I think OpenAI deserve credit for adjusting their approach after the last math drop. I think Terrance Tao shares some credit here for galvanizing the response.
Including the relevant mathematicians and scientists before dumping this onto the public is sure to go a long way to relaxing fears in those communities. I'm not sure how they will deal with accusations of favoritism but at least they are demonstrating a willingness to participate in a process.
I hope the community can express some positivity that will encourage OpenAI and other foundation labs to continue with this collaborative approach.
Well said. Attribution is important, for obvious human reasons and to direct people to valuable context. But our names will fade fast from history, what matters is the journey of human knowledge.
> It is amazing what you can accomplish if you do not care who gets the credit.
> It is amazing what you can accomplish if you do not care who gets the credit.
-- Me, 2026
Jokes aside, this is a thought provoking quote. I do think it's easy to say as a well known figurehead, when the quote and other actions below you may easily be attributed to you, at least partially. But I do agree somewhat, and I think the context around Truman's time (WWII) is a good example of what humanity can accomplish when the stakes at hand supersede the considerations of workers' rights or credit/ownership. Of course this is also a very favorable phrasing for a period of enormous human suffering. I think there is an arms race analogy to be drawn here, though the current situation and stakes are much different.
Funnily enough, there’s a good chance Truman never said those words; the earliest known attribution of the quote to him is from a Time magazine article written in 1988. It’s not in any of his personal writing or speeches.
For example, one generally don’t name cells in their body, and cells works generally pretty well to sustain a whole self.
On societal level, obviously no one is aware of who authored the biggest mass of benefit that is granted by the mere fact of living in society. No specific individual have credits for example regarding the language they use daily¹. For the most recent ones, it’s now often possible to attach some name to invention of some cultural/technological artifact, but of course Zeitgeist-wise perspective tend to consider it’s rarely a single punctual individual that should be considered the critical pivot point.
On the other hand, even Marx that was so critical of individual alienation couldn’t refrain to make the intellectual equivalent of pissing territory and attach a name referring to its own ego within all of his works. Ego is a masterpiece of illusion: it will bring the foundational leveraging to build every possible societal hegemony under the sun while convincing the deluded that serving the ego is the indispensable first step of everything to best serve every individual selfs.
¹ Even for constructed languages, such as Esperanto that I practice myself, whoever construct them obviously don’t start from cultural blank slate.
'A man may do an immense deal of good, if he does not care who gets the credit for it.'
"QI hypothesizes that a Jesuit Priest named Father Strickland deserves primary credit for this family of sayings. This claim is based on the diary entry dated September 21, 1863 written by Sir Mountstuart E. Grant Duff and multiple attributions thereafter. However, similar maxims may have been developed independently."
I don't understand. The AI-math guidance counsel has recently explicitly argued against what you say in paragraph two, and which is why for the Kakeya problem there was no group formed. The different communities have given different suggestions. I'm not sure how much I agree with the first paragraph either given these facts.
> We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies
1. They identified relevant experts for each of the discoveries they wanted to release
2. They collaborated with them to determine how to release the results
The effect for the first two (maths, comp sci) was the same because the experts they chose made that decision.
The downside here is actually the third case (physics). Since they only chose 5 scientists (David Gross, Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov) and those scientists chose to keep the results private until they finish their own work. Everyone else who might be interested has to wait.
That was my favouritism aside. Scientists working on similar or adjacent problems that are not included in these kinds of private releases might feel they are not getting a fair chance.
This is a process that will change, but it seems they listened to the anger from the math community and made changes that help address the concerns.
I think, sadly, these fields will collapse further into a tighter in-group. Anyone outside that group will find it increasingly hard to penetrate. Talent won’t get you there anymore.
The hubris in the framing is astonishing, amusing and sad all at the same time. Many say, that's what HN attracts. Anyone who talks about inevitability except of death in these times is just someone who considers they'd be in the "have" rather the "have not" category.
I still can't see what this means for human understanding and how open would this knowledge be. Who'd ultimately get access to this stuff going forward and what kind of power games would ensue hereon.
I'm not sure why you think that someone who believes that this is inevitable is just someone who assumes they'd be in the "have" rather the "have not" category.
It seems like you are doing exactly what you accuse the parent commenter of doing.
The only difference, perhaps, is that the commenter seems to extrapolate past progress to future progress, while your own assumption is a lot more tenuous.
It’s always weird when people are so quick to use a single positive thing to trivialize the many historically questionable things an entity has done up till this point. It certainly doesn’t come from a place of rationality.
What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
> What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
This is like going to a politician's event and seeing him giving away lollipops to toddlers and kissing babies. But it's the same politician caught in corruption and embezzlement scandals.
You look at that and praise how they have redeemed. Other people look at that and see a corrupt politician handing out candy.
People tell us who they are through their actions. If you prove yourself to be capable of morally reprehensible things to the point throughout the years you build a reputation for all the morally reprehensible things you do, does your potential to do morally reprehensible things change if you go way out of your way to let people know that sometimes something's you do aren't morally reprehensible?
Wrong, these companies have shown time and time again they are irredeemably misaligned with our interests.
Right now you and everyone else is doing the "leave the multi billion dollar company alone" version of the neckbeard "leave the lady be" meme. And you deserved to be chastised for it.
Its the equivalent of you going around telling everyone in the office about how your boss is turning it around because they bought donuts today, after they spent three years telling you you're worthless, they're entitled to taking all the credit for your work and they plan on firing you as soon as they can.
This is going to be a very big deal in some of my circles, which include KITP (the Kavli Institute of Theoretical Physics), where David Gross was the Director for many years and now holds the Chancellor's Chair Professor in Theoretical Physics.
I’m not sure what meaningfully changed except for the QCD / yang mills result (where leaders in the field are going to do a review and write up first).
After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
> After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
This.
The truth if the matter is that this is still a publicity stunt. This investment of time, effort, and compute is aimed at building a reputation that this company's model is better than the competition's. The previous publicity stunt brew up in their face, and now they seem to be trying to adapt and double down.
I would have found it a lot more impressive if they just empowered scientists through some kind of grant rather than to internalize all this. It sets the tone for what's to come and inevitably if what they find is juicy enough it will not be published. For instance once they start working on materials science (if they aren't already).
They don't necessarily deserve credit, it is just the bare minimum courteous behaviour in academia to reach out to other researchers in the same area before publishing a result.
I'm also unaware of the quality of the consultation with the relevant mathematicians, colour me skeptical there.
> "The external collaborations will follow four principles... until Anthropic, Google, DeepSeek, Google and dozens of Chinese companies all make the same offering, in which case we will rediscover and re-implement all of the things you hate about traditional academic advancement and publishing, except because our companies are run by people even more mercurial and self-absorbed than university presidents, we'll wind up supporting even fewer people to do it."
ETH is false! I really would not have believed it.
[0]:
> We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula. For every fixed k ⩾ 2, on a formula with n variables and m clauses, our algorithm uses $poly_k (n, m)$ space and runs in time $(n + m)^{O_k(1)} 2^{n/(\log n)^{1/2 + o(1)}}$
The dominant term there is (roughly) $2^{n / (\log n)^{0.5}}$, so still superpolynomial but less than exponential. Fucking wild.
Yeah, this really blew me away. I took a graduate computational complexity course and the original prof had to drop out due to health reasons. We got another professor from a completely different field who was basically as green in the field as we were, but with far better research chops. It turned the course into kind of a long-running lab where we just kind of did whatever we wanted. We spent a lot of time just philosophizing (mostly uselessly) about the ETH. I am very chuffed to see this result.
It's not AI that has created those open problems. It has simply shown that some conjecture is false and that leads naturally to problems in a sense already posed.
A very different thing is proposing a framework and in it identifying interesting questions. So far AI has only worked within existing frameworks addressing problems already posed. That may change in the future, but it would be a genuine and qualitative evolution.
ETH is false! This is one of the big long-standing hypotheses that I actually remember from university, and afaik, it was reasonable to assume it would be true; reasonable enough for a lot of papers and results to assume this for bounds or hardness proofs of common problems in parameterized complexity, anyways.
Also glad to see that one of the authors of the paper stating it is also in the loop here.
What will happen to fine grained complexity if ETH is false? So there will be a solution for the Clique problem? So our social networks will evolve differently after this?
Interesting that the physics result is not released yet. From the limited wordings, “unified completion of the Standard Model of particle physics” is probably SU(5), which is experimentally disproved as one describing our universe. So the wording sounds a bit confusing, ie they are not proving or discovering Grand Unified Theory (GUT), and there’s no Higgs. So it is not really the standard model, but models that’s closely related to that. It is still an impressive result. IMO at the level of the recent quasi Riemann hypothesis level result, shy of getting the millennium prize (no mass gap) but is actually a very meaningful result, both of which arguably more impactful that the one they solved a millennium prize problem (Navier–Stokes).
Capital allocation to advance mathematics and sciences is undoubtedly a good thing. But this does beget a few questions:
1. How much did it cost?
2. Would an equivalent allocation to incentivize human effort have borne the same results?
3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
#2 is particularly important to answer, given research grants have been getting cut elsewhere: see U.C.L.A's research grant cuts from last year and Terrence Tao's attempt to raise funds (about $60k) by collaborating with Open AI earlier this year [1].
As for #2, personally I feel like OpenAI's results are superhuman at this point. I don't think there's enough human talent alive right now to produce these sorts of results within a few years, no matter the budget.
There are lots of reasons I believe this. One is that frontier AIs are very clearly superhuman on shorter problems (as can be seen on benchmarks like FrontierMath, or just by playing with them yourself if you have a source for difficult problems)--so these achievements are "just" scaling that advantage to research-sized problems. Second, AI is also clearly superhuman in its knowledge across fields, so it can make connections and combine techniques that humans won't be able to access. Third, these problems are famous and have withstood efforts of many of the best humans, in many cases being considered unapproachable.
Altogether it seems like AI is not only smarter but also more cost efficient that humans for math-adjacemt problem solving. I'm not saying that AI is superior at everything in math, eg theory building, but I think it's really unlikely you could get any group of humans that could compete with OpenAI's latest results.
It depends on the definition of superhuman. For example my car is superhuman as it allows me to get in one hour to a place where it would take me ten hours to go by walking, and can I walk ten hours non stop? However it is not superhuman enough to carry me where I could not walk too. And there are places where I can walk to but I cannot reach by car. Maybe that applies to AIs as well. A plane is in the same category of superhumanity of cars, with more speed. But I could still get to Australia by walking and rowing with my hands on a log (and probably die, I know.)
However there are truly superhuman vehicles, spaceships and submarines, that can carry humans where no human could go alive.
I don't know if these AIs are superhuman in the way of planes or in the way of spaceships. I also don't know if they can do everything we can do by talking or if they have some limits.
I mean, a plane can fly but not reach a branch and stay still on it. A small drone, almost. My coding agents sometimes can't do the obvious but is it because of some intrinsic limitation or because my instructions are bad? So far my experience is that they are superhuman like cars or trains, not yet as planes.
>2. Would an equivalent allocation to incentivize human effort have borne the same results?
Unlike their initial massive brute-force run that solved the Navier–Stokes Millennium Prize Problem, most of their subsequent breakthrough results in mathematics have come from a single prompt to a single AI agent, using approximately 3 hours of ChatGPT Pro-equivalent thinking compute per result on average.
