Sharing AI progress in mathematics

I was a graph theory junkie long ago and even moved to Budapest for awhile to study among the greats. While I was there I started working on Barnette's Conjecture which came to occupy my thoughts over the next 24 years of my life, on and off as I worked in many different fields. Last summer I even thought for a few days that I had actually solved it.But it's supposedly proven here - problem 180. I don't know what to think exactly. I spent thousands of hours on that problem. I really enjoyed it. Hearing that it is solved somehow makes me sad in a far-off way, like hearing an ex-girlfriend died suddenly in a car crash. I don't know, there's probably a lot of people feeling odd emotions tonight.There's no Lean proof for this one so I'm digesting the paper. On the surface it looks like an approach I considered 24 years ago and abandoned.I revisited the problem this summer, along with my partial solutions, when the previous round of stunning proofs came out. Several hours of work with Fable simply convinced me it wasn't yet solvable and reinforced how hard of a problem it was. reply: I have very little understanding of higher math, so I ask you:...

As Kevin Buzzard recently said:> In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question. reply: The other crucial part to this is the ability to actually encode and test the theorem (via Lean). Otherwise, we would be swarmed with a billion lines of theorems that no one will be able to ever understand and verify anyway.

As a TCS/scheduling person, this one is definitely of lesser importance than UGC, but it has been an open problem since the book of Garey and Johnson in 1979:A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 https://github.com/openai/math/blob/main/preprints/A-polynom... reply: The largest I've seen [1] is an exponent of 10^12, which I suppose still counts as polynomial time.I'm sure all of these super small or large constants will improve over time, but it's still amusing. It is entertaining to see the exponents directly rather than have them hidden as n^c or epsilon or O(1).[1]: https://github.com/openai/math/blob/main/preprints/Determini...

This includes a proof of Barnette's Conjecture, which is one of the graph theory conjectures that I tried attacking with SOTA models a few months ago. I like it because it is easy to understand with a basic knowledge of graph theory. I spent quite a bit of time on it and failed. Their proof looks approachable at first glance.https://github.com/openai/math/blob/main/preprints/Paired-st... reply: I've been messing with that problem since 2002. I'm curious if you were trying the dual spanning tree direction (which is what the purported proof is using) or working with cycle construction on the original graph. I was working heavily with edge-Kempe swaps but couldn't quite get there.I am now very interested in the explicit calculation of Hamiltonian cycles in the non-bipartite case, and/or the calculation of their absence. If P=NP I think that's going to be a great route of attack.

Levent Alpöge (Anthropic mathematician) comment on the significance:> Sure, mathematical history features a lot of incredible developments, like the invention of proof, zero, or the computer, and on the great problems our progress has been over timelines measured in decades or centuries. Obviously this technology didn’t appear today, but blurring our eyes a bit to combine the past ten years, with today a measurement of those developments, there is nothing comparable. reply: Crikey it’s a pretty charitable vibe given the whole Navier-Stokes thing, OpenAI trying to stiff him out of co-authorship. I guess any of that sentiment is outweighed by a sense of optimism for where this goes

Unique Games Conjecture [0] is a seminal conjecture in Complexity Theory, and is an underlying assumption for many, many inapproximability results. A valid proof is a big deal![0] https://en.wikipedia.org/wiki/Unique_games_conjecture [1] https://github.com/openai/math/blob/main/preprints/The-Uniqu... reply: I also don't think there was general consensus on which way this would resolve prior to this (or is that a little out dated?) unlike some of the other major problem resolutions. I heard rumors that there would be a big result in TCS and speculation it would be UGC that or P neq PSPACE but I'm still a bit shocked.

“I think at the heart of this issue is that humans have two competing natures: a tendency to compete and a capacity to appreciate beauty,” said Kai Shaikh, a graduate student in mathematics at the University of Toronto. “To me this seems to be a case of the former attempting to strangle the latter.” [1]Beauty can be appreciated even when it is vast, even when it is beyond one's comprehension. I don't think this release should be primarily viewed as an outcome of competition. Instead it is revealing truths about the universe that were always there and always beautiful, even if we hadn't seen them yet. I believe there are infinitely more such beautiful truths currently hidden and waiting for us to discover.[1] https://www.nytimes.com/2026/10/06/science/openai-math-probl... reply: That's a fantastic quote. I definitely personally feel this tension.Not that I could ever "compete" on the frontier of math in the first place. But our nature to compete derives from our need to survive against other capable forces. And results like these make me feel very nervous about humans' capability to remain the dominant force in the universe.

