Machine Heart Editorial Team
Just this morning, the AI world and the mathematics world genuinely exploded. Many researchers and netizens were, quite literally, slumped in their chairs, shaken by an intelligence explosion.
It is hard to imagine that this GitHub repository holds not one or two heavyweight mathematical results, but a full 722 of them.
It is a list of mathematical papers: 722 manuscripts, grouped into 372 result families, all produced by an as-yet-unreleased internal frontier model.
The names on the list are enough to make anyone who has taken graduate-level mathematics courses stop and look twice: the quasi-Riemann Hypothesis, the four-dimensional Kakeya conjecture, the Unique Games Conjecture, the isomorphism problem for factors of free groups, Hilbert's tenth problem over the rationals, the Hodge conjecture for CM abelian varieties, Mahler's conjecture… Some of these problems have stood open for more than half a century; others were extension directions of Fields Medal–level work.
Sam Altman says we are entering a new era of discovery.
An explosive release
According to OpenAI, this batch of results came out of an evaluation of the mathematical capabilities of an internal model.
Because the model had already saturated existing mathematics benchmarks, the team extended the evaluation to open research problems, some of the results building on output the model had produced earlier. Throughout the evaluation, the model was fed roughly 4,000 research questions; after outputs were merged into result families by relevance and then filtered through a "sufficiently important" threshold, what remained was the current 372 families and 722 manuscripts.
A "result family" usually contains one main result plus supporting arguments, corollaries, or alternative proofs. By OpenAI's own subject labeling, these 372 families span 17 fields. The largest counts are theoretical computer science (40 families), combinatorics (37 families), algebra and complex geometry (36 families), and number theory (31 families), followed by probability and statistical mechanics, differential geometry, mathematical physics, operator algebras, and others; mathematical logic has the fewest, at 6 families.
On compute, OpenAI says: the vast majority of results went through the same fixed pipeline, consuming on average roughly the equivalent of three hours of ChatGPT Pro thinking per result.
Another detail worth noting: the manuscript folders end with dates, and almost all of them cluster between September 23 and October 5, with September 23 and 24 alone accounting for nearly 200 papers each. In other words, the bulk of this list appears to have been generated in batches over less than two weeks.
Alongside the papers, three categories of "proof artifacts" were released: Lean formal proofs, 10 condensed summaries of the model's reasoning process, and statistics on compute usage and the number of attempted problems. Based on our count of the repository structure, 235 of the 372 families come with a Lean formalization documentation page — about 63 percent.
But OpenAI is equally candid that not all manuscripts have been formalized, that unformalized results "may have problems," and that the team will correct them as quickly as possible and keep adding more.
At present, the project's star count is skyrocketing — in just three hours it has already passed 2k.
Repository: https://github.com/openai/math
Riemann and Kakeya
Among the 722 manuscripts, the first to go viral on social media was No. 003: the quasi-Riemann Hypothesis.
The Riemann Hypothesis asserts that all nontrivial zeros of the Riemann zeta function lie on the line with real part 1/2. That goal is too distant, so number theorists settled for something weaker: the "quasi-Riemann Hypothesis" — proving that there exists some constant θ less than 1 such that there are no zeros in the entire half-plane with real part greater than θ would itself be a huge advance.
For over a century, every zero-free region people could prove grew closer and closer to the line with real part 1 as the imaginary part increased, and never managed to "hold open" a vertical strip of fixed width. OpenAI's conclusion here is θ = 7/8, holding uniformly for the zeta function and all Dirichlet L-functions, and Lean formalization has already been provided. The accompanying paper also gives a uniform exclusion result for Landau–Siegel zeros.
Another item that sparked heated discussion is No. 074: the Kakeya problem.
On July 23 of this year, Hong Wang won the Fields Medal for the proof of the three-dimensional Kakeya set conjecture completed in 2025 with Joshua Zahl, becoming only the third woman ever to receive that honor; at the time, the "next steps" roundup in Quanta Magazine still prominently listed the four-dimensional Kakeya conjecture.
This family in OpenAI's list simultaneously claims to have solved the three-dimensional Kakeya "maximal function" conjecture and the four-dimensional Hausdorff dimension conjecture.
