OpenAI on the morning of Oct. 7 uploaded 722 mathematical manuscripts produced by an unreleased internal model to GitHub, organizing them into 372 “result families” spanning 17 areas, including number theory, geometry, theoretical computer science and mathematical physics. The MarsBit article, adapted from the WeChat account Jiqizhixin, set aside the backstory of the release and focused instead on what the repository actually contains.
The piece opens with three caveats. First, the 722 manuscripts are grouped into 372 families, and the review selects the best-known and oldest problems from each area. Second, the list includes both claimed proofs and claimed disproofs. By the article’s rough count, about 50 result families contain words such as “disproof” or “counterexample” in their abstracts, meaning some long-standing conjectures are presented as having been overturned rather than established. Third, and most important, every statement that a problem was “proved” or “solved” is OpenAI’s own wording. Most of the work has not been peer reviewed. About 60% of the result families come with Lean formalization, meaning a computer checked the proof line by line; where no formalization exists, the article treats the result as a claim only.
Number theory: from the quasi-Riemann hypothesis to BSD and Hilbert’s tenth problem
Number theory is described as one of the most startling parts of the repository. Out of 31 result families in the area, several are tied to some of the most famous names in mathematics.
Repository item No. 003 is a quasi-Riemann hypothesis result. The article recalls that Bernhard Riemann proposed in 1859 that the nontrivial zeros of the zeta function all lie on a single vertical line. Since the full conjecture remains out of reach, mathematicians have long tried to prove at least that there are no zeros to the right of some line. OpenAI claims to place that line at 7/8, simultaneously for the zeta function and for all Dirichlet L-functions, and says the result has Lean formalization.
The Birch and Swinnerton-Dyer conjecture, another Millennium Prize problem, also appears in the list. According to the article, OpenAI’s result families No. 002 and No. 006 together claim the full BSD formula for “almost all” quadratic twists of every rational elliptic curve. No. 006 is also said to prove Goldfeld’s conjecture, proposed by Dorian Goldfeld in 1979. Neither item is formalized.
No. 004 points to Hilbert’s tenth problem, one of the 23 problems David Hilbert posed in 1900. In 1970, Yuri Matiyasevich completed earlier work showing that there is no general algorithm to decide whether a polynomial equation with integer coefficients has an integer solution. The rational-solution version has remained open for more than half a century. OpenAI now claims the answer there is also negative, without formalization.
The article also singles out two statements that are easier to describe to non-specialists. One concerns Catalan’s constant, the alternating sum 1 - 1/9 + 1/25 - 1/49 and so on; OpenAI claims it is irrational. The other concerns how well π can be approximated by rationals. The best previously proved upper bound on its irrationality measure was about 7.1, while OpenAI claims the optimal value is exactly 2. The article says that result also settles the long-circulating question of whether the Flint Hills series converges. Both come with Lean formalization.
Algebraic geometry: a corner of Hodge, rationality, and quantum geometric Langlands
Algebra and complex geometry account for 36 result families, the third-largest share in the repository. The most eye-catching item in the article is No. 032: a claimed proof of the rational Hodge conjecture for all complex CM abelian varieties. The Hodge conjecture is itself a Millennium Prize problem. OpenAI also says that, using earlier work by James Milne, this result implies the Tate conjecture for all abelian varieties over finite fields. The repository README notes that this item did not follow the model’s standard workflow and is not formalized.
No. 054 concerns the rationality problem for cubic fourfolds. Alexander Kuznetsov proposed a conjectural criterion phrased in categorical language. OpenAI claims to construct examples satisfying that criterion that are still not rational, which would refute the conjecture.
Further up the abstraction ladder, No. 069 is a claimed quantum geometric Langlands correspondence. The article notes that Dennis Gaitsgory and collaborators proved the geometric Langlands conjecture in 2024 through a series of papers running close to 1,000 pages. OpenAI says it has established the quantum version at irrational parameter values as well.
No. 008, meanwhile, is presented as a solution to the Deligne-Drinfeld conjecture on the structure of the Grothendieck-Teichmüller Lie algebra, with Lean formalization attached.