> 3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
I think the scarier alternative may be OpenAI forever gating their super-frontier models, capturing a significant portion of the sciences and leveraging it for equity in businesses that build on scientific advances. e.g. "Hey CATL, how about you guve us 5% equity in exchange for a battery chemistry breakthrough our AI has found?".
It's roughly equivalent to the playbook Palantir had for the public sector when they marketed themselves as a company that had cracked data science. At least that's what they've been pitching back then. I can't find a lot on public disclosed deals based on that.
I think you are interpreting 2 as offering some large amount of money as a prize for discovering a solution. But I think it was meant more as funding a team of mathematicians to work on the problem.
3. no, but models will be better, it'll be cheaper to do
It's not a snark. The scale of VC philanthropy is unprecedented and it fuels itself. All of it will eventually, one way or another, become open. Long story short – there is nothing to complain about.
In 2025, Hong Wang won the Fields Medal for proving the Kakeya conjecture in three dimensions. Here, OpenAI is claiming to have solved a stronger problem that proves the Kakeya conjecture in all dimensions!
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
To be honest this doesn't feel as such impressive as the others. It was recently proven in 3 dimensions, so it could be "just" an extension of that. I haven't read the paper anyways so this is pure spectulation.
The question for some is, is whether the original authors would have produced their work without building off from the work of other mathematicians.
(which is somewhat valid).
Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
IMO that there is any work at all to build from is the result of a long tradition of taking the next step. Professor to student and so on. The danger of AI here is we get to take a leap, say, 10 steps - but then there is no more tradition to have the fortitude and mentoring necessary to continue the progress.
It all gets derailed while betting on RSI and never having to do math research again (but if an advanced theorem is proven, and no one alive is competent enough to comprehend it, does it even matter that we get to call it solved?)
I think the question is rather, what will happen to human beings when any possible personal intellectual accomplishment will no longer be seen as interesting or relevant by other human beings when AI can do most of it?
It's all well and good to say that we should just go on these journeys for our own enrichment, but even so they will just be doing it alone...or else not do it at all because they need a shared human effort surrounding them to do it, and therefore end up depressed and immersed in drugs or media all day.
Of course we're awed at what AI can do now, because all of us observing it have had the luxury of having such opportunities before AI. I got my PhD before AI and I found satisfaction in participating in a shared math community, which I wouldn't have done if I were starting my PhD now, because I would find the current academic environment unacceptable and useless.
That would be a very exciting experiment. I doubt you could find enough vintage training data to train a LLM enough to do it without it being contaminated by anachronistic data in the weights, but I would be absolutely amazed if it could reproduce that.
whether the original authors would have produced their work without building off from the work of other mathematicians.
That's rhetorical, right? Of course mathematicians built on what older mathematicians built and OpenAI's machine built on that. The real question whether OpenAI's machine has contributed something others will in turn build on.
I mean, the models (starting with Bel clearly and Astra probably--though Astra probably is worse than top humans in some areas) are superhuman at math.
I understand that the Exponential Time Hypothesis being false doesn't necessarily mean P=NP, but for people who are a lot more knowledgeable about complexity theory, does this update your priors significantly in the direction of P=NP?
It's been a long time since I seriously thought about complexity theory, but I'd say no. For similar reasons to why complexity class E is not closed under polynomial-time reductions.
The most important component in the time bound is of the form 2^{n / polylog(n)}, which is subexponential. But if you have reduction from another problem that increases instance size quadratically, the algorithm for the other problem is no longer subexponential. Which means we are still essentially in the same neighborhood as before.
It's a huge result if true, and probably the first major AI result that affects the papers I read. But since the boundary in the hypothesis was kind of arbitrary, breaching it should not significantly change our intuition on complexity classes.
I think the bigger issue (for someone hoping for P=NP) is that there's a huge gap between "polynomial" and "exponential." Even if it were the case that all NP problem classes had subexponential decision algorithms, that would not get us close to P=NP.
How much effort has been expended on optimizing those reductions though? I have a harder time believing that one can hang one's belief in P!=NP on that particular hook.
The challenge is the initial reduction from an arbitrary problem in NP to the problem you want to prove NP-complete.
In the usual approach, you assume a nondeterministic Turing machine (or equivalent) that solves the problem and a time bound (which can be an arbitrarily large polynomial). You then create a boolean expression that can be satisfied if and only if the Turing machine accepts the original problem instance. Unfortunately all constructions so far have included the time bound as a factor of the size of the expression.
I wonder if there are proofs regarding what neural networks like the transformer can solve. Because fundamentally, Astra is just a bunch of matrix multiplications and a few well defined non-linear functions.
Anyways, most of the times, SAT can be solved quickly, but not always, so getting the answer from Astra in polynomial time doesn't prove much unless you can also prove that it always does, no matter the input.
Iirc, there's a big difference between the computational complexity of what a transformer can solve and what it can solve in practice; what it can solve in practice tends to come from things like singular learning theory (which describes the forms of generalizations that a given ML algorithm can make), but I think (iirc) that transformers can compute a pretty broad class of O(N^2) functions?
Imagine you're trying to prove that water is wet (P!=NP). You lack a formalism that could make sense at building a proof even though pretty much everyone agrees from empirical evidence that the water is indeed wet. A promising attempt at building one such formalism was just demonstrated by a model to be a dead end.
I like this, but the actual finding is a bit different. Trying to keep the water is wet makes this awkward, but I believe it's more like:
Everyone agrees that water is wet (P!=NP), but no one can prove it. In the meantime, a whole field of computer science was built on a stronger belief (ETH): that water, at its worst, soaks. Turns out, even at its worst, water only makes things damp. It's still wet. That said, this finding did produce a new measurement of wetness.
No, it is famously a pointless internet debate with no settled answer. There is no scientific definition of what can or can't be wet, saying that wetness only applies to surfaces is just an opinion. It's an entirely linguistic question, and most people would say water is wet. If you want more than descriptivism, the common definition is something like 'containing or being covered with liquid', and I think most people would agree that a mass of water contains water.
Water is not wet. Human lacks water sensor. All the sensors we have a temperature ones so we sense "wet" as a combination of temperature and visual information. Close your eyes, touch water of your skin temperature and you won't sense it. It's not wet.
I think it would have to if you're honest with yourself. Any proof of a novel implication of P=NP, such as !ETH, should increase your belief in the two sets' equality. Just as any novel implication of P≠NP should adjust your belief the other way.
My thesis was around showing implications of few different hardness conjectures. So grounded was ETH that I introduced it in a side note as Conjecture 0 after Conjectures 1 and 2. I don't think it's easy to express how substantial of a Bayesian update this is all around.
No. Scott Aaronson said that it changed it from 3% to 5% for him, which is a lot more generous than I would be (but he knows far more about the subject than I do).
> does this update your priors significantly in the direction of P=NP?
My answer is no ... and Aaronson's answer would also be no ... clearly he doesn't view a change from 3% to 5% to be significant ... his response downplayed the significance of !ETH in re NP?=P. So apparently you downvoted me for simply stating my own opinion while you completely misunderstood what it was an opinion of.
For me, this result is significant. It basically adds much more hope P is equal NP (that majority seems being against). I believe one day AI will explain a major flaw in modern mathematics, so a kind of correction in axioms will be suggested to prove Rieman's hypothesis (or a completely new apparatus)
That's not that how that works btw. The incompleteness is a property of the algorithm which does the proofs, not the theorems. If a theorem is true it has a proof and a program which proves it (printf("the proof verbatim")). But no computer program can successfully prove all theorems. The space of potential proofs is too big to search systematically, but you can still get lucky.
Either P=NP or not. Hoping one way or the other changes nothing. Nor does it change how efficient computing is, nor does it have any effect on cryptography. Discovering a vastly more efficient means of factoring primes--perhaps via quantum computing--would do that. But just learning that P=NP does nothing to further such a discovery. Almost certainly, if P=NP, the exponents and/or constants in a polynomial factoring algorithm would be galactic. (FWIW, Knuth believes that P=NP and that it has no practical consequences.)
> Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026, which led us to develop the collaboration described here
Yes, people did. Well at least I did. We knew we can use QCD to compute loops and to some degree masses in the lattice. But there were serious doubts the underlying mathematical theory was sound.
For many it didn't even matter. It would just mean that at those energies the theory is not even valid.
I am not certain on the extend of this proof but QED is probably not sound on it's own and we have been using that even longer. lambda phi^4 behind symmetry breaking and the higgs is strongly believed to be unsound on its own.
Yes, but it has also been known for 50+ years that QED has a Landau pole and might be UV trivial or inconsistent, whereas QCD has asymptotic freedom and is thus controlled at UV.
More to the point, QCD has been experimentally tested both at UV (deep inelastic scattering) and at IR, while the physical meaning of testing QED at UV Landau pole regime is unclear.
Think openai should start including visual and interactive explainers in these where the viewer can toggle level to get a more accessible/technical explanation as needed. Would also be a great demo, and perhaps increasingly essential for us to know that explainability is not lagging behind provability.
I think at this point openAI does not really understand the results themself and any easy to understand visualisation would imply understanding. I would rather wait for the work group to produce some explainers or at least approve visualisation
Lots of progress recently in formally verifiable domains. Really curious if this is a product of the machine learning folks behind most large language models being much more likely to have a background in math, physics, etc. Or if it is that large language models have some fundamental limitations in the more "scruffy" sciences like biology.
My guess is that in formally verifiable domains you can use RL-esque processes to get lots of good high-quality training data with reward signals (does it verify?), but the same does not apply to "scruffier" work. I would also imagine that the data volume of high-quality scientific work in "scruffy" domains is too low for the large language models to have a good manifold for "high-quality biology research". Would love to be proven wrong if anyone knows of anyone doing great scientific work with large language models in non-verifiable domains though.
Many of the harnesses used to search for solutions rely on quick/cheap correctness checks to discard incorrect hypotheses. With something like biology, you might need to run a wet lab experiment to reject an idea. This can add days per iteration, slowing progress down by a factor of 100 or more.
The sad reality is that many computational approximations for biological phenomenon are relatively weak and lack the predictive power we need.
Is there a path where compute gets poured into developing, and then eventually using, computational models that supersede the shortcoming of the existing computational approximations you mention, which thus makes computational analysis and developments possible at rapid scale?
In an area I am more familiar with though still not my domain: when the stiffness matrix and computational linear algebra methods collide, it unlocks a world of computational structural design and analysis by commoditizing the underlying mathematics. This is a simple example, but you get the idea.
Obviously with CFD, mechanics, etc there is always massive work being done to try to unlock new computational methods, faster/parallelizable computational methods, differentiable ones, etc etc, but the general feeling is that the core physics are well modeled (NS millennium problem not withstanding) and are not lacking in predictive power for useful applications. I would have assumed that the same would be generally true for computational biology etc, so just curious to hear you elaborate more. It seems plausible to me that turning the brute force of “the models” (as in, swarms of AI compute) to work on/develop/solve/address whatever representational problems there are in the computational approximations/numerical methods you mention, particularly if that is a bottleneck to performing more extensive methodological/experimental exploration in silico to enable better exploitation/use of limited resources in wet lab.
Again, I know nothing about computational biology, just stating my intuition and curious what your take is. Probably the limits of my knowledge can be demonstrated by the fact that the only modeling methods that come to mind as being vaguely relevant would be, like,… molecular dynamics, maybe discrete element method in a weird way but doubtful, and various pop science understandings of protein folding etc.
