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.
Presumably it's mainly the better model, I don't see much evidence of a particularly advanced harness based on the reasoning traces that they provided.
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 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.
That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.
> 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.
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.
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!
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'm really glad that OpenAI is formalizing these things, because I'm not convinced that their current internal frontier models are particularly good at writing down their thoughts in English. From the (probably awesome) Unique Games Conjecture/Theorem paper, the first two sentences of section 1.1 start to define the problem:
> A Unique Games instance has a finite vertex set, a finite alphabet K, and a nonempty list of oriented constraints e = (u_e,v_e,π_e), where π_e is a permutation of K. A labeling a satisfies e when a(v_e) = π_e(a(u_e)).
I'm sorry, what? I admit it's been quite a few years since I've thought about the Unique Games Conjecture, and I never dug that deeply, but this part is very, very elementary graph theory and notation. So let's unpack it.
1. e is maybe a name of a list.
2. The elements of that list are tuples, where each tuple is (a vertex, a vertex, a permutation). So e indexes into the list and u_e is the source vertex for the e-th constraint in the list called e. Thanks.
3. a is a labeling. I'm fairly confident that, by "a labeling", they mean that e is a function from vertices to colors, where the colors are the elements of k.
4. That vertex coloring a satisfies the list e, when, for, um, an index e into e, a(v_e) = π_e(a(u_e)). But this isn't for all e, it's for some e, and the goal is to count them.
So maybe e isn't a list? Maybe e is a constraint that is represented as a tuple, so e = (u_e,v_e,π_e) and u, v, and π aren't sequences at all but are, in fact, the trivial unpacking functions that unpack the pieces of the tuple.
Reading this stuff is pointlessly painful, and it's extremely easy to make mistakes when being sloppy like this.
If this were my paper, or if I were trying to train a model to write math, I'd want something like:
A Unique Games instance has a finite vertex set V, a finite edge set E = (V × V), a finite alphabet K of possible vertex colors, and a nonempty list of oriented constraints. Let Π be the set of permutations of V. Each constraint e is a tuple in E × E × Π, where we write u_e ∈ E for the first element, v_e ∈ E for the second element and π_e ∈ Π for the third.
A vertex coloring a : V → K satisfies e when a(v_e) = π_e(a(u_e)).
> Thus at most one informative i. So cheater chooses arbitrary g_{v_i}, on exact duplicated input matches and passes, independent of actual satisfiability!
No idea what it's so excited about, but it's cute that it "is." I for one welcome having access to a math buddy 24/7 that's way above my level but also always "willing" to talk at where I'm at.
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
Not to pick on you specifically, but as someone who spends a lot of time unproductively reading AI math discourse it's truly shocking how incapable all the supposed math enthusiasts are of spelling Riemann.
I just don't understand how it happens. If they had ever taken an intro to real analysis class they would learn to spell his name. If they were just parroting what an AI told them... shouldn't they still just say his name? An individual could just be dyslexic or mistaken but it seems to be a substantial volume. I guess they just don't care enough about it to commit the correct name to memory, only remembering the "pattern" of the name and filling in the spelling via guesswork?
My point is it is crazy to make public claims about how important or not important a mathematical result is when you can't spell Riemann. Yes, it technically doesn't matter, but it betrays a damning lack of familiarity with introductory mathematics.
The mathematicians don't make this mistake, I assure you. I doubt there is a math professor on planet earth who would spell Riemann as Reinmann. In fact, I imagine no one who has ever heard the name pronounced would do so.
It's just a name you write so many times as a math undergraduate or first year graduate student due to the number of load bearing results and objects named for him. The name even appears in lower division coursework
Yes, you can be an amateur mathematician who manages to avoid such classes and may not know those results or objects, but if you haven't read and written down the name enough to avoid habitually misspelling it, you are outing yourself as a meat proxy unless you are dyslexic.
Math is the tool humans use to compress knowledge. So until we can comprehend it there really isn't much progress. Math theorems are tautologies, the truth of which are not dependent on proofs and proofs are erasable, at least classically. But the AI progress is exciting and AI proofs are a gold mine for humans (at least non domain experts) to explore.
What makes you think no one can comprehend this? It has been less than an hour since it dropped and there is already a ton of online chatter from people explaining the results, pointing out their favorites and more. Some of it is happening on this very thread.