Analysis and geometry: Kakeya, Mahler, and Koebe’s circle domain conjecture
Classical analysis and geometry are packed with old headline problems as well. No. 074 is the Kakeya problem. The article traces it back to a 1917 question by Japanese mathematician Soichi Kakeya about how much area a needle must sweep out to rotate through a full turn in the plane. In higher dimensions, the problem became central to harmonic analysis. In July this year, Wang Hong received the Fields Medal for work with Joshua Zahl on the three-dimensional Kakeya set conjecture. OpenAI now claims to go beyond that, proving a stronger three-dimensional statement and the four-dimensional dimension conjecture. This item is not formalized, and the article says it is among the results most urgently in need of expert reading.
No. 087 is Mahler’s conjecture, posed by Kurt Mahler in 1939, on the minimum possible product of the volumes of a convex body and its dual. The article notes that the symmetric three-dimensional case was only resolved in 2020 by Japanese mathematicians. OpenAI claims to settle both the symmetric and general versions in all dimensions and to characterize all minimizers, with Lean formalization.
No. 071 goes back to 1908. Paul Koebe conjectured that every planar domain can be conformally mapped to a circle domain whose boundary components are circles or points. OpenAI claims to prove the existence part of that conjecture, again with Lean formalization.
Two other geometric highlights are also mentioned: No. 345, which claims infinitely many closed geodesics on higher-dimensional spheres under arbitrary Riemannian metrics, and No. 344, which claims a metric version of the Blaschke conjecture. Neither is formalized.
Theoretical computer science: Unique Games and graph coloring hardness
Theoretical computer science is the largest category in the repository, with 40 result families. The article treats No. 102, the Unique Games Conjecture, as the heaviest item in the section.
Subhash Khot proposed the conjecture in 2002 and later received the 2014 Nevanlinna Prize for the work around it. Its importance lies in approximation hardness: if true, it pins down the best possible approximation thresholds for a wide range of optimization problems, including Max-Cut. The article notes that Khot and collaborators proved a weaker “2-to-2” version in 2018, but the full conjecture remained open. OpenAI now claims a complete proof and says it derives optimal approximation thresholds for Max-Cut, vertex cover and related problems. The main result is said to have Lean formalization.
No. 106 is another standout. It claims that for a graph known to be 3-colorable, coloring it with any fixed number of colors is NP-hard. The article describes the problem as deceptively simple and says it has troubled the field for decades. This result also comes with Lean formalization.
Combinatorics and graph theory: a claimed refutation of Hadwiger
Combinatorics contributes 37 result families. The article calls No. 157 the most surprising of them all: a claimed disproof of Hadwiger’s conjecture.
Proposed by Hugo Hadwiger in 1943, the conjecture is widely regarded as one of the most important open problems in graph theory and as a far-reaching extension of the four-color theorem. Before this, it had only been proved in small cases. OpenAI claims to construct arbitrarily large counterexamples, and not just to the original statement: the article says the failure already appears in a weaker fractional-coloring version. This item has Lean formalization. If it survives scrutiny, the article argues, its impact on graph theory could rival any positive proof in the repository.
No. 158 is another coloring result. It claims that the plane cannot be colored with only five colors under the rule that points at distance 1 must receive different colors. This is the classic chromatic number of the plane problem. For decades, the answer has been trapped between 4 and 7. In 2018, Aubrey de Grey used computer assistance to raise the lower bound to 5. OpenAI now claims to raise it to 6, leaving only 6 or 7 as possibilities. The result is formalized in Lean.
No. 159 concerns one of Paul Erdős’s best-known conjectures: if a set of positive integers has a divergent sum of reciprocals, then it must contain arithmetic progressions of arbitrary length. The article recalls that Ben Green and Terence Tao proved in 2004 that the primes contain arbitrarily long arithmetic progressions, a special case of the conjecture. Erdős had offered a $5,000 prize for the full problem. OpenAI claims a proof, with Lean formalization.
The article also notes No. 179, which is presented as a solution to Herbert Ryser’s 1963 circulant Hadamard matrix conjecture.