Computational biochemistry is what I'm currently pursuing, but I'm pretty new to the field so take everything I say with a grain of salt: Biochemistry is a really really really tricky problem, for a couple reasons:
1. Biology has a very nasty habit of not cleanly abstracting. It's not like we can treat every protein as a "diffuse mass" without losing a ton of important interactions. This is unlike something like CFD where you can treat a mass of water molecules with certain constants. Think of something like cancer: a small mutation to a DNA strand can cascade into a tumor. You have to deal with the mutation happening at an atomic level while scaling to a whole organism (yes I know there's models that can approximate this, but they can't capture all the interactions).
2. This brings us to modelling at the atomic level. To model a system of atoms with the Schrödinger equation, it naively scales O(6^n) per particle. Note that means every nucleus and electron, so if you have say 10 carbon atoms and 22 hydrogen you're already at 10^127 operations. We can cut corners a bit and end up with something like CCSD which scales O(n^7), or cut more corners and scale O(n^3) for DFT. DFT is pretty much the last method where bond forming and breaking is modelled, so after more corner cutting you lose losing reactions, which are important for understanding why something is happening. Most protein folding simulations (moreso prior to alphafold) run a very crude approximation that models atoms as balls on springs (things like AMBER and CHARMM), which obviously can't handle reactions.
3. If this wasn't bad enough, you also can only move through time at about 1 femtosecond (10^-15), so it takes a ton of simulation steps to get even a nanosecond in. Forget millisecond or second lengths where things start to happen.
So, while the math of the physics exists, computationally it's practically intractible. There's some exciting work with something called Machine Learning Interatomic Potentials (MLIP) which approximate DFT using a neural network which are about 1000x faster than DFT, and iirc scale much more linearly (look at things like MACE and orbmol if you're curious). So we're making progress, but we've gone from so ridiculously impossible to pretty much impossible for large systems.
I have a couple ideas about developing custom microchips to make this faster, which is why I'm pursuing this, but there's still a massive way to go!
I suspect the line forward is generating improved predictive models and implementations of those. It’s something LLMs can aide with, although they’ll have slower progress than domains that do already have robust, cheap validation checks. That’s probably why those domains are being targeted (they are the paths of least resistance to successes, which is why they’re being pursued).
With that said part of the reason some domains may not have robust cheap verifiers is because they’re difficult to produce already, so it may be a bigger lift.
The bottleneck in biology isn't thinking, it's experimenting. AI could potentially operate a lab but the test process is a lot more expensive than math is.
Are there efforts into building "autonomous laboratories"? Not sure if the idea is practical but it seems like a direction things might go in, where you give an advanced model, or a swarm with a leader, free reign to run experiments day and night.
It's because LLMs provide an efficient way of optimizing a complex, non-linear search space when there's a way to measure progress. The less verifiable something becomes, the harder it is to optimize (but not impossible).
Super cool research! I think this is still a formally verifiable domain but different than the kind of formal verification I was thinking. Maybe if LLMs continue to be great at this human science turns into making “fuzzy domains” much more formally verifiable (although I don’t think this is possible in some areas).
Obviously it's easy to understand why having strong abilities in maths and coding can help with solving problems in physics. But it's also true of knowledge and mental models more broadly.
For example, say somehow you managed to train a frontier model without any data about elephants. If you did this it would be relatively easy to later train elephants into the weights. You'd basically just need to train it about elephants at a very superficial level. After all, it it already knows what a mammal is. It knows what legs are. It knows what eyes are. It understands how it's digestive system works. All the things that makes the biology of elephants difficult doesn't need to be learnt from scratch.
I assure you people on HN will be surprised at how quickly frontier models will become competent in domains that are harder to RL. They won't need anywhere near as much data as people currently believe.
>We give a deterministic algorithm for exactly counting the satisfying assignments of
an arbitrary k-CNF formula [in superpolynomial but subexponential time]
Polynomial is something like exp(n log x), where x is the problem size. Exponential is something like exp(x). Superpolynomial but subexponential is roughly when the exponent is between n log x and x. An exponent of (log x)^2 would do the trick.
I glance at the HN headlines but rarely have time to dig into the details (and the few times I read 30 mins or more, a lot goes over my head).
Are there any good accounts to follow (platform agnostic) that give occasional updates on the intersection of AI and math that aren't too dumbed down, i.e. that a layperson can appreciate?
This is so far above my head it isn't even funny. I just looked up ETH and was 9 links in before I realized even the explanations of the parts of it were out of my grasp. Can someone explain the implications of this?
It depends on your background. If you don't have a computer science background the statement needs layers and layers of buildup that can't happen in a single hacknews comment. But the outside implication is still accessible.
The outside implication for what is going on is that OpenAI has very clearly demonstrated insane capability in coming up with clever algorithms. You cannot deny some real level of super human like abstract reasoning here.
K-sat is the hard theoretical computer science problem. Nothing else is close in terms of how much it has been studied. There are journals and conferences for SAT.
It is really hard to come up with faster ways to solve it. So much so that people were convinced that you really can't go faster than exponential time. OpenAI has just shown a breakthrough construction better then decades of research by elite specialists was able to come up with.
They do that to make it easier for readers to verify the statement (which is entirely contained in that file) separately from the proof (which is spread over thousands of files and github can't even show them all).
Thurston wrote about this in “On Proof and Progress in Mathematics,” including how his own approach changed from proving theorems to helping a community understand the mathematics.
Nicolas Le Roux, a Lagrange Prize winner / prof / strong mathematician, turned me onto that around the time I finished my PhD in ML last year. I have been thinking about it a lot, lately, for obvious reasons.
At first, it felt silly to grapple with what science is only after finishing a PhD. This publication from OpenAI finally pushed me over the line to publishing an essay about this struggle between AI science and human science: https://news.ycombinator.com/item?id=50253759
I am not sure if I side with Tao on this, or 3B1B. It's already causing quite a kerfuffle (to say the least) in the academic communities. This is the new norm now.
Well, this is certainly different, can't say they ignored the criticism. Wonder if this is a monkey paw curls moment; if the findings continue to accelerate, most of it just isn't public now.
I'll give you an example from my field (nuclear fusion). I have a small team working on an idea which I think would really advance the field. Our initial concept paper was published in 2019. Progress on it has been slow because we don't have enough manufacturing capabilities, budget, or time on the facility. Will AI solve that? Don't get me wrong, I'm pretty bullish on AI, and it's been phenomenally useful so far (I definitely feel like a 10x physicist in the last 6 months), but I don't see a swarm of agents putting fusion energy on the grid tomorrow.
Also, fwiw I have a personal test problem I give to every frontier model on release, which is to ask it to final a realatively obvious solution (in hindsight) to a minorly important open problem in my field. I know the answer, but havent gotten around to writing it up yet (it's really not that important). Every frontier model (so far) just regurgitates existing solutions.
How much effort do you put into prompting it? Are you asking it one shot with a simple prompt? Are you giving it to multiple agents and giving them a workspace to collaborate in? You might give it to a trusted colleague who does not know the solution along with an agreement to jointly publish, ask them to go through the process of solving it with agentic support, and then compare your solutions.
Edit: to be clear, I completely believe your experience - just pointing out that it might be interesting for you to actually “try” to solve it thoroughly with the tools, though it is obviously hard to do that in a fair way when you know the solution.
Iirc some of the prior math discoveries involved very very very heavy handed guidance to force the AI to discard prior assumptions, and to try many avenues with MAX effort. So ya models don’t behave like ground breaking theoreticians by default.
Ha, I had a similar little test problem, but the frontier models are now smart enough to read enough online of what I posted here and there to piece together the answer. And someone else has written the paper in the meantime and got it accepted for publishing.
Definitely. It will let some smooth-talker with a vibe-coded website raise billions for "Nuclear Fusion AI". Then they can use AI to squeeze your team with lawsuits until you agree to sell cheap, and maybe keep you on as an employee to build it.
I mean, you never know. If we could have simulation models with excellent fidelity, then AI could do all the research and iterations and come up with something very plausible. Cutting down physical world experimental iterations/prototyping.
So effectively, you would spend the first 1000 experiments inside a computer simulation and move to the physical world only when you had enough confidence.
I've got frontier models doing pretty sophisticated nuclear reactor design and analysis using open-source nuclear fission reactor tools using open-source reactor designs as baselines. It is WILD how good they do.
The thing that I've not fully groked here (might be relevant) is that OAI is generating "paper" proofs first, and only then formalizes them in Lean as a separate step.
They've as far as I'm aware not said anywhere how many write ups failed formalization due to a bug in the proof. If we assume that number is low, models would be quite good at one-shotting math proofs. Then it depends if you can replicate the math RL training for chemistry/biology/medicine. Is there enough data out there for that? No idea.
I would trade (as if I have any say lol) every single breakthrough they have made in math/CS for _any_ good progress towards a universal cancer cure (very theoretical, I know).
I'm kinda hoping they are investing just as much into that type of research. Its their money but man... I'd say 999/1000 people are unaffected by all these breakthroughs. They probably have no idea tons of legendary math/CS conjectures were proven true/false. If a breakthrough in cancer research were to be made, it would very quickly start effecting everyone on earth.
The time required to verify is much higher, so some combination of vastly speeding that up or vastly increasing the quality of individual "thought steps".
It’s accelerating. We can automate scientific discovery. Imagine the amount of knowledge we may have over the next several years. It could be staggering.
How are people getting these results? Are they correct?
I use these latest available public models daily in max effort modes, and they barely seem to work and need extreme guidance and as much context and constraints as possible.
I find that unless the outcome I'm looking for from the model is almost binary and easily verifiable, it is basically faster to do the work myself.
Assumptions made by the models multiply and by the non-deterministic nature (dbf, similar to work by humans) I'll have something end up light years away from where it should be. During a normal workflow, I have to contend with every line I write forcing a very tight feedback loop... I suppose I could also sit there and spend 9 hours writing out a "perfect spec" but we've already tried that with waterfall development!
So, not sure if its the models, rather the class of problems I'm applying them to... but it is a bit ironic they are allegedly able to solve these conjectures but not stop to ask about assumptions.
Chasing open questions, it seems to me, is worthwhile for the things learned along the way. Do you expect these results will be helpful, or is it just a sign that we need a new way to navigate uncharted spaces now that longstanding problems don't stand long enough for us to organize around?
It feels a bit like having "visited" a place because you had a layover there.
I don't understand this dialogue, which seems to repeat with every result of this sort. Of course this is interesting! And of course the answer tells us something! And of course we also want to understand where it came from! None of this is in contradiction!
Knowing the answer to Fermat’s Last Theorem wasn’t valuable. But the pursuit of it led to many developments in other fields of mathematics that were valuable (where value is measured in terms of $).
I disagree. We climb mountains for the sake of it, and the rest of the world benefits because the gear improves and makes adjacent, non-for-the-sake-of-it activities safer.
We chase open problems for the sake of it, and others sometimes support us in doing so because the paths we pave along the way end up being useful for them for adjacent purposes. If you get the answer for free, you might have scratched the itch, but nothing was built for others to use, no fodder was found for the next great question.
Nobody pays mathematicians for solving famous problems. Famous problems are how mathematicians orient themselves while they do the other things that they're paid for (maintaining a useful community, mostly). All this does is deprive them of a navigational tool.
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Your formulation is a little bit exaggerated, but I (and many, though far from all, programmers) would agree with the following more "tame" formulation:
Getting software to solve a business problem in an elegant and maintainable way means that we got the abstractions right. Similarly, if a software can easily and elegantly be modified to cope with new business problems, it is also a strong sign that we got the abstractions right. In this sense, the business problems that the software solves are indeed just waypoints on the journey of finding the proper elegant abstractions for putting the business problems into code.