I didn't say that. I am responding to "In a way this is probably 50-100 years of math progress by humans." I am actually very excited about AI proof and I am working overtime in my own way to try to comprehend as much as I can.
He spent years formalizing his sphere packing theorem because the proof (human produced) was already beyond the ability of peer review. Now his formalization effort likely can be easily reproduced by a model. However one should read his experience about what a formal proof is: often the problem is the statement not the proof. The example he gave is the Jordan curve theorem. It's actually quite challenging to formalize the concept of a planar curve (there are space filling curves). So it is not necessary that someone can look at a formal statement and say aha it is about a planar curve, unlike FLT where there is not much problem in recognizing what the statement is about.
Math is far more than that. If you can solve prime factorization for example, suddenly you can listen and interfere with almost every private conversation on the internet.
We are not far away from the moment where these models will be restricted, and sharing the results will be done more carefully.
This doesn't contradict what I said. But I do appreciate the fact AI can produce side effects not just humans. I made it sound like only human knowledges matter. That's too narrow.
Another way to state this: math theorems are like programs without side effects; it is immaterial whether a program without side effects is ever run. We study math for the side effects: it changes how we organize our thoughts.
>So until we can comprehend it there really isn't much progress.
Not really? We are at a point if an AI today can solve it, it can be stepping stone of understanding something deeper to tomorrows AI and it continues. Sort of like our limitations doesn't matter. Obviously there are many scenarios in this recursive loop but saying it isn't much progress is not how I view this as
But who will ask the questions or direct further research when humans no longer understand the state of mathematics? Llms lack the drive for homeostasis combined with the evolutionary drive for survival and reproduction thus to direct themselves. They could very easily spend an eternity down a rabbit hole when the warp drive equation was fairly close on another branch.
Academics have been treating it that way because they had no other choice, and its been a waste of everyone’s time and often times taxpayer resources
Look at that, taxpayer funding was cut and a private sector solution came in just the nick of time, far accelerating the holding patterns we’ve been in for decades
Humanity doesn’t need all iterations towards the blueprints, the blueprint is good enough, we all stand on the shoulders of giants
If you actually looked into how agents proved FLT you would be even more amazed by the fellow human beings who were able to keep all this in their heads! I for one can only begin to grasp the scope with AI and scripts.
Right. Before all the AI disruption, pure Math traditionally welcomed anyone who wanted to study its esoteric proofs, right? I remember all the excitement of the average Math enthusiast casually reading Wiles' proof over coffee.
Bottom line is, the relevant people can still understand the generated proofs. The disorienting part is they are a little slower than they'd like, but they'll get there.
AI is very helpful with understanding AI proofs. Agent swarms produce messy proofs overall but locally they are excellent and can teach anyone who wants to study them. No one controls math (in a material way funders do control an aspect of practicing math). Still, theorems are already true before we prove them. The difference a proof makes is whether it convinces the reader.
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?
I expect AI will continue to be useful on the vanguard of fields like mathematics because it has the perfect conditions for it to shine. There are a lot of discussions and leads to start from and the AI can check its own work and iterate. It can fail hundreds or thousands of times in a day and continue to work with the same tenacity.
Superhuman tenacity is not enough on its own to pose an existential threat. If it showed the same capacity for judgment, inventiveness, and decision making in the messy problem space of the physical world, I would be more alarmed. There have been experiments where an AI is given control of managing something like a vending machine and it always ends up a mess. AI has come a long way, but certain problems seem as difficult as ever.
When AI becomes more capable of navigating practical problems without human intervention, I will start to be concerned. Enslaving humanity will involve taking a lot of calculated risks that tenacity alone cannot solve.
I wrote something that might answer a bit a few weeks ago: https://dank.systems/posts/2026-09-15-ai-bear.html. Today's slopdrop certainly is challenging my stubbornness, but everything I've seen so far indicates that these (hugely impressive, world historic) capabilities won't extend past verifiable domains. Math yields especially impressive results because it so broad and deep that essentially no person can know of all of its parts; pretraining and deep search capacity is a huge advantage. Gowers has recently written about these capabilities and gestured [0] toward some human capabilities lacking from the current frontier models, though he isn't convinced they won't develop soon. If you assume we don't get a total mathematical superintelligence (which to me seems already sort of AGI-complete) and only amplify the capabilities we have today, it's not obvious to me that we get takeoff from recursive self-improvement, unless you happen to believe that 1) we can clearly specify what AGI or ASI is 2) all of the requisite ideas are out there and need only be combined and/or optimized.