The heavyweight results go far beyond these two.
— In theoretical computer science, No. 102 claims to prove the Unique Games Conjecture proposed by Subhash Khot in 2002, and on that basis delivers at once the optimal approximation hardness thresholds for classic problems such as Max-Cut and Vertex Cover;
— In algebraic geometry, No. 032 claims to prove the rational Hodge conjecture for all complex CM abelian varieties;
— In number theory, No. 002 and No. 006 together give a complete BSD formula on a family of quadratic twists of density one for every rational elliptic curve; No. 004 gives a negative answer to Hilbert's tenth problem over the rationals.
All three touch on Millennium Prize Problems or landmark problems of the twentieth century.
In operator algebras, No. 287 claims to resolve the isomorphism problem for factors of free groups, concluding that free group factors of different ranks are isomorphic to one another, with Lean formalization attached.
In addition, the list also includes results proving that the irrationality measure of π is exactly 2, that Catalan's constant is irrational, that Mahler's conjecture holds in all dimensions, and that the plane cannot be colored with five colors so as to give a unit-distance coloring, along with counterexample constructions for the Kaplansky zero-divisor conjecture and the Hadwiger conjecture.
As a supplement, below is a ranking by former OpenAI researcher Will Depue, who used GPT 6 Pro and Fable 5.1 to rank all discoveries of the past three years. Blue denotes human discoveries, red denotes AI discoveries (before October 6), and green denotes AI discoveries from the OpenAI/math repo. The stunning part: 81% of them were released only today.
A mysterious internal model
Throughout the entire release, the model itself remained absent. OpenAI did not disclose its name, referring to it in the blog only as an "internal frontier model," and said it is moving forward with its "responsible release."
On September 8, OpenAI issued the Navier-Stokes announcement. It stated that an internal model — "significantly stronger than GPT-6 Astra," which began training on August 28 and is still training — used roughly 10,000 parallel agents over 88 hours of work to prove that three-dimensional incompressible fluid under smooth external forces can develop a singularity in finite time starting from a state of rest, with GPT-6 Astra spending an additional 17 hours to complete the Lean formalization.
Related link: https://openai.com/zh-Hans-CN/index/navier-stokes-solution/
On the eve of the release, an OpenAI spokesperson had already confirmed that this new internal model had solved more than 100 long-standing open problems "across most areas of mathematics."
Mathematicians up in arms
Over the past two-plus months, friction and pressure between AI labs and mathematicians have climbed steadily.
On August 1, OpenAI announced ten advances by its next-generation model Astra in mathematics and theoretical computer science. Just days later, mathematicians pointed out that among the counterexamples disproving the Connes rigidity conjecture, one group did not satisfy the required conditions. On August 11, Anthropic also disclosed progress by its unreleased model on directions related to the Riemann Hypothesis.
According to Wired, in August OpenAI privately convened about 40 mathematicians to discuss what to do if AI surpasses humans in mathematics.
Attendees said OpenAI hinted at the time that its model had already solved hundreds of long-standing hard problems and sought opinions on how to release them; one attendee said the company also promised not to release them all at once, but an OpenAI spokesperson said the company "was not aware" of any such promise.
A Northwestern University mathematician recalled that the mood in the room was a mix of excitement and fear, and that attendees raised on the spot the idea that results should no longer be released only through blog posts and tweets as in the ten-problem August batch, but written up as papers so mathematicians could digest and use them.
According to reports, OpenAI only mobilized thousands of agents to attack the problem after hearing "rumors" that someone was about to solve it. Tristan Buckmaster of New York University accused OpenAI of jumping ahead of work he had been collaborating on privately with Levent Alpöge; Alpöge is himself an Anthropic employee, and the two had not yet published their results, though they had been using OpenAI's tools to assist their research.
Related link: https://www.wired.com/story/openai-is-pissing-off-a-bunch-of-mathematicians-again/
Mathematicians say, "People feel these leading AI companies have a kind of gangster way of operating."