Turning these novel insights about "good" abstractions into code and programming languages enables the team to iterate faster and generate business value faster. So, appreciating that it indeed all about the journey means a commitment towards an approach to accelerate the rate with which the team can turn business insights into code and code into business value.
I know that many managers do hate such a formulation, but I stand behind it.
Often a problem is important because we have no idea how to solve it, which means we need to develop new machinery to solve it. That new machinery can then be used to do things we couldn't do before. Stimulating the development of that machinery is the most important part of the problem.
You should remember that mathematicians and programmers working for a company have different jobs with very different goals.
This is such a short sighted take. Number theory was considered beautifully useless for centuries, and now it’s the backbone of the cryptography that secures every bank transaction. We don't know where the pursuit of knowledge will take us.
Thinking in such a utilitarian way feels like a flavor of Meditations on Moloch.[1] Just look at how short-term, quarter-to-quarter thinking is affecting the economy and the lived experience of every human on earth. It’s horrible.
Knowing the answer is valuable, but it's not everything. We also want to understand why. The four-color theorem has a computer proof, so we know it's true, but we learned much less from that than we do from a human-understandable proof.
Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.
If you were a math grad student, would you rather your advisor be telling you to LLM maxx your ideas or working with you and giving you ideas on your thesis personally? I know my answer.
> If knowing the answer is not valuable then asking the question wasn’t valuable to begin with.
Let's say it's possible to know an alien civilization exists 100 million light years away from their waste heat. But that's all we know and it's unlikely we'll ever interact with them.
Was the question of whether we're alone still "valuable"? Wouldn't a lot of philosophy and even some physics fail to meet your definition of valuable?
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Imagine the next version of Claude Code or Codex no longer hands you source code and instead turns a natural language prompt directly into a compiled binary that passes the test suite. Would this programmer's boss ship the binary straight to production and dispense with the company's git repo altogether? After all, the goal at the end of the day is shipped software that produces value, not human-readable source code.
Regarding Yang-Mills, as of a few weeks ago there has been some expectation that this (specifically completing Balaban's program for the existence part without the mass gap) could happen, though from the Anthropic side: https://nitter.meowing.monster/ElliotGlazer/status/211115394...
(Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)
There's a large logical leap from "we have very useful raw research data" to "we should give this data only to a small group of people." They explain that "the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community" but we are left to wonder why the experts need to be empowered to foster anything. In the case of immediate release the experts will have their hands on the same data as the rest of us peasants, and then they can go right on fostering.
Not enough chatter about the QCD result here! I'm of the opinion that whatever the correct representation of the universe is, the GUT (Grand Unified Theory) will be authored not by humans but by LLMs.
I wonder why the physicists are the only group to not want the full information released upfront? As a casual fan of physics[0] it seems like their completion of the standard model is possibly the biggest deal of all these breakthroughs, am I wrong?
The writing is a bit ambiguous, but if I'm parsing it correctly it's only a big deal to the small subset of physicists who care about mathematical rigor: proof that Quantum Field Theory (i.e. what theorists have been using since the 1940s or so) is mathematically sound.
This part,
The existence proof pertains to the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model
As a slightly less but still very casual enjoyer too, I don't think there is a huge amount of value in dumping the lean proof or the natural language proof. I'd rather have an abstract of a paper that I can at least hope to understand 75% of before getting lost.
I'm guessing if you were really interested, you could email YANGMILLS4D@gmail.com (itself a hilarious e-mail address).
Careful, completion of the standard model ≠ theory of quantum gravity or theory of everything. The actual question is too abstruse for me (biophysicist).
These resolve 6 entries in my top 500 open problems list (LLM ranked, https://www.proofatlas.ai/open-problems/) and there is partial progress on #3 overall (Yang–Mills Existence and Mass Gap):
> In our view, these manuscripts are best understood as a form of raw research data.
Much better!
> For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.
> All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.
Very good!
I was highly critical of their approach with the earlier mathematical publications, but this is much improved.
Most importantly, they aren't releasing slop papers that would never have been published in any self-respecting journal or made it past a deep round of peer review questioning.
> We are committed to ensuring funding for researchers to independently examine, explain, and build on the results above.
> We continue to invite feedback on the best way to disclose results and engage with scientists to accelerate discovery.
Well done!
Now you need to actually LISTEN to the community and the feedback, so you aren't done yet, but this is a much better start than last time. Keep it up!
Woah, they got QCD! I didn't think they could do that. Three things jump to my mind:
1. Is it really true? Did they get the basic conditions right?
2. Would I ever be able to understand the proof
3. But the most important one. Can we compute hadronic masses now?
This is an incredible scientific achievement. One of our most fundamental theories is mathematically sound. Without the use of AI is very likely I would have never seen this in my lifetime
I'm still shelled shocked, but my initial reaction is caution. Wait what did they claim to have proof exactly?
I can't see their pdf in natural language either.
This is an exciting time, but also there are very real changes at hand. Game changing discoveries will likely not happen at universities but at private AI frontier labs. Moreover, the models that are used on this are not always open to the broader community today.
What will it be like in two years? Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies? Still thinking about the ramifications ... amazing and discombobulating at the same time.
I disagree with that. Commercial entities are very different from Universities. They are sharing now, but they don't have to continue to share in the future, that all depends on what they find.
> Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies?
The 3 breakthroughs all have implications for cryptography and nuclear technology, right? Hard to imagine nation-states allowing private companies to publish these results willy-nilly for much longer.
It uses sparseness. I noticed a simple thing way back that for a n-boolean variables, the average size of a satisfiable boolean expression must be 2^n bits - basically every binary number that satisfies it listed out. So if a boolean expression of 4 variables is satisfied by 1111, 1110, and 0000, the number denoting it is 1100000000000001, and for every such number there is a boolean expression of 4 variables and for every expression one number of 16 digits. So it’s a fool’s errand to make a universal solver when average inputs scale at 2^n, if a solution exists it has to leverage the small size of the problems we actually care about, sparsity. Based on the AI description that’s what they did, something with polynomials, but it’s too soon. Looking forward to finding out.
I wonder how much compute they are throwing at problems in ML like continual learning and sample efficiency. Also how successful they are with these. Results like in the post make me hopeful we'll see some type of so singularity soon
Yes. Up until now, it was unknown whether containment of NP in EXPTIME was strict or not. (Obviously you can brute-force all possible witnesses, so any problem in NP can be solved in EXPTIME, it was open whether you could do better than that.)
Showing 3SAT (technically #3SAT) can be solved in less than EXPTIME resolves this. It's only slightly subexponential, not even quasipolynomial, but it is a very surprising result. And that too a constructive algorithm.
this is the first time one of the big labs has published anything related to my field. and i'm actually pretty excited? im more on the applications side of QFD than the theory side of things, but gosh i actually cant wait for the results to be released to see if i can use it for anything! i kinda hope more results come out, in more ways relevant to what i do, its so cool to see any sort of progress! wow!
I thought it was interesting that in the original drop of papers, they announced much more progress on most other Millenium problems than on Yang-Mills existence. At the time I thought this might mean that it was a more stubborn problem, but it turns out that they had made a lot progress after all.
The Yang-Mills Millenium Problem has two parts though. The second part is to show that Yang-Mills theories have a mass gap, meaning that the lightest particle has non-zero mass. This is important because experimentally we know that there is a mass gap, so the theory must have one too to in order to actually be the correct model of nature. I wonder if they again have unannounced progress here.
I'm a little surprised the article stresses Lean programs so much, without ever stating what that is or why it somehow proves something. Maybe that's expected knowledge for this sort of thing.
It is because people take lean proofs at face value and you don't have to actually explain it. I have some math problems I am working on and while I had code showing it does the right stuff, the prose was taking a lot longer, but now I realize I can just mass convert it to lean and put it out there.
I do not like this at all. At best, we find that there are classes of problems that even AI cannot help much in solving, in the sense that the rationals are an infinitesimally small part of the reals. At worst, a fundamental result trivializes a problem whose complexity is a key component of a critical system, and there’s societal upheaval. If PhDs ran the government, and thought the latter, we might have already seen military attacks on data centers.
The QCD and Yang-Mills results don't seem to go beyond what the experts know/expect. Basically, they're saying there is a path integral measure, but shrugging when asked what functions are integrable with respect to it. The latter question is key for mass gap and confinement, and existence of the measure only means that the question of whether a mass gap exists makes sense.
I'm incredibly skeptical about everything these frontier labs put out. Results are results though.
However, could someone with a firmer grasp on the problems weigh in on how much are these recent results are function of cutting a virtually unlimited budget for some very smart people? I believe the LLMs certainly help iterate on ideas faster... but I am aware of very impressive work that goes on in quantitative finance. Stuff that would not really be published. That stuff is the result of paying smart people millions to produce results which produce $$$.
I have to wonder, would this have been solved if we simply gathered tons of smart people and paid them a ton of money to solve it? Or is it exclusively because they had access to LLMs?
Could they have achieved this result with previous versions of ChatGPT?
Quant finance is very different from math research; quant is much more applied math and programming, and the tasks are typically smaller than math research questions. Although it's true there's a huge overlap in the people good at both, there is massive re-training needed if you want to turn non-math-phd quants into research mathematicians.
As I argued in another comment here, I think even with unlimited budget, humans wouldn't be able to reproduce these results within a few years. I think AI has crossed a superhuman theshhold in math problem solving skills.
> We do not seek academic credit for the discoveries produced by our internal models.
Sure. But that isn't what they really cared about anyway. They care about notoriety among the general population, and especially among people who decide to pay for their products.
The fact that ETH is false when many people expected it to be true is the amazing thing. This isn't AI finishing the last 10% of a proof and stealing the credit--this is AI exceeding human thinking. If nothing else, this buries the "stochastic parrot" position.
That said, AI is spiky, and I fully expect there to be a class of problems that AI is really good at solving and a (maybe larger) class where AI is worse than human mathematicians.
> In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach
Mathematics and computer science are turning into an empirical science
If all you care about is whether the statement is true, then yes.
But what they're getting at is that we also want to gain an insight of "what is really going on", and it will take time to distill this proof into a theory that gives us a more complete understanding.
I'm going to be the cynical one and offer this translation: "Our results are too important to wait for the pesky peer review, so we are dumping a whole ton of money in a fund and giving some elites a breathless first look so our buzz keeps getting jacked up."
The reality that Science is a grind and there are thousands of new results every year discovered by regular humans who use a regular human process to cross-check and review.
Treating every result from the AI like a gift from god is getting tiring.
These are not run of the mill results from AI, those are surely being posted to arxiv constantly by normal researchers without any fanfare.
These are really big results from AI, if a human had come up with them we would also be making a really big deal out of them too (maybe not as big of a deal, part of the excitement is that this was from AI and more results of this magnitude may soon follow).
Regular human processes, it turns out, have been a bottleneck that AI is now removing. AI is now routinely popping problems that have routinely stumped humans for decades or centuries.
Honestly, I’m a bit scared but also a bit excited.
I’m currently reading Africa by John Reader which covers a lot of humanity storyline. From HN I get the humanity achievements that are happening right now.
The feeling digesting both daily is in order of magnitudes bigger than what any really exciting tv show did to me.
I honestly at this point don’t know anything about what could be possible and what couldnt.