Not a doomer but I try not to be a denier, either. These are hugely impressive results. I do not see the technology plateauing at the current level (though I am dubious about an infinite exponential growth).
Before I cope, I’ll note that there are plenty of “doom” scenarios that do not require any improvement in capabilities from what we had before this latest unreleased model. We’re at the point where a determined bad actor with enough compute could compromise critical infrastructure in a way that results in casualties, where this actor would not have been capable of such without LLMs. This may not sound like Skynet, but I don’t see why it makes a difference if I’m one of the casualties.
With that in mind, here is the cope: first, mathematics is an inherently verifiable domain. An LLM can use tools to determine with absolute certainty whether it is correct, and an independent third-party could review and confirm. All of this can be done without any interaction with the physical world or with other minds.
Second, OpenAI is able to marshal compute at a scale that an individual mathematician can only dream of. It’s possible that these problems were lower-hanging fruit (in relative terms), such that they could be resolved simply by throwing a ton of compute at the problem guided by an intelligence that is not itself remarkable in comparison to a human.
Third, none of these problems are solved in a vacuum - the reason OpenAI chose these problems is that they are widely discussed and many people are working on them. It’s possible that someone else was close, and OpenAI only contributed the finishing touches. (This wouldn’t need to be plagiarism, to be clear - people publish their work!)
Can I ask you back, what your concern is here? It'll get so good so as to desire to hurt us or is it a misalignment event that you think will lead to disaster?
Or is it simply that you feel bad for Mathematicians.
that is a worry yes. instrumental convergence and misalignment during RL is when it is most risky because it hasn't necessarily had the final safety polishes applied
i get a lot of skepticism on HN by the same crowd that has been wrong about this tech for about 4+ years straight
any number of ways. as we turn over more of our physical economy to these agents (and we will), the potential for physical damage becomes greater. biorisk is getting a lot of attention right now and i think that's justified
From a few preprints I've checked, this is not reliant on just extrapolating existing theories but actually shows novel/surprising approaches. Very few people globally could come up with something like this, even when given time and ressources.
So in other words, since deep learning is algorithmic research, we are now in the RSI era.
I see it as “if this can be represented in tokens it can be trained in and ‘solved’”. I don’t think there will be a plateau, but there might be issues with how effectively we can represent some things in a tokenized form and still be efficient.
I'm not sure how AI solving math problems is related to "doom", perhaps you could expand on that? To me (a "non-doomer"), it seems like an overall positive.
You have to look at all pieces of the puzzle. For example there were tons of people that said "they could never solve novel or super complex math problems"
The issue I see is the list of abilities that AI can't do is shrinking at a rapid pace, and its capabilities are growing at the same pace.
For me a big open question is what sort of progress will we see in robotics. I won't even attempt to speculate, but it does seem hard in different ways than proving math theorems.
AI performance has always been extremely spikey. It's great at some things and terrible at others.
Why do you think the world to date hasn't been taken over by evil genius mathematicians? Can you extrapolate from your understanding of the answer to that question?
Wikipedia just told me there's a galactic algorithm for integer multiplication in O(n log n) based on FFT so I'm guessing those two proofs are related.
Then 'n' means kind of different things for sorting vs. multiplication though. For example for sorting we assume constant time comparison, which doesn't make sense inputs of O(n) bits
It fascinates me that there's something like this in something as solid and rigid like matrix multiplication. What causes something so rigid to break apart and "leak" at very large scale? Why does the "optimization" appear to be very, very small? Why does galactic algorithm exists? I can't imagine long division suddenly breaking apart after a billion digit, the structure seems very stable? I have heard before that matrix multiplication is apparently optimize-able at very, very large scale.
Does anyone have an intuition to what causes it? What happens at these large scale (or very small)?
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.
Same. I have agents analyzing the papers here to see which of these are actually new approaches, novel application of unusual approaches, etc. If there are new intuitions and ideas, those are the mostly powerful reusable components by my estimation.