On September 11, 25 Fields Medalists, including Terence Tao, signed a joint statement. The statement argues that conquering famous problems was originally a tool and a proxy metric for advancing conceptual understanding, and that rushed announcements leave no time to write good papers or do proper citations, creating serious problems of attribution and plagiarism.
Related link: https://terrytao.wordpress.com/2026/09/11/a-severe-misalignment-of-ai-in-mathematics/
Before that, the Leiden Declaration, signed by more than 4,000 mathematicians, had already demanded that AI companies comply with the mathematical community's standards, covering disclosure of AI use, peer review, citation of prior work, and maintaining open access; Bryna Kra is one of its drafters.
On September 29, the Independent Advisory Group on Mathematics and AI at the Institute for Advanced Study (IAS), after aggregating more than 600 community surveys, published recommendations for releasing AI-generated mathematical results. This is precisely the document OpenAI specifically cited in its tweet and blog post this time.
https://agmai.org/general-sep29/
The IAS recommendations come in two steps.
The first step is "initial release": labs should search for and cite relevant prior work, and write proofs in a form consistent with conventional mathematical paper practice; they should disclose the model name, prompts, a summary of the chain of thought, time spent, and compute cost estimates; they should formalize as much as possible; they should deposit results in a neutral repository "not controlled by any AI lab," with persistent identifiers and a revision history; and they should report failure rates on comparable problems.
The second step is "supporting understanding": funding conferences, workshops, and expository writing, and providing broad, equitable model access to the global mathematical community.
Measured against this list, OpenAI's report card is arguably "selectively passing." It discloses reasoning summaries, compute estimates, and the number of attempted problems; it commits to funding a series of workshops and conferences around major AI results; it commits to version tracking and issuing revisions as new versions; and it has formalized more than 60 percent.
There are also many points of non-compliance. Most notably, the model's name is not disclosed. The repository sits under OpenAI's own GitHub organization, and OpenAI says it is still exploring community-hosted alternatives. OpenAI also acknowledges in its blog that future releases will keep improving on citations, mathematical exposition, and presentation.
The bottleneck shifts from "proof" to "understanding"
If the September Navier-Stokes announcement proved that AI can chew through one of the hardest problems, this release proves something else: across a fairly wide mathematical frontier, producing checkable new results can now be done at scale, at a pace of "a few hours per problem."
This makes the judgment Terence Tao offered in September all the more pointed: the real conflict is not human reasoning versus machine reasoning, but rather a mismatch between an abundance of proof output and a scarcity of expert attention.
Of the 372 result families, 235 come with Lean proofs, which genuinely lowers the cost of verifying "whether the proof is correct"; but the Connes rigidity conjecture episode has already reminded everyone that formalization can guarantee that the reasoning is error-free, yet cannot automatically guarantee that the proposition proved is the one the mathematicians had in mind. In the Lean documentation page prepared for each family, OpenAI explicitly states "how far the formalization covers and how far it does not" — for example, the quasi-Riemann Hypothesis family notes that the later application sections of the paper fall outside the scope of formalization. That in itself shows that checking the statement level still requires humans.
As for those heaviest results that have not yet been formalized — the four-dimensional Kakeya problem, the Hodge conjecture for CM abelian varieties, Hilbert's tenth problem over the rationals — whether they can withstand line-by-line expert scrutiny in the coming months is the real report card for this release.
A question even more worth asking: when the supply of new theorems is no longer scarce, how will the mathematical community reallocate its most precious resource — the time to read and understand them?
Sam Altman says we are entering a new era of discovery. The first half of that sentence already has 722 pieces of evidence today; for the second half, "discovery," to truly hold, humans still need to read these results, catch them, and pass them on.
Reference links
https://openai.com/index/sharing-ai-progress-in-mathematics/
https://github.com/openai/math/blob/main/lean/docs/003.md
https://quantamagazine.org/hong-wang-wins-2026-fields-medal-the-third-woman-ever-20260723
https://www.remio.ai/post/terence-tao-ai-math-warning-challenges-the-benchmark-race
© THE END
For reprints, please contact this official account for authorization.
Submissions or reporting inquiries: liyazhou@jiqizhixin.com