I understand that this is really freightening, but also especially ETH being false gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits. I know I’m the minority, but these answers really interest me.
Nobody is solving colonization of planets in this lifetime unless we find a way to accelerate mass faster than the speed of light (and decelerate it at the other end) or extend the lifespan/radiation resistance or can preserve bodies in suspended animation for long periods. Even then, that just removed one of the many technical hurdles. We don't know which planets to target- are you going to send people to solar systems when you don't know if they are habitable? Modify people to be able to inhabit otherwise hostile worlds? How do you deal with boredom on a 1,000+ year trip?
Even with aging, our efforts are better spent increasing quality of life across humanity than helping a few rich people live to 150.
> Nobody is solving colonization of planets in this lifetime
Eh, speak for yourself.
I'm not sure how anyone can say "nobody is doing x.." or "we'll never be able to do y.." after the advances of the last few years. Not sure you understand the implications of what we're seeing here and what RSI/ASI will bring about.
Even just imagining the technological acceleration we'll experience when open source models at the Opus level can be run without insane hardware (I guess in a year or so) or if we just stopped development of AI today - is still insane.
If there's a will, there's a way. There are engineering solutions to every problem you mentioned.
"Sufficiently advanced technology is indistinguishable from magic."
I've worked in ML since 1994. I've been very impressed with the progress of models lately. I read "The Coming Technological Singularity" when it was first written and dedicated my career to seeing these sorts of things coming true.
That doesn't mean that their exponential improvement will change faster than light travel. Therefore, to deliver a human being to the surface of another planet (let's say Proxima Centauri at ~4 ly away) with enough resources to create a stable colony means we'll travel at less than the speed of light. Even with a solar assist, the best we can do right now is less than 1% the speed of light in an unmanned probe. People are working on alternatives, but those are not the sort of technology you just pop in a space ship and send for 400 years to a neighboring star. How do you support a population for 400 years?
As an alternative, let's just make humans like tardigrades, capable of shrivelling up and surviving high-radiation, low-moisture environments. Then we can take 10K years to get to a remote location. As a bonus, the resulting humans can survive in a much wider range of conditions so we have many more target systems to consider. Oh, but that's human engineering- are you sure that we'll just be able to genetically modify people (technologically and morally) at that level any time soon? A single person died in an early gene therapy trial and it shut down the field for 10+ years.
Basically, what you're arguing is that exponential growth in AI (not a given) will unlock huge material sciences and biological advances. While possible, it's unlikely to combine in a way that suggests "we should send people to other solar systems".
How about sending robots to a habitable planet with biological material of humans, programmed to tell us the story of earth?
Re life quality: Sure technical advancement can lead to cheaper electricity, better materials discovered, disease prevention and a lot of other stuff. A lot is hold back by complexity.
I don’t want to make a call that this will happen and that it’s a net-positive for society. I’m just saying the technical advancements is interesting and enablin.
biological material of humans? you'll have to expand your argument here. Are you reconstituting the material on the other end? If you have that technology, do you need a robot to tell a human the story of earth? Why not just upload it directly?
> gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits.
There are infinitely many more ways to kill and torture a human than to save one.
I looked over the announcement for any substantial contribution, and found they're donating $5 million to a research foundation.
$5 million might sustain 5-10 PhD students for 5 years.
Maybe it's just me but I think they should contribute more in exchange for forcing the scientific community into paying for tokens to generate slop and then pouring their time into verifying or falsifying the slop.
I said the same thing in another thread and someone compared it to no one forcing you to own a smart phone. I will preempt that argument by saying the community is motivated to verify these results because they are important to the field.
"In our view, these manuscripts are best understood as a form of raw research data."
They are staying on top of the exact formulations that spread like memes. I think this one was unfortunate. It's an analogy for a new type of thing that stops attempts to figure out what thing that is.
That said, uh, let's go? Let's see if OpenAI earns some goodwill here. It probably can't go worse than the last math drop!
It is raw research data though. Human math is way more refined. LLMs are superhuman at deduction, yet cannot create their own pre-training (where most learning happens) data.
Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.
ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.
QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.
Is there any obvious path from these results to any sort of physics, chemistry, or biology breakthrough? From what I can tell: no (but I lack the physics and chemistry expertise to say anything about the QCD finding).
Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?
From the physics, not without more breakthroughs, and even then I can't imagine what that would look like. The particle physics that you can use industrially has long been understood, and the field has long been long stuck because the only things we don't understand are also insanely expensive to experimentally access. Maybe once we move up the Kardashev scale, though.
ETH sounds more like there's a path it could become important? In general, the recent complexity results suggest that a) our intuitions about that world are unreliable, and b) the truth seems to be insanely complicated. I mean, new faster multiplication algorithms in 2026, what the hell? How can we say anything about the full space of all possible algorithms?
I wonder if the best statement we can make for future AI discoveries is that, ideally, they will allow us to move up the Kardashev scale faster.
The new faster multiplication algorithms are both galactic, are they not?
It does not seem surprising to me at all that AI can find new galactic algorithms that are not of any direct use (at best they can guide us to find better solutions that are non-galactic).
What would be surprising and interesting is if AI can find a new floating point multiply-add that is 2X faster per operation than the current fastest chip implementations. That would immediately have billion+ dollar implications. The teams who already work on this have excellent approaches and invest enormous amounts into tiny improvements already; there are probably not much low-hanging fruit there (proponents of using AI for chip design improvements have impressive claims, but unclear that any of those really worked out in reality).
Glad to help (I only really understand the implications of ETH well enough to summarize myself without assistance haha). I figured if I wanted simplified context, others might too.
Title is no doubt crafted to get ahead of the usual online backlash, but here’s the actual announcement:
> We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
1. Local smoothing and the Kakeya conjecture
2. The Exponential Time Hypothesis is false
3. Existence of quantum chromodynamics and completions of the Standard Model
In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."
(Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)
We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies:
[...]
For the physics result, form an external group of collaborators and defer to them on the release of materials.
[...]
The group requests that all communications with any and all members of the group regarding this effort be directed to the following email account: YANGMILLS4D@gmail.com.
>>> We reached out to external experts and asked them to act as shepherds
Sure, if you meet some resistance to your method of undermining a field, the best go-around is to co-opt the top professionals to help pull the ladder up. Astonishing, the arrogance of whoever didn't bother with this before.
I think OpenAI deserve credit for adjusting their approach after the last math drop. I think Terrance Tao shares some credit here for galvanizing the response.
Including the relevant mathematicians and scientists before dumping this onto the public is sure to go a long way to relaxing fears in those communities. I'm not sure how they will deal with accusations of favoritism but at least they are demonstrating a willingness to participate in a process.
I hope the community can express some positivity that will encourage OpenAI and other foundation labs to continue with this collaborative approach.
Well said. Attribution is important, for obvious human reasons and to direct people to valuable context. But our names will fade fast from history, what matters is the journey of human knowledge.
> It is amazing what you can accomplish if you do not care who gets the credit.
--Harry S. Truman
> It is amazing what you can accomplish if you do not care who gets the credit. -- Me, 2026
Jokes aside, this is a thought provoking quote. I do think it's easy to say as a well known figurehead, when the quote and other actions below you may easily be attributed to you, at least partially. But I do agree somewhat, and I think the context around Truman's time (WWII) is a good example of what humanity can accomplish when the stakes at hand supersede the considerations of workers' rights or credit/ownership. Of course this is also a very favorable phrasing for a period of enormous human suffering. I think there is an arms race analogy to be drawn here, though the current situation and stakes are much different.
Funnily enough, there’s a good chance Truman never said those words; the earliest known attribution of the quote to him is from a Time magazine article written in 1988. It’s not in any of his personal writing or speeches.
Attribution is a fascinating topic.
For example, one generally don’t name cells in their body, and cells works generally pretty well to sustain a whole self.
On societal level, obviously no one is aware of who authored the biggest mass of benefit that is granted by the mere fact of living in society. No specific individual have credits for example regarding the language they use daily¹. For the most recent ones, it’s now often possible to attach some name to invention of some cultural/technological artifact, but of course Zeitgeist-wise perspective tend to consider it’s rarely a single punctual individual that should be considered the critical pivot point.
On the other hand, even Marx that was so critical of individual alienation couldn’t refrain to make the intellectual equivalent of pissing territory and attach a name referring to its own ego within all of his works. Ego is a masterpiece of illusion: it will bring the foundational leveraging to build every possible societal hegemony under the sun while convincing the deluded that serving the ego is the indispensable first step of everything to best serve every individual selfs.
¹ Even for constructed languages, such as Esperanto that I practice myself, whoever construct them obviously don’t start from cultural blank slate.
Yeah, and it’s amazing that we chose to build our society around maximizing the benefits of taking credit regardless.
(From [0])
'A man may do an immense deal of good, if he does not care who gets the credit for it.'
"QI hypothesizes that a Jesuit Priest named Father Strickland deserves primary credit for this family of sayings. This claim is based on the diary entry dated September 21, 1863 written by Sir Mountstuart E. Grant Duff and multiple attributions thereafter. However, similar maxims may have been developed independently."
[0] https://quoteinvestigator.com/2010/12/21/doing-good-selfless
> If you can't handle me at my worst, you don't deserve me at my best
--Albert Einstein (Director, Los Alamos Laboratory)
I don't understand. The AI-math guidance counsel has recently explicitly argued against what you say in paragraph two, and which is why for the Kakeya problem there was no group formed. The different communities have given different suggestions. I'm not sure how much I agree with the first paragraph either given these facts.
The approach hasn't changed all that much. It's more that the scientific community is slowly starting to accept the inevitable.
From the article:
> We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies
1. They identified relevant experts for each of the discoveries they wanted to release
2. They collaborated with them to determine how to release the results
The effect for the first two (maths, comp sci) was the same because the experts they chose made that decision.
The downside here is actually the third case (physics). Since they only chose 5 scientists (David Gross, Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov) and those scientists chose to keep the results private until they finish their own work. Everyone else who might be interested has to wait.
That was my favouritism aside. Scientists working on similar or adjacent problems that are not included in these kinds of private releases might feel they are not getting a fair chance.
This is a process that will change, but it seems they listened to the anger from the math community and made changes that help address the concerns.
I think, sadly, these fields will collapse further into a tighter in-group. Anyone outside that group will find it increasingly hard to penetrate. Talent won’t get you there anymore.
I'm happy to relax for six months, and ask Qwen-8-E2B on my Nintendo 3DS for the solution.
The hubris in the framing is astonishing, amusing and sad all at the same time. Many say, that's what HN attracts. Anyone who talks about inevitability except of death in these times is just someone who considers they'd be in the "have" rather the "have not" category.
I still can't see what this means for human understanding and how open would this knowledge be. Who'd ultimately get access to this stuff going forward and what kind of power games would ensue hereon.
I'm not sure why you think that someone who believes that this is inevitable is just someone who assumes they'd be in the "have" rather the "have not" category.
It seems like you are doing exactly what you accuse the parent commenter of doing. The only difference, perhaps, is that the commenter seems to extrapolate past progress to future progress, while your own assumption is a lot more tenuous.
It’s always weird when people are so quick to use a single positive thing to trivialize the many historically questionable things an entity has done up till this point. It certainly doesn’t come from a place of rationality.
What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
> What is weird is criticizing people who acknowledge good things because of past bad things when nothing anyone said here, that I can see, anything close to "they are redeemed rejoice!"