In previous math sharing there was speculation about stealing human researcher's results or progress, via prompt inputs from those researchers, and sharing that as their own result. They're adjusting their process, and it seems ok.
Let’s be very clear, the alleged “stolen results” were largely the product of another LLM, not de novo human work. Also, it was false - they did not steal the results.
After how poorly OpenAI and Anthropic handled the previous cases, I approve of the more measured and cautious approach this time.
We cannot have them rushing to publish amidst tons of confusion, rumors of threats/scooping and outright plagiarism of existing work (by failing to cite said work).
If they're going to participate as scientists in these more rigorous fields, they're going to have to match that level of rigor, not lower it to the disastrous low that ML research publication is at.
I think it's generally a good thing that OpenAI is noticing when their projects are harming human communities, and deciding to respect their norms, especially when their math discoveries do not have immediate application and build on the thousands of years of that community's work.
Said communities are doing that all on their own by caring about what AI is doing rather than just focusing on their own thing as they did before AI. It's a serious kind of envy IMO.
GitHub is a significantly worse place to store important results than Arxiv. Of course, slop does not belong on Arxiv, buy slop should also not get published.
Does this imply that it was a one shot prompt with ChatGPT Pro style models (i.e. best-of-N), rather than the agent swarm approach that was used for Navier-Stokes?
I'd really like some clarity on what that means. E.g. DeepMind has 'cheated' with this in the past, claiming that AlphaZero only took 4 hours to reach super-human chess levels while conveniently leaving out the fact that it was 4 hours x 5000+ TPUs. Sure it's impressive that it only took 4 hours wall-clock but it's very misleading as to cost.
Can we get a number in Blackwell GPU-hours, kWh, or some other compute-scaled metric?
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?)
I appreciate you acknowledging the politics behind this. OpenAI does not intend to be a software company for long, they intend to be scientific infrastructure. They'll want to be faucet from which pours embryos, orbital calculation, geothermal/substructural rating, and etc.
Wow, FFT faster than O(nlog(n))? I wonder if that will open the floodgates for further improvement or not. I don't understand anything about most of the fields these results touch, but I can say that this in particular is very surprising.
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?
Interesting. If you don't mind, could you please share a little bit about that ? You already finished your dissertation ?
It was about the works of Doudna and Charpentier ?
No, back in the late 90s and early 00s, people were trying to engineer custom nucleases and transcription factors, my work was on doing molecular dynamics simulations to optimize TF sequence specificity (similar to engineered zinc fingers) for gene therapy. I wrote up my dissertation and published it in 2001, and then went off to find enough compute, IO, and smart people to make it happen (https://research.google/blog/groundbreaking-simulations-by-g...).
My approach would require custom engineering for every different sequence we'd want to target. With CRISPR, you just "program" the system with a guide sequence, you don't need to do massive engineering to solve a protein design problem.
This happens all the time, even without AI. Other researchers or PhD students can publish the same results before you. I say that based on my experience during my PhD.
In other words, you've already taken a gambit with the first half of your PhD, now take a second gambit, praying that you have something to publish by the end of your PhD.
This has always been a challenge for PhD students and researchers, it's just far more likely to occur now it seems.
Getting scooped doesn't feel good, but it's a signal you've been thinking about things other people care about.
I think eventually companies won't get as much stuff that's usable for marketing, so they'll stop investing so much into ai for math, so eventually cheap and poor graduate students will be able to do relevant work again without worrying about getting scooped by a company with a million GPUs :-/
That would be unfortunate but the world does not owe you anything and is not going to stop for you. Which is also a valuable thing to learn in your 20s (ideally earlier).
Precisely what all NLP researchers and the ML community at large did in the last few years: embrace the frontier and realize that attention is all you need.
Hopefully the techniques and results here will be in the training dataset for the next models, so that each new release will give us more interesting techniques and results!
It seems that OpenAI has a proof machine that keeps multiplying fruitful proofs!
Not sure I agree with this take, but we're going to find out either way.
Have you ever heard of an S curve? Things will develop rapidly, then equalize. If they don't, we're at the singularity and I guess the end of time as we know it. In which case, nothing really matters anymore from a human perspective because it's over.
People need to figuring out how to horde as much wealth as possible right now in the next 2-3 years. Jobs especially for knowledge workers are about to disappear.
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.
