a heated debate in the field over research practices, ethics and how labs credit the human mathematicians whose work their systems build on.

What's next: mathematicians now get to read, check and argue. With 722 papers, that will take a while, and the version history is where you'll see the outcome.

• You can read it all: the repository is public and free, and the license lets anyone reuse the material.
• You can't use the model: it's an internal, unreleased system, so ChatGPT won't solve these problems for you today.
• Look for the Lean proofs: a paper with a formal proof has been checked by a computer. One without it is still waiting for human review.
• Watch the corrections: OpenAI keeps old versions public, so any fixes will be easy to track.