This is like going to a politician's event and seeing him giving away lollipops to toddlers and kissing babies. But it's the same politician caught in corruption and embezzlement scandals.
You look at that and praise how they have redeemed. Other people look at that and see a corrupt politician handing out candy.
People tell us who they are through their actions. If you prove yourself to be capable of morally reprehensible things to the point throughout the years you build a reputation for all the morally reprehensible things you do, does your potential to do morally reprehensible things change if you go way out of your way to let people know that sometimes something's you do aren't morally reprehensible?
Wrong, these companies have shown time and time again they are irredeemably misaligned with our interests.
Right now you and everyone else is doing the "leave the multi billion dollar company alone" version of the neckbeard "leave the lady be" meme. And you deserved to be chastised for it.
Its the equivalent of you going around telling everyone in the office about how your boss is turning it around because they bought donuts today, after they spent three years telling you you're worthless, they're entitled to taking all the credit for your work and they plan on firing you as soon as they can.
The history isn't particularly long.
This is going to be a very big deal in some of my circles, which include KITP (the Kavli Institute of Theoretical Physics), where David Gross was the Director for many years and now holds the Chancellor's Chair Professor in Theoretical Physics.
I’m not sure what meaningfully changed except for the QCD / yang mills result (where leaders in the field are going to do a review and write up first).
After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
Allegedly.
> After their effort to take credit for the research of others last time, it's within OpenAI's best interests to go the extra mile to convince skeptics to not dismiss new research findings.
This.
The truth if the matter is that this is still a publicity stunt. This investment of time, effort, and compute is aimed at building a reputation that this company's model is better than the competition's. The previous publicity stunt brew up in their face, and now they seem to be trying to adapt and double down.
I would have found it a lot more impressive if they just empowered scientists through some kind of grant rather than to internalize all this. It sets the tone for what's to come and inevitably if what they find is juicy enough it will not be published. For instance once they start working on materials science (if they aren't already).
Attribution is very important. Though AI hasn't really done anything yet so it hasn't affected actual mathematicians yet.
Yeah it’s just one Millenium problem, no big deal. I was gonna crack quasi-Reimann next year anyway, just hadn’t gotten around to it.
All AI is good for is remixing frontier mathematics, there’s no way it could ever make its own. Right?
I hope it gets us somewhere new. You can hope that and appreciate it hasn't yet though. (I'm not part of their IPO)
It's parasitic on humans and it's not an infinitely deep well. Wrong?
They don't necessarily deserve credit, it is just the bare minimum courteous behaviour in academia to reach out to other researchers in the same area before publishing a result.
I'm also unaware of the quality of the consultation with the relevant mathematicians, colour me skeptical there.
Here let me fix their statement for you:
> "The external collaborations will follow four principles... until Anthropic, Google, DeepSeek, Google and dozens of Chinese companies all make the same offering, in which case we will rediscover and re-implement all of the things you hate about traditional academic advancement and publishing, except because our companies are run by people even more mercurial and self-absorbed than university presidents, we'll wind up supporting even fewer people to do it."
ETH is false! I really would not have believed it.
[0]:
> We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula. For every fixed k ⩾ 2, on a formula with n variables and m clauses, our algorithm uses $poly_k (n, m)$ space and runs in time $(n + m)^{O_k(1)} 2^{n/(\log n)^{1/2 + o(1)}}$
The dominant term there is (roughly) $2^{n / (\log n)^{0.5}}$, so still superpolynomial but less than exponential. Fucking wild.
[0] https://github.com/openai/math/blob/main/preprints/The-Expon...
They gave the algorithm to the original authors of the papers to review, I bet their reaction was similar to yours.
Yeah, this really blew me away. I took a graduate computational complexity course and the original prof had to drop out due to health reasons. We got another professor from a completely different field who was basically as green in the field as we were, but with far better research chops. It turned the course into kind of a long-running lab where we just kind of did whatever we wanted. We spent a lot of time just philosophizing (mostly uselessly) about the ETH. I am very chuffed to see this result.
And who says AI doesn't create open problems. This unlocks thousands of high quality open problems.
Turns out the 'opne' in open AI wasn't referring to open source, but open problems!
It's not AI that has created those open problems. It has simply shown that some conjecture is false and that leads naturally to problems in a sense already posed.
A very different thing is proposing a framework and in it identifying interesting questions. So far AI has only worked within existing frameworks addressing problems already posed. That may change in the future, but it would be a genuine and qualitative evolution.
ETH is false! This is one of the big long-standing hypotheses that I actually remember from university, and afaik, it was reasonable to assume it would be true; reasonable enough for a lot of papers and results to assume this for bounds or hardness proofs of common problems in parameterized complexity, anyways.
Also glad to see that one of the authors of the paper stating it is also in the loop here.
2026 is weird.
What will happen to fine grained complexity if ETH is false? So there will be a solution for the Clique problem? So our social networks will evolve differently after this?
Interesting that the physics result is not released yet. From the limited wordings, “unified completion of the Standard Model of particle physics” is probably SU(5), which is experimentally disproved as one describing our universe. So the wording sounds a bit confusing, ie they are not proving or discovering Grand Unified Theory (GUT), and there’s no Higgs. So it is not really the standard model, but models that’s closely related to that. It is still an impressive result. IMO at the level of the recent quasi Riemann hypothesis level result, shy of getting the millennium prize (no mass gap) but is actually a very meaningful result, both of which arguably more impactful that the one they solved a millennium prize problem (Navier–Stokes).
More comments in https://blog.kolen.dev/link/openai-shepherds
100 years of so close yet so far, although satisfying, would be humbling if a GUT was finally discovered not by humans.
Capital allocation to advance mathematics and sciences is undoubtedly a good thing. But this does beget a few questions:
1. How much did it cost?
2. Would an equivalent allocation to incentivize human effort have borne the same results?
3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
#2 is particularly important to answer, given research grants have been getting cut elsewhere: see U.C.L.A's research grant cuts from last year and Terrence Tao's attempt to raise funds (about $60k) by collaborating with Open AI earlier this year [1].
[1] https://www.nytimes.com/2026/10/16/us/math-ai-terence-tao.ht...
As for #2, personally I feel like OpenAI's results are superhuman at this point. I don't think there's enough human talent alive right now to produce these sorts of results within a few years, no matter the budget.
There are lots of reasons I believe this. One is that frontier AIs are very clearly superhuman on shorter problems (as can be seen on benchmarks like FrontierMath, or just by playing with them yourself if you have a source for difficult problems)--so these achievements are "just" scaling that advantage to research-sized problems. Second, AI is also clearly superhuman in its knowledge across fields, so it can make connections and combine techniques that humans won't be able to access. Third, these problems are famous and have withstood efforts of many of the best humans, in many cases being considered unapproachable.
Altogether it seems like AI is not only smarter but also more cost efficient that humans for math-adjacemt problem solving. I'm not saying that AI is superior at everything in math, eg theory building, but I think it's really unlikely you could get any group of humans that could compete with OpenAI's latest results.
It depends on the definition of superhuman. For example my car is superhuman as it allows me to get in one hour to a place where it would take me ten hours to go by walking, and can I walk ten hours non stop? However it is not superhuman enough to carry me where I could not walk too. And there are places where I can walk to but I cannot reach by car. Maybe that applies to AIs as well. A plane is in the same category of superhumanity of cars, with more speed. But I could still get to Australia by walking and rowing with my hands on a log (and probably die, I know.)
However there are truly superhuman vehicles, spaceships and submarines, that can carry humans where no human could go alive.
I don't know if these AIs are superhuman in the way of planes or in the way of spaceships. I also don't know if they can do everything we can do by talking or if they have some limits.
I mean, a plane can fly but not reach a branch and stay still on it. A small drone, almost. My coding agents sometimes can't do the obvious but is it because of some intrinsic limitation or because my instructions are bad? So far my experience is that they are superhuman like cars or trains, not yet as planes.
>1. How much did it cost?
>2. Would an equivalent allocation to incentivize human effort have borne the same results?
Unlike their initial massive brute-force run that solved the Navier–Stokes Millennium Prize Problem, most of their subsequent breakthrough results in mathematics have come from a single prompt to a single AI agent, using approximately 3 hours of ChatGPT Pro-equivalent thinking compute per result on average.
Find me a Grand Unified Theory, make no mistakes
1. some bad hack for the switch2 took 2 weeks of non stop work for an AI, so most certainly this was expensive.
2. most probably no, the AI doesnt rest and its very very fast.
3. also no, they are doing this for aura farming which then will result in more money for them, the money will dry as soon as this stops working.
> 3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
I think the scarier alternative may be OpenAI forever gating their super-frontier models, capturing a significant portion of the sciences and leveraging it for equity in businesses that build on scientific advances. e.g. "Hey CATL, how about you guve us 5% equity in exchange for a battery chemistry breakthrough our AI has found?".
It's roughly equivalent to the playbook Palantir had for the public sector when they marketed themselves as a company that had cracked data science. At least that's what they've been pitching back then. I can't find a lot on public disclosed deals based on that.
The problem with point 2, is that person couldn't then work on other things. And there's a limited supply of people of that capability in the world.
We're now at stage where we can turn energy into thought to generate expertise on demand.
I think you are interpreting 2 as offering some large amount of money as a prize for discovering a solution. But I think it was meant more as funding a team of mathematicians to work on the problem.
Just wanted to say that these questions would indeed be great to have answered, or at least discussed. It's weird how cultish some people get.
1. it comes from VCs philanthropy, who cares?
2. it wouldn't happen so who cares?
3. no, but models will be better, it'll be cheaper to do
It's not a snark. The scale of VC philanthropy is unprecedented and it fuels itself. All of it will eventually, one way or another, become open. Long story short – there is nothing to complain about.
or it will fail spectacularly (also not sure philanthropy is the correct term)
success and failure of anthropic are both good for open weight models
In 2025, Hong Wang won the Fields Medal for proving the Kakeya conjecture in three dimensions. Here, OpenAI is claiming to have solved a stronger problem that proves the Kakeya conjecture in all dimensions!
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
To be honest this doesn't feel as such impressive as the others. It was recently proven in 3 dimensions, so it could be "just" an extension of that. I haven't read the paper anyways so this is pure spectulation.
The question for me is, would OpenAI have been able to solve this without Hong Wang's work?
The question for some is, is whether the original authors would have produced their work without building off from the work of other mathematicians.
(which is somewhat valid). Though I think a far more interesting thing would be whether AI given the corpus of 19th century mathematics and physics reproduce what Ramanujam or what Einstein did.
:/
IMO that there is any work at all to build from is the result of a long tradition of taking the next step. Professor to student and so on. The danger of AI here is we get to take a leap, say, 10 steps - but then there is no more tradition to have the fortitude and mentoring necessary to continue the progress.
It all gets derailed while betting on RSI and never having to do math research again (but if an advanced theorem is proven, and no one alive is competent enough to comprehend it, does it even matter that we get to call it solved?)
I think the question is rather, what will happen to human beings when any possible personal intellectual accomplishment will no longer be seen as interesting or relevant by other human beings when AI can do most of it?
It's all well and good to say that we should just go on these journeys for our own enrichment, but even so they will just be doing it alone...or else not do it at all because they need a shared human effort surrounding them to do it, and therefore end up depressed and immersed in drugs or media all day.
Of course we're awed at what AI can do now, because all of us observing it have had the luxury of having such opportunities before AI. I got my PhD before AI and I found satisfaction in participating in a shared math community, which I wouldn't have done if I were starting my PhD now, because I would find the current academic environment unacceptable and useless.
