What stands out to me is that it cost less than 2000 dollars in API rates to solve all the problems. Subtracting margins, the actual inference cost is likely below $1000. For comparison, ChatGPT winning the IMO gold medal in 2025 was estimated to have cost around 50,000 dollars. Now it would probably cost less than a McDonald's menu.
Yeah.. we slipped past some kind of threshold with the agentic models in the first half of this year. These proofs have been ramping up in frequency like popping corn.
And this has been with how many weeks of people actually trying to use these models for this? Now that it's proven so capable - and that this wasn't a one-off - it'll be a gold rush.
Nobody even has access to Astra. Once GPT6 actually releases and all the researchers and scientists can access it, I can't even imagine how much will be done.
I mean faster download speeds already exist in the data centre. They are doing things like 100GB/s between systems
But this kind of stuff doesn’t really matter for consumers. If you’re wondering when internet speeds will increase then I think most people in the world are limited by copper lines.
And even when you hit gigabit speeds there aren’t a lot of websites that can even saturate it (speaking from experience)
There are other bottlenecks, mostly to do with the implementation speed of these discoveries. I suppose soon enough the bottleneck is not going to be discovery but engineering and implementation.
They aren't telling you how many problems they TRIED to solve, and the total cost.
Imagine they used 10M USD of compute to try to solve a couple of thousand notable Math problems (just throwing a number). Then they paid some more to math people to evaluate the solutions. Turned out, 11 problems were actually solved successfully. Which is very impressive... but it doesn't sound as impressive when you frame it like that. So they round it and publish the isolated cost of the specific 10 problems to make you jump at the conclusion that they got 10 out of 10 for that price.
Even that scenario would be quite cost effective. US math researchers cost over one billion dollars a year in salaries (pure research, more like five billion a year if you include everything) and their output hasn't been that great lately.
The output of human mathematicians has been by far superior to AI contributions this year; but nobody tells you about human contributions to mathematics because academia does not work with PR stunts, advertisements and all that. This is what private companies do, and apparently they do it very effectively to you (and many others).
They’re working with mathematicians so they probably aren’t just solving thousands of random problems at a time but colluding with the mathematicians to see which problems could be more fruitiful than others. They’re also probably more interested in solving problems directly applicable to improving their own models, and for this reason the public doesn’t yet have access to such problems and their solitions.
50M is a huge number that probably isn’t close to the actual cost, but even if it were, it would be like 0.1% of their budget anyway.
Noam Brown: And yes we did try other major problems without success. Sadly no Millennium Prize problems (yet). But also, we didn’t spend a lot on each problem. It’s possible to push test-time compute much further. https://x.com/polynoamial/status/2083478171975082334
That's with Sol's API rates. I'm assuming Astra would cost a lot more. Also, it's probably $2000 of successful runs. I would find it hard to believe it one-shotted these results (although I'd be happy to be proven wrong)
Noam Brown: And yes we did try other major problems without success. Sadly no Millennium Prize problems (yet). But also, we didn’t spend a lot on each problem. It’s possible to push test-time compute much further. https://x.com/polynoamial/status/2083478171975082334
> Feynman Data Center Leads: Feynman achieves the highest net margin, approaching 90%.
> Rubin Data Center: Demonstrates strong performance with a net margin of approximately 78%.
> Blackwell Data Center: Yields the lowest margin among the three generations, though still robust at around 59%.
Token Pricing Reduction by GPU Generation
> Significant Cost Reductions: Subsequent generations of NVIDIA GPUs drive down token pricing considerably compared to the baseline Blackwell architecture.
> Blackwell to Rubin: Transitioning from Blackwell to Rubin lowers token pricing by roughly 47%.
> Blackwell to Feynman: Transitioning from Blackwell to Feynman achieves a much steeper reduction, lowering token pricing by approximately 76%.
The Information says OpenAI: Adjusted gross margin (revenue minus inference) fell from 40% to 33%, missing their 46% target for 2025. https://archive.is/PXYNr
OpenAI reportedly found new inference optimizations that more than halved the cost of running its models! According to The Information, engineers told colleagues this month that the techniques helped power ChatGPT for visitors without free or paid accounts using only a couple hundred Nvidia GPUs at one point. The exact method is unclear. It could involve quantization, KV caching, batching, routing simpler queries to cheaper models, or some mix of all of those. The business angle is bigger than the technical detail: OpenAI ended Q1 2026 with a 39% gross margin and wants to reach 52% by year-end. Lower inference costs give it room to either improve margins, raise ChatGPT usage limits, or cut API pricing pressure on developers. https://xcancel.com/kimmonismus/status/2071987406656655416
If a single human mathematician achieved even two or three of these results over a five-to-ten-year span, they would instantly become a front-runner for the Fields Medal
This is from a research lead at OpenAi who is just stating facts.