In Stellaris you can play as a civilization of robots who keep their biological creator race alive as "bio trophies". The bio trophies don't do anything meaningful besides by existing satisfy the need their ancestors placed in the robots to take care of them. Starting to wonder if that's the best we can hope for, if these things will be, if they aren't already, better than us at anything that matters. reply: In the Culture series, the hyperintelligent Minds that run civilization are described as keeping human citizens happy as a competition with eachother, where they compare their approval rates. One character likens it to people keeping a beloved aquarium.

This is significant progress and released without all the drama. Some very important progress in Reinmann, Hodge and unique games theorem. Point the repo to your agent and ask for the significance! In a way this is probably 50-100 years of math progress by humans

A quick check shows that this list claims to fully solve 90 of the top 500 open problems in math (https://proofatlas.ai/open-problems/).The highest ranked would be:| 22 | Hilbert’s tenth problem over ℚ || 29 | Unique Games || 31 | Anderson-model extended states || 37 | Spacetime Penrose inequality || 48 | Nonexistence of Landau–Siegel zeros || 52 | Baum–Connes || 78 | Abundance || 80 | Hadwiger || 87 | Bose–Einstein condensation || 92 | Two-dimensional entanglement area law |

It’s fascinating to read through the reasoning traces: https://github.com/openai/math/tree/main/reasoning_tracesLook at one of their examples of an initial prompt: https://github.com/openai/math/blob/main/reasoning_traces/re...

As an AI "doomer" can I ask the non-doomer people here how you interpret the significance of results like these, and what kind of progress you expect to see in the next 1-5 years?Like do you see the technology plateauing at the current level, do you expect progress will continue but only in mathematics, I'm interested to know why others are not concerned?

From their github: "The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking." That's pretty crazy.

Can someone with a math background explain the significance of these and previous problems that have been solved by AI?Does some real problem get solved in physics, chemistry, biology, materials, etc? Or are these fun puzzles for mathematicians with not much real world impact? Eg solving the 8 queen problem in leetcode.

Some of these are interesting ngl.109. Integer multiplication below n log nSurprising that this is possible.158. The Euclidean plane cannot be colored with five colors.Only 6 and 7 remain!376. Universal computation in forced Navier–Stokes flows.Morning coffee proven turing complete

Two weeks ago, I heard rumblings that UGC, RL=L, and a matrix multiplication lower bound (even lower than the one here, but so unbelievably lower I think it was a typo), and a few others were about to be proven by someone at OpenAI. I find myself saying "big if true" a lot lately.Even those on their own were enough to make your head spin. But seeing about 100x that? Geeze.

I'm a software engineering/biology/ML guy who loves when clever math ideas get turned into real solutions (https://en.wikipedia.org/wiki/Compressed_sensing). I am curious if any of the results have immediate applications in any kind of engineering or science.It's fine if not, but it'd be great if even just one of these helped us solve a long-running problem.

Extremely unfortunate that gate keeping got to the point where they felt the need to ask for permission to share math.

Verified Riemann Zeta in Lean: https://github.com/davegoldblatt/openai-zeta-proof-check

It may be useful to publish a formalization of ALL known mathematics at this point. Like every book ever printed, every paper on arXiv etc.How many Gigabytes would that be, compressed? Wikipedia once fit on a DVDThis might also allow for some interesting meta-mathematics

From the GitHub description: “On average, each result used 3 hours of ChatGPT Pro thinking compute”

What's missing for me for each result are the following:* Explain the result to me as if I'm a 10-year-old. * Create the infographic for this result. * Make a Khan Academy-style video to teach me this result.

Let's hypothetically say I'm a PHD student who is half way through my studies and I have a halfway written version of one of these "preprints" - what do I do?Seems like a lot of PHD students are doing to have to pivot the entire structure of their PHD studies? Or just produce something which is already written by OpenAI?

I think they should put human names on the papers as someone who has reviewed the result, even if just a preliminary review. (I’m assuming they didn’t just pipe their model output directly to the internet and these had some amount of review?)

These results are wild. Several individual findings are crazy good and use mostly unexplored methods (the improvement over Riemann for example)... I'm pretty sure some of these results would have been Fields-worthy.But having so many of them at once? Damn. We really live in the future.

So happy this is shared on GitHub rather than some gatekeeping paid journal. Truly a new age for science.

So they set up a whole "Advisory Group on Mathematics and Artificial Intelligence", filled it with renowned academics, promised to listen to the group on "review and communication of emerging results"...and then a week later went nah, we are just going to dump it all on github. OpenAI truly is a special company - in the best and worst way.

Unique games conjecture and matmul <= 2.25. What the hell.