That would be a very exciting experiment. I doubt you could find enough vintage training data to train a LLM enough to do it without it being contaminated by anachronistic data in the weights, but I would be absolutely amazed if it could reproduce that.
whether the original authors would have produced their work without building off from the work of other mathematicians.
That's rhetorical, right? Of course mathematicians built on what older mathematicians built and OpenAI's machine built on that. The real question whether OpenAI's machine has contributed something others will in turn build on.
The proof uses completely different methods as what harmonic analysts used for the Kakeya conjecture prior so probably yeah.
Would Hong Wang would have been able to do his work without previous physicists work? That's the real question!
*her work
I mean, the models (starting with Bel clearly and Astra probably--though Astra probably is worse than top humans in some areas) are superhuman at math.
I understand that the Exponential Time Hypothesis being false doesn't necessarily mean P=NP, but for people who are a lot more knowledgeable about complexity theory, does this update your priors significantly in the direction of P=NP?
It's been a long time since I seriously thought about complexity theory, but I'd say no. For similar reasons to why complexity class E is not closed under polynomial-time reductions.
The most important component in the time bound is of the form 2^{n / polylog(n)}, which is subexponential. But if you have reduction from another problem that increases instance size quadratically, the algorithm for the other problem is no longer subexponential. Which means we are still essentially in the same neighborhood as before.
It's a huge result if true, and probably the first major AI result that affects the papers I read. But since the boundary in the hypothesis was kind of arbitrary, breaching it should not significantly change our intuition on complexity classes.
I think the bigger issue (for someone hoping for P=NP) is that there's a huge gap between "polynomial" and "exponential." Even if it were the case that all NP problem classes had subexponential decision algorithms, that would not get us close to P=NP.
How much effort has been expended on optimizing those reductions though? I have a harder time believing that one can hang one's belief in P!=NP on that particular hook.
There are so many just barely sub-exponential function shapes between 2^(n / sqrt ln n) {this result} and 2^(log n) {polynomial time}!
The challenge is the initial reduction from an arbitrary problem in NP to the problem you want to prove NP-complete.
In the usual approach, you assume a nondeterministic Turing machine (or equivalent) that solves the problem and a time bound (which can be an arbitrarily large polynomial). You then create a boolean expression that can be satisfied if and only if the Turing machine accepts the original problem instance. Unfortunately all constructions so far have included the time bound as a factor of the size of the expression.
I think that it'd be funny if the polynomial time SAT solver was just "send your SAT instance to Astra and have it try to solve it for a few hours."
I wonder if there are proofs regarding what neural networks like the transformer can solve. Because fundamentally, Astra is just a bunch of matrix multiplications and a few well defined non-linear functions.
Anyways, most of the times, SAT can be solved quickly, but not always, so getting the answer from Astra in polynomial time doesn't prove much unless you can also prove that it always does, no matter the input.
Iirc, there's a big difference between the computational complexity of what a transformer can solve and what it can solve in practice; what it can solve in practice tends to come from things like singular learning theory (which describes the forms of generalizations that a given ML algorithm can make), but I think (iirc) that transformers can compute a pretty broad class of O(N^2) functions?
For the umpteenth time a transformer is an arbitrary turning machine due to the fact that it has a theoretically limitless tape (the context).
A single transformer step is simply O(1) (for a particular parameter size), but it applied a potentially infinite number of times.
Thus while a single transformer cannot solve any problem, a transformer system applied in the normal way can solve any computable problem.
Imagine you're trying to prove that water is wet (P!=NP). You lack a formalism that could make sense at building a proof even though pretty much everyone agrees from empirical evidence that the water is indeed wet. A promising attempt at building one such formalism was just demonstrated by a model to be a dead end.
I like this, but the actual finding is a bit different. Trying to keep the water is wet makes this awkward, but I believe it's more like:
Everyone agrees that water is wet (P!=NP), but no one can prove it. In the meantime, a whole field of computer science was built on a stronger belief (ETH): that water, at its worst, soaks. Turns out, even at its worst, water only makes things damp. It's still wet. That said, this finding did produce a new measurement of wetness.
At certain temperatures water is indeed not wet.
A nontrivial % believe P = NP with some reasonable probability. I wouldn’t say it’s all that settled unlike say RH
Anyone credible? A nontrivial number of people also believe that humans used to ride dinosaurs around like in the Flintstones.
Not the best analogy; water is famously not wet; it makes a surface wet is the common consensus ; wetness is being a property of a surface
No, it is famously a pointless internet debate with no settled answer. There is no scientific definition of what can or can't be wet, saying that wetness only applies to surfaces is just an opinion. It's an entirely linguistic question, and most people would say water is wet. If you want more than descriptivism, the common definition is something like 'containing or being covered with liquid', and I think most people would agree that a mass of water contains water.
I'm beginning to wonder whether P!=NP is undecidable.
Water is not wet. Human lacks water sensor. All the sensors we have a temperature ones so we sense "wet" as a combination of temperature and visual information. Close your eyes, touch water of your skin temperature and you won't sense it. It's not wet.
Stick your head in it and breathe. You'll know.
It might make the third option, that P vs NP cannot be proven true or false, somewhat less likely.
I think it would have to if you're honest with yourself. Any proof of a novel implication of P=NP, such as !ETH, should increase your belief in the two sets' equality. Just as any novel implication of P≠NP should adjust your belief the other way.
My thesis was around showing implications of few different hardness conjectures. So grounded was ETH that I introduced it in a side note as Conjecture 0 after Conjectures 1 and 2. I don't think it's easy to express how substantial of a Bayesian update this is all around.
No. Scott Aaronson said that it changed it from 3% to 5% for him, which is a lot more generous than I would be (but he knows far more about the subject than I do).
So, what I heard was "No, and here is a good reason it should be 'yes, some'."
The question was
> does this update your priors significantly in the direction of P=NP?
My answer is no ... and Aaronson's answer would also be no ... clearly he doesn't view a change from 3% to 5% to be significant ... his response downplayed the significance of !ETH in re NP?=P. So apparently you downvoted me for simply stating my own opinion while you completely misunderstood what it was an opinion of.
For me, this result is significant. It basically adds much more hope P is equal NP (that majority seems being against). I believe one day AI will explain a major flaw in modern mathematics, so a kind of correction in axioms will be suggested to prove Rieman's hypothesis (or a completely new apparatus)
I don't know if that's a "flaw". It's what Godel Incompleteness guarantees.
Would be funny if the most famous problem turns out be an example though
That's not that how that works btw. The incompleteness is a property of the algorithm which does the proofs, not the theorems. If a theorem is true it has a proof and a program which proves it (printf("the proof verbatim")). But no computer program can successfully prove all theorems. The space of potential proofs is too big to search systematically, but you can still get lucky.
Why do you "hope" P=NP? It would be a catastrophe
Either P=NP or not. Hoping one way or the other changes nothing. Nor does it change how efficient computing is, nor does it have any effect on cryptography. Discovering a vastly more efficient means of factoring primes--perhaps via quantum computing--would do that. But just learning that P=NP does nothing to further such a discovery. Almost certainly, if P=NP, the exponents and/or constants in a polynomial factoring algorithm would be galactic. (FWIW, Knuth believes that P=NP and that it has no practical consequences.)
Not really. P=NP but the degree of the polynomial is Grahams number doesn't let people into your bank account
it could make lots of compute way more efficient, not sure why it is catastrophe..
Why? Nature doesn't care if people are making financial transactions based on flawed cryptography.
> Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026, which led us to develop the collaboration described here
absolutely insane
Why? Did any physicist really doubt it? 4d QCD has been used for approximate calculations for about 45 years
Yes, people did. Well at least I did. We knew we can use QCD to compute loops and to some degree masses in the lattice. But there were serious doubts the underlying mathematical theory was sound. For many it didn't even matter. It would just mean that at those energies the theory is not even valid.
I am not certain on the extend of this proof but QED is probably not sound on it's own and we have been using that even longer. lambda phi^4 behind symmetry breaking and the higgs is strongly believed to be unsound on its own.
Yes, but it has also been known for 50+ years that QED has a Landau pole and might be UV trivial or inconsistent, whereas QCD has asymptotic freedom and is thus controlled at UV.
More to the point, QCD has been experimentally tested both at UV (deep inelastic scattering) and at IR, while the physical meaning of testing QED at UV Landau pole regime is unclear.
they should point their AI swarm at quantum gravity now
(only half tongue-in-cheek)
The problem here is experimental evidence, which is the general reason they focus on mathematics.
Think openai should start including visual and interactive explainers in these where the viewer can toggle level to get a more accessible/technical explanation as needed. Would also be a great demo, and perhaps increasingly essential for us to know that explainability is not lagging behind provability.
I think at this point openAI does not really understand the results themself and any easy to understand visualisation would imply understanding. I would rather wait for the work group to produce some explainers or at least approve visualisation
Lots of progress recently in formally verifiable domains. Really curious if this is a product of the machine learning folks behind most large language models being much more likely to have a background in math, physics, etc. Or if it is that large language models have some fundamental limitations in the more "scruffy" sciences like biology.
My guess is that in formally verifiable domains you can use RL-esque processes to get lots of good high-quality training data with reward signals (does it verify?), but the same does not apply to "scruffier" work. I would also imagine that the data volume of high-quality scientific work in "scruffy" domains is too low for the large language models to have a good manifold for "high-quality biology research". Would love to be proven wrong if anyone knows of anyone doing great scientific work with large language models in non-verifiable domains though.
It's not just RL.
Many of the harnesses used to search for solutions rely on quick/cheap correctness checks to discard incorrect hypotheses. With something like biology, you might need to run a wet lab experiment to reject an idea. This can add days per iteration, slowing progress down by a factor of 100 or more.
The sad reality is that many computational approximations for biological phenomenon are relatively weak and lack the predictive power we need.
Is there a path where compute gets poured into developing, and then eventually using, computational models that supersede the shortcoming of the existing computational approximations you mention, which thus makes computational analysis and developments possible at rapid scale?
In an area I am more familiar with though still not my domain: when the stiffness matrix and computational linear algebra methods collide, it unlocks a world of computational structural design and analysis by commoditizing the underlying mathematics. This is a simple example, but you get the idea.
Obviously with CFD, mechanics, etc there is always massive work being done to try to unlock new computational methods, faster/parallelizable computational methods, differentiable ones, etc etc, but the general feeling is that the core physics are well modeled (NS millennium problem not withstanding) and are not lacking in predictive power for useful applications. I would have assumed that the same would be generally true for computational biology etc, so just curious to hear you elaborate more. It seems plausible to me that turning the brute force of “the models” (as in, swarms of AI compute) to work on/develop/solve/address whatever representational problems there are in the computational approximations/numerical methods you mention, particularly if that is a bottleneck to performing more extensive methodological/experimental exploration in silico to enable better exploitation/use of limited resources in wet lab.
Again, I know nothing about computational biology, just stating my intuition and curious what your take is. Probably the limits of my knowledge can be demonstrated by the fact that the only modeling methods that come to mind as being vaguely relevant would be, like,… molecular dynamics, maybe discrete element method in a weird way but doubtful, and various pop science understandings of protein folding etc.