Algebra and operator algebras: a run of counterexamples
Where other sections lean toward claimed proofs, algebra and operator algebras are dominated by claimed disproofs. Irving Kaplansky proposed a family of conjectures on group algebras in the mid-20th century. One of them, the unit conjecture, had already been overturned in 2021 by a computer-assisted counterexample found by Giles Gardam.
OpenAI now claims in No. 196 to refute Kaplansky’s zero-divisor conjecture and in No. 197 to refute the direct finiteness conjecture, with Lean formalization for both. No. 294 is said to refute another Kaplansky conjecture on quasitraces in C* algebras.
No. 285 is presented as even more consequential. OpenAI claims a counterexample to the coefficient-free Baum-Connes conjecture and, at the same time, a refutation of the Kadison-Kaplansky conjecture. Since its proposal in 1982, the Baum-Connes conjecture has served as a central bridge between geometry, topology and operator algebras. The article notes that previously known counterexamples required extra conditions. This new claim is not formalized.
The one major “positive” result in the section is No. 287. OpenAI says it has solved the isomorphism problem for free group factors, asking whether the von Neumann algebras associated with free groups on different numbers of generators are in fact the same. The claimed answer is that they are all isomorphic, and the result comes with Lean formalization.
Mathematical physics and fluids: Haldane, BKT, and universal computation in Navier-Stokes
Mathematical physics and probabilistic statistical mechanics together account for 54 result families. The article says many of the statements here are things physicists have long believed but lacked fully rigorous proofs for.
No. 268 claims a proof of the Haldane conjecture, namely that the spin-1 antiferromagnetic Heisenberg chain has a spectral gap. Duncan Haldane proposed that prediction in 1983, and it formed part of the work recognized by the 2016 Nobel Prize in Physics.
No. 271 claims a rigorous proof of Felix Bloch’s 1930 law that low-temperature magnetization in a ferromagnet scales like T to the 3/2 power, together with the exact coefficient.
No. 216 concerns the BKT transition, another topic tied to the 2016 Nobel Prize in Physics. No. 221, the Mézard-Parisi formula, is linked to Giorgio Parisi’s spin glass theory, part of the work recognized by the 2021 Nobel Prize in Physics.
No. 215 claims to construct the continuum limit of the two-dimensional O(3) model and prove the existence of a positive mass gap, with Lean formalization. The article notes that this model is often treated as a toy version of Yang-Mills theory, whose own mass gap problem is a Millennium Prize problem.
Fluid dynamics also gets a separate mention. Beyond the Navier-Stokes singularity result disclosed in September, repository item No. 376 claims the construction of a Navier-Stokes fluid, driven by external force, that can execute any Turing machine program — in other words, universal computation in a fluid. This result has Lean formalization. The article adds that Terence Tao had suggested in 2016 that building a “computer” out of fluid flow might offer a route toward understanding singularities in the Navier-Stokes equations.
No. 362, meanwhile, claims a global smooth solution for the three-dimensional relativistic Vlasov-Maxwell system with large initial data, also with Lean formalization.
The repository is public; verification is the real next step
The article ends on a restrained note. Its most immediate impression after going through the list is density: in any one subfield, a graduate student may only be able to name one or two classic open problems, yet this repository places several in nearly every direction, and many of them are older than the mathematicians who posed them.
That same density is also why the article urges caution. Several of the hottest items on social media — the four-dimensional Kakeya claim, the CM case of the Hodge conjecture, Hilbert’s tenth problem over the rationals, and the Baum-Connes counterexample — still lack Lean formalization. Even where formalization exists, experts still need to check whether the formal proof faithfully matches the original statement of the conjecture. By the standards of the mathematics community, the article says, it could take months or even years for any single one of these claims to be genuinely accepted.
It closes by citing a line quoted by OpenAI CEO Sam Altman: “The sea is vast, and my boat is so small.” The list is now public. The next question is how many of the 722 manuscripts will survive scrutiny, and how long it will take mathematicians to read them.
The original Chinese article was credited to the WeChat account Jiqizhixin (ID: almosthuman2014), authored by “Jiqizhixin covering AI” and edited by Panda.