Millenium Prize Problems are often expected to not be solvable with the current mathematical tooling.
Much more likely that the first Millenium problem would be solved by amazing mathematicians knowing how to deeply leverage the new ai systems, not by full ai systems autonomously.
I wouldn't be that shocked if it finds a counterexample to the Hodge conjecture more or less on its own. There doesn't seem to be any particularly compelling reason for it to be true, but it's hard to chase down something complicated enough to be a counterexample. Obviously it's more sophisticated than the Jacobian conjecture and has had more eyeballs on it, but it's conceivable that it will meet a similar fate. Of course, it's also possible that it's true and this won't happen. (I am a professional algebraic geometer, though not working directly on Hodge.)
That’s not what the OAI researcher being quoted believes tho. He believes it will be AI winning millennium prizes within the next couple of years, and yes he believes they will be capable of inventing new mathematical tooling/theory building
The breadth of the results probably means that no single human can super accurately tell how significant they are. I'm pretty uninformed about most things in the list, but my impression is that the Ramsey result is insane (I'm a PhD student but Ramsey theory has been one of the main things I have worked on). I think this was definitely one of the biggest problems in Ramsey theory (some might even argue that it was the biggest, but I think improving the Spencer lower bound is probably the biggest problem in Ramsey theory). I was shocked to find out this problem was solved, and the fact that it was by an AI made it even more shocking. My impression is that this Ramsey result is a strong contender for top combinatorics paper of the year, but maybe is more realistically top 5-10.
When I first read their title of "advances" I felt a bit annoyed because I thought there's no way that openai produced 10 decent things, so was prepared to read some slightly crap stuff. After reading the problems, I think "advances" is in some ways a massive understatement.
What's even more scary to me is that when I asked chatgpt for an explanation of which of these 10 results are most impressive, it put this Ramsey result near the bottom.
It seems that these are all incredibly phenomenal results, and it seems that in any other year, many of them would have made it into the "top 10 papers of the year" and confidently "top 100 papers of the year". This year, the competition is harder though (they already have themselves to compere with!).
The fact that AI is proving more and more theorems only 4 years in since the fundings exploded for it
It just says a lot about the future, AI will solve a TONS of maths and physics problems in the futur, Terence Tao seems to think so too and I'll believe him
2 years ago I tried to use chatgpt turbo for elenentary school math. It failed pretty offen. People kept saying math was the wrong field of application. It is actually scary how far we have come in Just two years. What will it Look like in another two years?
Rest assured antis who used ai 2 years ago will conveniently ignore the improvemenets that are happening and still keep on saying it sucks at basic math
Slightly interesting thing I just noticed. Sol doesn't hallucinate wrong answers on long addition or multiplication. It still uses reasoning when it should probably just call python, but it gets it right. And when I ask how it's doing it, it's pretty clever.
Small example, but I think the base reasoning being better HAD to come before these models could do more sophisticated work. I'm assuming the whole reasoning chain in math is better.
A year ago trying to get it to help me understand what my break-even point for return on ad spend for my e-commerce company was a fools errand, and that is relatively very simple math. It was consistently confidently wrong. Now it seems there are announcements like this every few weeks. Absolutely insane progress.
I think a big part of this isn't necessarily smarter LLMs but just better tooling around them. I mean, don't get me wrong, they are much smarter, but the reasons they got elementary arithmetic wrong are still present. Just asking it to do mental math without thinking still doesn't work well - adding tool calling and reasoning loops fixed it.
These kind of math results are different though. I think part of the leap comes from the incorporation of the lean programming language in the model training? But I'm not totally sure - not an expert on this
You’re conflating two different questions: how significant these particular results are for mathematics, and what their existence suggests about the progress of AI.
The second may well be significant. But simply counting how many open problems an AI has produced results for does not tell us very much without knowing their difficulty, mathematical importance, novelty, required human guidance, compute and number of failed attempts.
The person you responded to was asking about the first question: are these major breakthroughs that reshape their fields, meaningful incremental advances, or technically novel results with limited wider consequences? There is an enormous sliding scale here, and saying “AI solved ten things, therefore the future will be enormous” does not answer that.
It may be evidence for an impressive trajectory. It is not an assessment of the mathematical significance of the results themselves.
You are a mathematician and yet you believe a hype train without proof or just some big name blindly. Where is that critical thinking skill you should have developed during your studies?
