We might be in the math singularity.
OpenAI has released 722 AI-written math manuscripts, and CEO Sam Altman has picked out four of them as the biggest. In a post summarizing the release, Altman described them as claims that have not yet been confirmed by outside mathematicians. They are a proof of the quasi-Riemann hypothesis, a proof of the Unique Games Conjecture, a proof of the Hodge conjecture for a special class of shapes called CM abelian varieties, and a resolution of the free group factors problem.
Altman says the first, second and fourth have Lean proofs, meaning a computer has checked their logic. The Hodge result does not have one yet. Here is what each claim says, in plain English, and how much weight to put on it for now.
The quasi-Riemann hypothesis #
The Riemann zeta function is closely tied to how prime numbers are spread out. Its “nontrivial zeros,” the inputs where it equals zero, all sit in a vertical strip of the complex plane. The Riemann hypothesis, the most famous open problem in mathematics, says every one of them lies exactly on the strip’s center line, at real part 1/2.
Nobody has been able to prove that. The quasi-Riemann hypothesis is a much weaker statement: that there is some fixed distance from the right edge of the strip that zeros can never cross. The best earlier results left a sliver near the edge that narrows as you go higher up the strip, but never ruled out zeros creeping arbitrarily close to it. The claimed theorem says there are no zeros with real part above 7/8. Because zeros come in mirror-image pairs, that means all of them are confined between 1/8 and 7/8.
What it would mean. This is not the Riemann hypothesis, which would need 1/2, and it doesn’t earn a Millennium Prize. But a fixed zero-free strip is a long-standing goal in its own right, and in principle it gives much sharper control over how many primes there are up to a given size than has been possible. The manuscript says the same zero-free half-plane holds for every Dirichlet L-function, which are cousins of zeta, and a companion paper in the same family gives a different proof of a slightly weaker region, real part above 11/12. A third manuscript in the family is titled as a uniform exclusion of Landau–Siegel zeros, a separate long-standing obstacle in the field.
This is also one of two results OpenAI’s readme lists as an exception to its usual production procedure. It doesn’t say how these two were produced differently, and the 11/12 writeup was edited by humans for readability. AI’s involvement with the Riemann hypothesis isn’t new: earlier this year, Claude couldn’t prove it but improved a longstanding lower bound for a related function from 41% to 67%.
The Unique Games Conjecture #
Many practical problems are too hard to solve perfectly, so computer scientists settle for algorithms that guarantee an answer within some factor of the best possible one. A natural question is how good such guarantees can get. Take Max-Cut: split the nodes of a network into two groups so that as many connections as possible run between the groups. The best known algorithm, from 1995, is guaranteed to get roughly 88% of the best possible cut. Another example is Vertex Cover, which asks for the smallest set of nodes that touches every connection, where the known guarantee is a factor of two.
In 2002, Subhash Khot proposed the Unique Games Conjecture: that a certain puzzle, where each constraint between two items pins down exactly one allowed label for the second item given the first, is hard to solve even approximately. If true, it would mean those known algorithms for Max-Cut, Vertex Cover and many others are the best anyone can do unless P = NP, which is generally believed to be false.
What it would mean. As Altman puts it, a proof means the best known algorithms for problems like Max-Cut can’t be beaten unless P = NP. The conjecture has been central to the field for more than two decades because so many results depend on it. The catalog’s entry also lists direct reductions for some of the consequences, such as the hardness of beating the Max-Cut and Vertex Cover guarantees, and it separately claims the related 2-to-1 Games Conjecture with perfect completeness. It’s one of the ten results for which OpenAI released a reasoning summary.
The Hodge conjecture for CM abelian varieties #
Mathematicians study shapes defined by polynomial equations, such as curves, surfaces and their higher-dimensional cousins. Such shapes have topological features, loosely speaking “holes” of various dimensions. The Hodge conjecture is about which of those features can be explained by smaller pieces of the shape that are also cut out by polynomial equations, rather than by arbitrary geometry. It is one of the seven Millennium Prize problems, and it is wide open in general.
CM abelian varieties are a special, unusually symmetric family: higher-dimensional relatives of the elliptic curves behind much of modern number theory, with extra arithmetic structure called complex multiplication. The claim is that the (rational) Hodge conjecture holds for all of them, in every dimension. OpenAI’s catalog says that, combined with earlier theorems by James Milne, this also yields the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic. A companion result covers products of K3 surfaces.
What it would mean. It would be a major advance on a famous problem, but only a special case, so it does not claim the full prize. Clay’s rules require a solution to be published, to stand for two years, and to be accepted by the community before it considers an award. This is the other result the readme lists as an exception to the standard procedure, and it’s the one Altman says doesn’t have a Lean proof yet. When OpenAI said on September 10 that it had made “substantial progress” on another Millennium Prize problem without naming it, speculation pointed to the Hodge conjecture.
Free group factors #
A “von Neumann algebra” is a mathematical structure of operators, the kind of object used to describe quantum systems, and one can be built from any group. A free group is the simplest kind of group, built from a set of generators with no relations between them. The free group on two generators and the free group on three are clearly different groups. The question, which Altman says has been open since the 1940s, is whether the algebras built from them are nevertheless the same.
The claimed answer is yes, and not just for two and three: all of the nonabelian free group factors are isomorphic, including the one built from infinitely many generators. The catalog adds that their common algebra has “fundamental group” equal to all positive real numbers, a technical consequence.
What it would mean. It would settle one of the field’s best-known open questions. It’s another one of the results Altman says has a Lean proof, and OpenAI also released a reasoning summary for it.
What “Lean proof” does and doesn’t mean #
Lean is a proof assistant: you write a proof in code, and the software checks every logical step. A Lean proof rules out a flaw in the argument, but a person still has to check that the formal statement matches the problem mathematicians care about, with the definitions set up correctly. Proofs generated by AI have been checked this way before: Claude worked autonomously for 11 days to generate the first computer-checked proof of Fermat’s Last Theorem.
OpenAI’s readme is also direct that “many, but not all” manuscripts have been formalized, that some unformalized results “could have issues,” and that corrections will be recorded as new versions. These claims have not been checked by outside mathematicians, and I haven’t verified the Lean status Altman describes.
The backdrop: a community that isn’t convinced about the pace #
This release arrives after a run of OpenAI math claims since the spring. The company said in September that an internal model had produced a proof of the Navier–Stokes Millennium Prize problem, which came with a credit dispute involving mathematician Tristan Buckmaster. It later said its models had resolved more than 100 long-standing problems beyond that one, and an informal tracker counted 234 problems solved with OpenAI’s models as of early August. Fields Medalist Cédric Villani called the Navier–Stokes claim “a cataclysm unlike anything math has known.” OpenAI president Greg Brockman had said that AI could solve a Millennium Prize problem in two to five years.
Not everyone is cheering. Terence Tao has warned that AI proofs can be a net negative for mathematics if they arrive as opaque black boxes. On September 11, 25 Fields Medal winners, including Tao, signed a declaration saying rapid AI proofs are harming mathematics in a “severe misalignment,” with results announced in a rush without proper verification. OpenAI says it has consulted an independent advisory group on mathematics and AI at the Institute for Advanced Study on how to release the work.
What to watch #
The four claims differ a lot in how soon they can be tested. The ones with Lean proofs can be checked mechanically in short order, and mathematicians can then read the formal statements against the originals. The Hodge result has no such shortcut and will need expert readers, and a proof of this scale will likely take months to assess. Any one of these confirmed would be a landmark result; all four would be extraordinary. For now, they are claims, and the verdict rests with outside mathematicians.