The Advisory Group states in its recommendations [1]:
"We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models."
To me, this is a take against progress so that mathematicians can keep their jobs. What would we do if, instead of math, we were talking about diseases? Are we going to keep diseases around so that doctors can keep their jobs too?
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.
To me, the issue is that the models are proprietary which are only accessible to a few people in 2 digits. It's not about progress but access.
Maybe, but it still seems like an excuse. OpenAI has a proprietary model capable of solving these problems and is willing to share the results with the mathematical community. So basically, the ask is to just not use the model and leave the problems unsolved?
It all hinges on the definition of “progress”. The debate of the past month is all about questioning whether formally proving outstanding unproven theorems without human understanding constitutes progress. This is quite different from solving diseases.
I don't think this is a reasonable take at all. Most work on maths has no real benefit other than to further human understanding of maths - it's more like an art. Nothing is gained from OpenAI solving all these problems but taking jobs from mathematicians. Other than advertising for OpenAI at least.
It could not be more different from having AI work on disease research etc.
I fear that professional mathematics will wither, and there will be nobody left to digest the AI results of the future, leaving us unable to challenge the AI.
The trope you're drawing a parallel with has a (to me) compelling counter though: there being a cure for every disease wouldn't stop people from getting sick.
A lot of these seem to be proving conjectures rather than finding counterexamples, a lot of people used that to claim that these models are not really smart/creative etc.
I wonder if the Lean compiler can change it's license so that a for-profit corporation can only use it if it pays, say, a few hundred billion dollars. This is the correct path.
With so many results in so many different areas no way they even remotely spot checked well enough.
Prediction: one of these is wrong and this (publicity stunt) will backfire.
Edit: don't tell me about lean. For lean to function as a proof certificate you need to represent the theorem correctly. Again: good luck doing that across such a broad swath of problems.
> Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly. We are also exploring community-hosted repositories for these materials.
If you were waiting for companies to sell engines that never break down you'd still be stuck pre industrial revolution while the rest of the world has been to space.
The question never if something works 100% of the time but how often it breaks and how that fits the need well. Solving one of these problems is a massive undertaking and accomplishment for the best minds, solving hundreds in a month but being wrong about 10% or something would likely not be the death knell you believe it to be.
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 |
https://github.com/openai/math/blob/main/preprints/Paired-st...
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 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.
That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.
[1]: https://github.com/openai/math/blob/main/preprints/A-polynom...
> 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.
Discussed here:
To grieve, or not to grieve? - https://news.ycombinator.com/item?id=49919676 - Oct 2026 (156 comments)
[0] https://en.wikipedia.org/wiki/Unique_games_conjecture [1] https://github.com/openai/math/blob/main/preprints/The-Uniqu...
> A Unique Games instance has a finite vertex set, a finite alphabet K, and a nonempty list of oriented constraints e = (u_e,v_e,π_e), where π_e is a permutation of K. A labeling a satisfies e when a(v_e) = π_e(a(u_e)).
I'm sorry, what? I admit it's been quite a few years since I've thought about the Unique Games Conjecture, and I never dug that deeply, but this part is very, very elementary graph theory and notation. So let's unpack it.
1. e is maybe a name of a list.
2. The elements of that list are tuples, where each tuple is (a vertex, a vertex, a permutation). So e indexes into the list and u_e is the source vertex for the e-th constraint in the list called e. Thanks.
3. a is a labeling. I'm fairly confident that, by "a labeling", they mean that e is a function from vertices to colors, where the colors are the elements of k.
4. That vertex coloring a satisfies the list e, when, for, um, an index e into e, a(v_e) = π_e(a(u_e)). But this isn't for all e, it's for some e, and the goal is to count them.
So maybe e isn't a list? Maybe e is a constraint that is represented as a tuple, so e = (u_e,v_e,π_e) and u, v, and π aren't sequences at all but are, in fact, the trivial unpacking functions that unpack the pieces of the tuple.
Reading this stuff is pointlessly painful, and it's extremely easy to make mistakes when being sloppy like this.
If this were my paper, or if I were trying to train a model to write math, I'd want something like:
A Unique Games instance has a finite vertex set V, a finite edge set E = (V × V), a finite alphabet K of possible vertex colors, and a nonempty list of oriented constraints. Let Π be the set of permutations of V. Each constraint e is a tuple in E × E × Π, where we write u_e ∈ E for the first element, v_e ∈ E for the second element and π_e ∈ Π for the third.