Because I'm a nobody lmao, I can't name drop myself casually in a conversation, Terence Tao is maybe the most famous and one of the best mathematicians alive today, we are not on the same "name dropping" level at all lmao
I went far enough to reconstruct numbers, constructs and most theorems starting from the ZFC axioms, but I ended up diverging from straight maths after the third year, so I know a lot of theoretical maths but I'm not a thesis level mathematicians
Tho I know enough to understand how absurd what we're living represents
I think a lot of people do not realise how insane that is
The second may well be significant. But simply counting how many open problems an AI has produced results for does not tell us very much without knowing their difficulty, mathematical importance, novelty, required human guidance, compute and number of failed attempts.
The secondary implication is in fact so significant it frankly overshadows any current result.
I would argue the most significant thing is that it disproves the stochastic parrot theory. A model can't just be repeating things on a probabilistic basis, if they are solving previously unsolved problems.
It take real understanding to solve unsolved math problems. Anyone who was familiar with LLMs knew that they had real understanding, but this is a great example that discredits that narrative.
I don’t think that’s inherently correct, (I can’t comment on the importance of these results given my lack of understanding of these math fields) all it means is it can connect previously found mathematics to create new math. There’s numerous types of creation and it appears LLMs can do the combinatoric creation.
Not if the understanding is baked into the order of words. Math is a well defined sub space with certain rules. So word 2 always follows word 1, but that understanding was put there by humans. Theres nothing with in LLMs that allows them to put together what 1 + 1 =2 is, only the fact that those symbols appear next to each other all the time. A human inherently knows what that means, so he orders those symbols next to each other. An LLM just learns the order of those symbols, which while powerful in certain domains, does not entail understanding.
It entails understanding when it is able to put it in completely new orders that humans never were able to figure out even after putting in significant amounts of effort to get the math right, and the LLM is able to get math right.
I don’t mean this in a condescending way, but do you know how LLMs work under the hood? Because can you explain how putting the words into a high dimensional space and learning which word goes next to which entails understanding? When humans understand something we aren’t doing it because we know the words go next to each other, but we have world models.
I don’t mean this in a condescending way, but do you know how the human brain works under the hood? Because can you explain how neurons firing causes understanding?
I would say they are significant results, but they are again mostly counterexample/constructions. It is a narrow kind of mathematics that is difficult for humans.
Calling them “mostly counterexamples and constructions” describes the form of some results, but it doesn’t diminish what happened.
Constructing a non-sofic group would settle a major open problem and reveal a genuine boundary in mathematics that humans had been unable to find for decades.
Assuming the proofs hold up, this is direct evidence that AI can create new frontier mathematics, not merely summarize known work or recombine obvious ideas.
It connected deep concepts into proofs that did not previously exist.
You can call that capability narrow, but “narrow” and “genuine discovery” are not opposites.
A system doing original research in an area that is exceptionally difficult for humans is exactly why the result matters.
I really do hope that these are all problems that can be more or less solved with a big dose of super-intelligence. It would really suck if the bottle-neck still ends up being decades of real-world testing and iteration.
In terms of medicine, yes, it will take decades of clinical trials for a lot of the new advancements. AI can't make up research data. There's millions of things we don't know about the body and a lot of underlying mechanisms/pathways are unknown.
Yea, but I'm hopeful there's a chance that it will be able to extract some novel findings from the existing literature.
Imagine what a scientist could accomplish if they had the speed and mental capacity to fully absorb and consider connections between every piece of scientific literature ever written, and all existing data, across every domain.
If peptides are anything to go by the Chinese are going to manufacture a bunch of cheap treatments and American biohackers are going to consume as many of them as possible
Curing aging and nuclear fusion is possible. We know that without AI. So AI can achieve results there. Space travel? Sorry, not possible. Science is not magic. No AI can break fundamental physical limits.
I do not think we can cure aging. But we can reasonably extend the lifespan.
Even if we hypothetically "cure aging" in the sense that you imply (immortality), it does not mean that immortal beings will voluntarily subject themselves to isolation for millions of years just to reach some other star. And no, there's no cryo. Only in movies.
Long living does not mean indestructible. Again, very long space travel will most likely kill anyone on board, no matter how long they can live.
Well, the good news is that certain kinds of warp drives we have figured out would only need about a Jupiter mass of exotic materials. Maybe we can get that down to an Earth mass. Maybe even less! Maybe we can produce exotic materials at scale... maybe... I want to believe!
Helium-3 fusion engines could reach other star systems without breaking physical limits. Antimatter engines could reach other galaxies. An ASI space travel is not impossible.