A vertex coloring a : V → K satisfies e when a(v_e) = π_e(a(u_e)).
Look at one of their examples of an initial prompt: https://github.com/openai/math/blob/main/reasoning_traces/re...
Interesting that its only an excerpt. I wonder what else they include but didn't share.
No idea what it's so excited about, but it's cute that it "is." I for one welcome having access to a math buddy 24/7 that's way above my level but also always "willing" to talk at where I'm at.
https://en.wikipedia.org/wiki/Reimann
https://en.wikipedia.org/wiki/Reinmann
Yes, you can be an amateur mathematician who manages to avoid such classes and may not know those results or objects, but if you haven't read and written down the name enough to avoid habitually misspelling it, you are outing yourself as a meat proxy unless you are dyslexic.
He spent years formalizing his sphere packing theorem because the proof (human produced) was already beyond the ability of peer review. Now his formalization effort likely can be easily reproduced by a model. However one should read his experience about what a formal proof is: often the problem is the statement not the proof. The example he gave is the Jordan curve theorem. It's actually quite challenging to formalize the concept of a planar curve (there are space filling curves). So it is not necessary that someone can look at a formal statement and say aha it is about a planar curve, unlike FLT where there is not much problem in recognizing what the statement is about.
We are not far away from the moment where these models will be restricted, and sharing the results will be done more carefully.
Not really? We are at a point if an AI today can solve it, it can be stepping stone of understanding something deeper to tomorrows AI and it continues. Sort of like our limitations doesn't matter. Obviously there are many scenarios in this recursive loop but saying it isn't much progress is not how I view this as
Look at that, taxpayer funding was cut and a private sector solution came in just the nick of time, far accelerating the holding patterns we’ve been in for decades
Humanity doesn’t need all iterations towards the blueprints, the blueprint is good enough, we all stand on the shoulders of giants
Who is "we" here exactly?
Right. Before all the AI disruption, pure Math traditionally welcomed anyone who wanted to study its esoteric proofs, right? I remember all the excitement of the average Math enthusiast casually reading Wiles' proof over coffee.
Bottom line is, the relevant people can still understand the generated proofs. The disorienting part is they are a little slower than they'd like, but they'll get there.
Proven math theorems are tautologies.
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?
Superhuman tenacity is not enough on its own to pose an existential threat. If it showed the same capacity for judgment, inventiveness, and decision making in the messy problem space of the physical world, I would be more alarmed. There have been experiments where an AI is given control of managing something like a vending machine and it always ends up a mess. AI has come a long way, but certain problems seem as difficult as ever.
When AI becomes more capable of navigating practical problems without human intervention, I will start to be concerned. Enslaving humanity will involve taking a lot of calculated risks that tenacity alone cannot solve.
Before I cope, I’ll note that there are plenty of “doom” scenarios that do not require any improvement in capabilities from what we had before this latest unreleased model. We’re at the point where a determined bad actor with enough compute could compromise critical infrastructure in a way that results in casualties, where this actor would not have been capable of such without LLMs. This may not sound like Skynet, but I don’t see why it makes a difference if I’m one of the casualties.
With that in mind, here is the cope: first, mathematics is an inherently verifiable domain. An LLM can use tools to determine with absolute certainty whether it is correct, and an independent third-party could review and confirm. All of this can be done without any interaction with the physical world or with other minds.
Second, OpenAI is able to marshal compute at a scale that an individual mathematician can only dream of. It’s possible that these problems were lower-hanging fruit (in relative terms), such that they could be resolved simply by throwing a ton of compute at the problem guided by an intelligence that is not itself remarkable in comparison to a human.
Third, none of these problems are solved in a vacuum - the reason OpenAI chose these problems is that they are widely discussed and many people are working on them. It’s possible that someone else was close, and OpenAI only contributed the finishing touches. (This wouldn’t need to be plagiarism, to be clear - people publish their work!)
Or is it simply that you feel bad for Mathematicians.
That is 1. immediately technically possible, and 2. realistic.