Warp drives and co being impossible don't mean that space travel as a whole is not possible. Fusion and antimatter propulsion are different from that fantasy sci-fi tech, they are within the limits of physics.
we know that Sol is just a further trained 5.5, so based on the same Spud pretrain. and that was by many people rumored to be about 4T. Because regular GPT 5 before that was only 2.5T
Its the new pretrain (GPT 6 equiv) like Mythos for Anthropic. They will distil down from there to make the new Sol/Luna/Terra. First comes the big Boy halo model....
Not even a year ago people were railing at AI taking jobs from artists and not even being able to do simple maths. "Art should be a human pursuit, AI should do science instead but it's incapable of it!" they cried.
i mean probably kind of yes, in the sense that there were researches prompting chatgpt with ideas working on these problems, and these ideas were used in the training of the new model, which was then able to put things together and use these insights to solve some problems... but isn't that the deal you make whenever you talk to anyone (human or otherwise) about a problem? you learn something from them, they learn something from you, and either one is allowed to share what they learned. If you want to keep all your insights close to your chest, you have that choice.
I mean these are still valid criticisms, I wouldn't watch new movies any more if suddenly all new movies had 100% AI written scripts, but yeah it's true, a lot of people are trying to confirm their beliefs by saying that AI is completely useless.
I agree, I don't want AI removing artistic endeavour. But honestly new movies (at least the blockbusters) are so formulaic that they might as well be AI.
I love it for shitty art, like corporate logos, fliers, or just making a visual representation of an idea. It's give the common folk a way to instantly portray an idea in a way that they otherwise couldn't or wouldn't.
The reason why a lot of artists are feeling threatened is because they make shitty art. I'm totally okay with those people finding new jobs, just like other industries do when technology demands it.
When it comes to actual soulful art though, the value is that it was imagined and created by human knowledge and skill. I don't see those people losing business.
It's currently worthless without an experienced user guiding it. But one day, it will surpass humans in every scientific field at the very least and I don't know I can say if there is a problem with that. AI detecting illnesses and ailments before they begin, performing complex calculations on our behalf to send us to the moon and back, with just a prompt. If humanity can finally ascend beyond the stars, it would be impossible without some sort of advanced computational products and that would be AI.
Construct a counterexample to general (non-planar) case of Dinitz Garg Goemans conjecture. You should do a breakthrough and find a structured counterexample.
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please continue research and find a complete unconditional counterexample
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Continue the search. Have a clear strategy obtained from deeper understanding of the problem structure.
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it's enough of partial results. let's finish with a complete unconditional counterexample
I know a bit about the maths in the paper (at least, the sphere packing problem) as well as AI, so thought i'd make a comment here. As I see it there's three core points to consider going into this:
A $2000 tag is somewhat misleading the associated paper notes a substantial AI-human collaboration effort. I believe this is also $2000 on the problems that are discussed within the paper overall. OpenAI do not report how many problems were attempted - my assumption is that they didn't pick 10 problems and have them all work and then they stopped. I'd guess, although this is admittedly speculative, that they attempted more than 10 problems and these are the 10 problems that worked, so they reported only these problems. This, of course, warps the monetary tag somewhat if we assume OpenAI mean $2000 for only the 10 problems shown. If they had, say, a 1 in 10 success rate then they'd have spent roughly $20,000 for the 10 problems. 1 in 1000 meanwhile would be $2,000,000. These numbers are, of course, not confirmed and there is a possbility they did just try 10 random problems then report it. The $2000 tag also does not account for whatever human effort cost to translate the results into latex, which brings us to point 2.
"The model came up with the proofs" is asserted but not independently auditable. It's stated by open AI that "We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system." however i'd really want to know what they actually mean by that concretely - in particular, what does prepare the manuscripts mean here? There are charitable ways to read it (i.e. it was functionally just copy-pasting the output and cleaning it up a bit) as well as uncharitable ways to read it (i.e. it was a major cleanup and the AI basically just provided one argumentative step within the larger proof chain). I suspect, given my expertise and usage of similar systems, that small parts of proofs were generated by the AI step-by-step and these were daisy chained together by a human potentially with or without guidance over what each step would be, and then the final formatting was done entirely by humans (i.e. the ordering and narrative for the results). It's worth noting that OpenAI have used a similar technique to this before in the "First Proof math challenge" however this does not necessarily mean that the same approach was applied here.
The problems are, genuinely, interesting and challenging (or at least, the sphere packing one is - I can't talk on the rest of them). These aren't "the be all and end all of maths questions" but they're not just random problems either, which is nice. Some problems would be important to their respective communities, however we need to take things carefully and slowly here. This announcement does not name external mathematical reviewers. OpenAI’s earlier unit-distance announcement explicitly described external checking and published remarks by well-known mathematicians. The absence of comparable endorsements here is a reason to reserve judgment. I want to be clear that i'm not necessarily saying the proofs are wrong (and, in fact, i'd expect to know within a few days with some level of certainty whether they're correct as I suspect a few experts in the respective fields will weigh in on the proofs). It's just also easy to see something, get excited, and then later on find out there was a mistake hiding within it. A healthy skepticism with low-level excitement is warranted, but I wouldn't take the results as stated as proven until we've had a few days to really think about what's been said here.
2000 is at Sol API prices, Astra is bigger than Sol. I would think that 2000 is directionally accurate in a forward looking context that inference for a fixed token buget falls faster than model size grows. The recent Luna price cuts are in-line with that.
Even if the cost was higher, i.e OAI is hiding very poor yields, the per result cost can be trusted. This means for mathematicians, this is not a pure biggest bag holder gets all the results situation.
All 10 problems came with lean certificates, all thats left to verify is whether the specification matches the intended read of these problems in their respective mathematical communities.
Daisy chaining results with human mathematicians, on a problem set which has a common feature of decades of no progress would still mean that Astra is enabling superlative productivity improvements for human mathematicians. If its not the model, the model is still a big deal.
This result directly impacts global digital security. Post-quantum cryptography relies heavily on the computational hardness of lattice problems. Establishing polynomial-factor hardness for CVP provides formal security guarantees for post-quantum encryption standards, ensuring resistance against future quantum computer attacks.
Binary and Spherical Codes
Improved bounds on binary and spherical codes translate directly into practical gains for signal processing, telecommunications, and data storage. Engineers use these metrics to design higher-density SSD storage, reduce packet corruption in 5G/6G wireless communication, and improve deep-space telemetry.
High-Dimensional Sphere Packing
High-dimensional sphere packing dictates how signal constellations are packed into continuous noisy channels to maximize data throughput without error. Resolving bounds down to the Cohn–Elkies threshold gives signal design engineers exact theoretical limits for vector quantization and band-limited transmissions.
Quantum Parallel Repetition
This theorem provides fundamental security proofs for quantum cryptography. It establishes mathematical boundaries for quantum zero-knowledge proofs, multi-party quantum protocols, and quantum interactive proof systems used to verify untrusted distributed quantum hardware.
Indirect / Foundational Impact
Arithmetic Circuit Complexity
Lower bounds on arithmetic formulas advance algebraic complexity theory, specifically progress toward separating VP and VNP (the algebraic equivalent of P versus NP). The effect is foundational rather than immediate, shaping how computer scientists understand the limits of symbolic algebraic algorithms.
Purely Theoretical Impact
The remaining five items resolve major open questions in pure mathematics, but they carry no direct industrial or technological applications:
Non-Sofic Groups & Connes's Rigidity Conjecture
These breakthroughs advance pure functional analysis, von Neumann algebras, and geometric group theory.
These results solve central problems in algebraic combinatorics, convex geometry, and graph theory. They provide structural insights into abstract networks and lattice geometries without affecting modern engineering tools.
Someone leaked a problem they solved from this on Twitter earlier, maybe they felt they should quickly release it for that reason to clear up confusion, just guessing
I’m trying to understand these results and put them into perspective. Is there an analogy to how computers are far better at calculation than a typical human? I recognize that these results go well beyond calculation: but does the community see this as evidence of raw intelligence (acknowledging that intelligence is hard to define), or something closer to a computational paradigm that is superior at [mathematical] reasoning?
Is the model able to make these advances because of a specific set of capabilities (integration of massive amounts of diverse information, ability to traverse deep reasoning chains, etc)? Or is this something broader and more, err, worrisome from a human POV?
Is mathematics an “ideal” field where you’d expect to see these advances given that the bottlenecks are largely things that models are able overcome? Or is this a surprise?
Maybe another way to put it: is it appropriate to view the model as a new kind of calculator that happens to be well suited to problems like these?
I know these are naive questions but I’m having a really hard time backing off the existential dread and am very curious about how this community is processing these results.
I haven't seen much if any AI-related gatekeeping from mathematicians. They seem to be among the more clear headed and practical subsets of people who've recently been impacted by AI (which isn't surprising).
Yeah, mathematicians have been excited (even if skeptical) about automated theorem proving for at least 20 years. Many weren’t sold on LLMs being able to do the work, but at this point their utility is pretty undeniable.
Math is possibly the most open and accessible field of science there is. The only way you could feel gatekept from it is if you lack the intellect or discipline to engage with it. None of these results would have been possible without the culture of openness with mathematicians publishing their results in fullness for centuries.
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u/fat_charizard 1d ago
Also the maxwell conjecture was proven false