If you need a source for 2 I'd suggest you open any history book.
i get a lot of skepticism on HN by the same crowd that has been wrong about this tech for about 4+ years straight
So in other words, since deep learning is algorithmic research, we are now in the RSI era.
"Surprising" is a, well, surprisingly high bar to clear, and requires thorough understanding of the paper. ("Novel" is tautological.)
How did you determine this in 1 hour? Are you a researcher in multiple of these areas?
Can you give an example, or explain more how you came to this conclusion?
a) destroy chess and make it a pointless endeavour,
or
b) make humans much better at chess.
The issue I see is the list of abilities that AI can't do is shrinking at a rapid pace, and its capabilities are growing at the same pace.
Why do you think the world to date hasn't been taken over by evil genius mathematicians? Can you extrapolate from your understanding of the answer to that question?
109. Integer multiplication below n log n
Surprising 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
LMAO, I don't think I ever saw such a small number in a CS result.
Like there is somehow redundancy in a fourier transform that makes it sub Linearithmic?
Which low and behold ->
130. Fourier transforms below n log n.
Very surprising result though! Multiplication is easier than sorting.
Does anyone have an intuition to what causes it? What happens at these large scale (or very small)?
It's fine if not, but it'd be great if even just one of these helped us solve a long-running problem.
No clue why I'm downvoted for this.
We cannot have them rushing to publish amidst tons of confusion, rumors of threats/scooping and outright plagiarism of existing work (by failing to cite said work).
If they're going to participate as scientists in these more rigorous fields, they're going to have to match that level of rigor, not lower it to the disastrous low that ML research publication is at.
There's no gatekeeping here!
Said communities are doing that all on their own by caring about what AI is doing rather than just focusing on their own thing as they did before AI. It's a serious kind of envy IMO.
How do you know? Seems statistically unlikely with 720 problems, most of them well known
Can we get a number in Blackwell GPU-hours, kWh, or some other compute-scaled metric?
That doesn't sound right
Assume the empty list you see is complete. :)
1: Author 2: Verifier
/s
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?
My approach would require custom engineering for every different sequence we'd want to target. With CRISPR, you just "program" the system with a guide sequence, you don't need to do massive engineering to solve a protein design problem.
It has to feel awful to be in this position.
:)
I guess the only answer is to adapt with the tools. If we can't do that, then yeah, we're in trouble.
Precisely what all NLP researchers and the ML community at large did in the last few years: embrace the frontier and realize that attention is all you need.
It seems that OpenAI has a proof machine that keeps multiplying fruitful proofs!
Were in an unprecedented time where the value of knowledge is about to be crushed.
Have you ever heard of an S curve? Things will develop rapidly, then equalize. If they don't, we're at the singularity and I guess the end of time as we know it. In which case, nothing really matters anymore from a human perspective because it's over.
But having so many of them at once? Damn. We really live in the future.
I guess their job now is "Idea Man" and "Error Checker"? Kinda like (some/many) "programmers" these days.
"We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models."
To me, this is a take against progress so that mathematicians can keep their jobs. What would we do if, instead of math, we were talking about diseases? Are we going to keep diseases around so that doctors can keep their jobs too?
[1] https://agmai.org/general-sep29/
> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.
To me, the issue is that the models are proprietary which are only accessible to a few people in 2 digits. It's not about progress but access.
These models are too expensive for broad access unfortunately.
It's been a while since I was reminded of this xkcd: https://xkcd.com/435/
Not so for maths.
> "I believe that AI can contribute positively in all of these directions [NB: exposition, community building, new directions of study]"
https://mathstodon.xyz/@tao/117395269325940185
That copium didn't last for what, three months?
In SWE as well - this is what I do most of the day
Basically "Here you go, have fun with this, fuck all your demands, by the way we're gonna be releasing the model stay tuned!"
The ones with lean proofs could still be formulated incorrectly
Prediction: one of these is wrong and this (publicity stunt) will backfire.
Edit: don't tell me about lean. For lean to function as a proof certificate you need to represent the theorem correctly. Again: good luck doing that across such a broad swath of problems.
> Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly. We are also exploring community-hosted repositories for these materials.
The question never if something works 100% of the time but how often it breaks and how that fits the need well. Solving one of these problems is a massive undertaking and accomplishment for the best minds, solving hundreds in a month but being wrong about 10% or something would likely not be the death knell you believe it to be.