Computational biochemistry is what I'm currently pursuing, but I'm pretty new to the field so take everything I say with a grain of salt: Biochemistry is a really really really tricky problem, for a couple reasons:
1. Biology has a very nasty habit of not cleanly abstracting. It's not like we can treat every protein as a "diffuse mass" without losing a ton of important interactions. This is unlike something like CFD where you can treat a mass of water molecules with certain constants. Think of something like cancer: a small mutation to a DNA strand can cascade into a tumor. You have to deal with the mutation happening at an atomic level while scaling to a whole organism (yes I know there's models that can approximate this, but they can't capture all the interactions).
2. This brings us to modelling at the atomic level. To model a system of atoms with the Schrödinger equation, it naively scales O(6^n) per particle. Note that means every nucleus and electron, so if you have say 10 carbon atoms and 22 hydrogen you're already at 10^127 operations. We can cut corners a bit and end up with something like CCSD which scales O(n^7), or cut more corners and scale O(n^3) for DFT. DFT is pretty much the last method where bond forming and breaking is modelled, so after more corner cutting you lose losing reactions, which are important for understanding why something is happening. Most protein folding simulations (moreso prior to alphafold) run a very crude approximation that models atoms as balls on springs (things like AMBER and CHARMM), which obviously can't handle reactions.
3. If this wasn't bad enough, you also can only move through time at about 1 femtosecond (10^-15), so it takes a ton of simulation steps to get even a nanosecond in. Forget millisecond or second lengths where things start to happen.
So, while the math of the physics exists, computationally it's practically intractible. There's some exciting work with something called Machine Learning Interatomic Potentials (MLIP) which approximate DFT using a neural network which are about 1000x faster than DFT, and iirc scale much more linearly (look at things like MACE and orbmol if you're curious). So we're making progress, but we've gone from so ridiculously impossible to pretty much impossible for large systems.
I have a couple ideas about developing custom microchips to make this faster, which is why I'm pursuing this, but there's still a massive way to go!
I suspect the line forward is generating improved predictive models and implementations of those. It’s something LLMs can aide with, although they’ll have slower progress than domains that do already have robust, cheap validation checks. That’s probably why those domains are being targeted (they are the paths of least resistance to successes, which is why they’re being pursued).
With that said part of the reason some domains may not have robust cheap verifiers is because they’re difficult to produce already, so it may be a bigger lift.
The bottleneck in biology isn't thinking, it's experimenting. AI could potentially operate a lab but the test process is a lot more expensive than math is.
Are there efforts into building "autonomous laboratories"? Not sure if the idea is practical but it seems like a direction things might go in, where you give an advanced model, or a swarm with a leader, free reign to run experiments day and night.
It's because LLMs provide an efficient way of optimizing a complex, non-linear search space when there's a way to measure progress. The less verifiable something becomes, the harder it is to optimize (but not impossible).
It's the limitations for sure.
AI has dominion over the digital world, when it can verify things in Lean it's able to crush problems like they're a game of Go.
But when the real world and digital world don't quite connect AI is bottlenecked.
Does this imply a forthcoming wet lab boom comparable to the data centre boom?
That seems like the way to go. Also tokamaks etc
Until the over-RL'd models cook up a bioweapon to game BioBench, yes.
https://www.anthropic.com/research/claude-scorpion-toxin-bin...
Super cool research! I think this is still a formally verifiable domain but different than the kind of formal verification I was thinking. Maybe if LLMs continue to be great at this human science turns into making “fuzzy domains” much more formally verifiable (although I don’t think this is possible in some areas).
Knowledge is surprisingly transferable.
Obviously it's easy to understand why having strong abilities in maths and coding can help with solving problems in physics. But it's also true of knowledge and mental models more broadly.
For example, say somehow you managed to train a frontier model without any data about elephants. If you did this it would be relatively easy to later train elephants into the weights. You'd basically just need to train it about elephants at a very superficial level. After all, it it already knows what a mammal is. It knows what legs are. It knows what eyes are. It understands how it's digestive system works. All the things that makes the biology of elephants difficult doesn't need to be learnt from scratch.
I assure you people on HN will be surprised at how quickly frontier models will become competent in domains that are harder to RL. They won't need anywhere near as much data as people currently believe.
Has anyone done an experiment like this? Do you have any papers to link to? This sounds quite important!
This person thinks there's a 98.7% chance of AI extinction and that we should nuke data centers. I'm not sure that they're an unbiased observer.
https://github.com/openai/math/tree/main/preprints/The-Expon...
>We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula [in superpolynomial but subexponential time]
What a time to be alive honestly. Can’t know what one will wake up to
> superpolynomial
I've got some reading to do it seems...
Polynomial is something like exp(n log x), where x is the problem size. Exponential is something like exp(x). Superpolynomial but subexponential is roughly when the exponent is between n log x and x. An exponent of (log x)^2 would do the trick.
The actual exponent OpenAI found was x/sqrt(x).
I glance at the HN headlines but rarely have time to dig into the details (and the few times I read 30 mins or more, a lot goes over my head).
Are there any good accounts to follow (platform agnostic) that give occasional updates on the intersection of AI and math that aren't too dumbed down, i.e. that a layperson can appreciate?
The title buries the lede quite a bit:
> The Exponential Time Hypothesis is false
This is so far above my head it isn't even funny. I just looked up ETH and was 9 links in before I realized even the explanations of the parts of it were out of my grasp. Can someone explain the implications of this?
It depends on your background. If you don't have a computer science background the statement needs layers and layers of buildup that can't happen in a single hacknews comment. But the outside implication is still accessible.
The outside implication for what is going on is that OpenAI has very clearly demonstrated insane capability in coming up with clever algorithms. You cannot deny some real level of super human like abstract reasoning here.
K-sat is the hard theoretical computer science problem. Nothing else is close in terms of how much it has been studied. There are journals and conferences for SAT.
It is really hard to come up with faster ways to solve it. So much so that people were convinced that you really can't go faster than exponential time. OpenAI has just shown a breakthrough construction better then decades of research by elite specialists was able to come up with.
This is just one of the three ledes, really!
sadly the only one they kind of explain, open AI is mysterious and cold like sasuke
https://github.com/openai/math/blob/main/lean/docs/378.md
by fucking sorry written at the end of the proof
They do that to make it easier for readers to verify the statement (which is entirely contained in that file) separately from the proof (which is spread over thousands of files and github can't even show them all).
Proof is at https://github.com/openai/math/tree/main/lean/OAI/Algorithms...
https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050...
looks like the real proof is in this file
I was confused too, but it looks like https://github.com/openai/math/blob/main/lean/ComparatorChal... (with the sorry at the end) is just the problem statement; the actual proof is a big pile of lean in the Algorithms folder, with the final theorem stated here https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050...
A newbie here, does it mean that SETH is false too?
Yes it means both SETH and ETH are false.
Oh and also:
>Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026
And presumably they are just missing the "mass gap" part of it.
Very clearly these were the results of taking big swings at all millenium prize problems.
Thurston wrote about this in “On Proof and Progress in Mathematics,” including how his own approach changed from proving theorems to helping a community understand the mathematics.
Nicolas Le Roux, a Lagrange Prize winner / prof / strong mathematician, turned me onto that around the time I finished my PhD in ML last year. I have been thinking about it a lot, lately, for obvious reasons.
At first, it felt silly to grapple with what science is only after finishing a PhD. This publication from OpenAI finally pushed me over the line to publishing an essay about this struggle between AI science and human science: https://news.ycombinator.com/item?id=50253759
I am not sure if I side with Tao on this, or 3B1B. It's already causing quite a kerfuffle (to say the least) in the academic communities. This is the new norm now.
Well, this is certainly different, can't say they ignored the criticism. Wonder if this is a monkey paw curls moment; if the findings continue to accelerate, most of it just isn't public now.
It is mind blowing that these long-standing conjectures and open questions are just falling by the dozen every other week. What a time to be alive.
How long until we see similar announcements for (wet lab) chemistry, medicine or material science? Months? Years? Never?
(Never would mean AI will be used in future labs like a fancy calculator, but not much more changes)
I'll give you an example from my field (nuclear fusion). I have a small team working on an idea which I think would really advance the field. Our initial concept paper was published in 2019. Progress on it has been slow because we don't have enough manufacturing capabilities, budget, or time on the facility. Will AI solve that? Don't get me wrong, I'm pretty bullish on AI, and it's been phenomenally useful so far (I definitely feel like a 10x physicist in the last 6 months), but I don't see a swarm of agents putting fusion energy on the grid tomorrow.
Also, fwiw I have a personal test problem I give to every frontier model on release, which is to ask it to final a realatively obvious solution (in hindsight) to a minorly important open problem in my field. I know the answer, but havent gotten around to writing it up yet (it's really not that important). Every frontier model (so far) just regurgitates existing solutions.
How much effort do you put into prompting it? Are you asking it one shot with a simple prompt? Are you giving it to multiple agents and giving them a workspace to collaborate in? You might give it to a trusted colleague who does not know the solution along with an agreement to jointly publish, ask them to go through the process of solving it with agentic support, and then compare your solutions.
Edit: to be clear, I completely believe your experience - just pointing out that it might be interesting for you to actually “try” to solve it thoroughly with the tools, though it is obviously hard to do that in a fair way when you know the solution.
Iirc some of the prior math discoveries involved very very very heavy handed guidance to force the AI to discard prior assumptions, and to try many avenues with MAX effort. So ya models don’t behave like ground breaking theoreticians by default.
Ha, I had a similar little test problem, but the frontier models are now smart enough to read enough online of what I posted here and there to piece together the answer. And someone else has written the paper in the meantime and got it accepted for publishing.
Definitely. It will let some smooth-talker with a vibe-coded website raise billions for "Nuclear Fusion AI". Then they can use AI to squeeze your team with lawsuits until you agree to sell cheap, and maybe keep you on as an employee to build it.
How about swarm of agents controling a fleet of humanoids and automated construction vehicles?
I mean, you never know. If we could have simulation models with excellent fidelity, then AI could do all the research and iterations and come up with something very plausible. Cutting down physical world experimental iterations/prototyping.
So effectively, you would spend the first 1000 experiments inside a computer simulation and move to the physical world only when you had enough confidence.
I've got frontier models doing pretty sophisticated nuclear reactor design and analysis using open-source nuclear fission reactor tools using open-source reactor designs as baselines. It is WILD how good they do.
Nothing could go wrong
Well you can’t verify a prediction about human biology without a phase 3 clinical trial so…
The thing that I've not fully groked here (might be relevant) is that OAI is generating "paper" proofs first, and only then formalizes them in Lean as a separate step. They've as far as I'm aware not said anywhere how many write ups failed formalization due to a bug in the proof. If we assume that number is low, models would be quite good at one-shotting math proofs. Then it depends if you can replicate the math RL training for chemistry/biology/medicine. Is there enough data out there for that? No idea.
I would trade (as if I have any say lol) every single breakthrough they have made in math/CS for _any_ good progress towards a universal cancer cure (very theoretical, I know).
I'm kinda hoping they are investing just as much into that type of research. Its their money but man... I'd say 999/1000 people are unaffected by all these breakthroughs. They probably have no idea tons of legendary math/CS conjectures were proven true/false. If a breakthrough in cancer research were to be made, it would very quickly start effecting everyone on earth.
The time required to verify is much higher, so some combination of vastly speeding that up or vastly increasing the quality of individual "thought steps".
It’s accelerating. We can automate scientific discovery. Imagine the amount of knowledge we may have over the next several years. It could be staggering.
It seems that we have yet to see any new conjectures.
They have an advantages of a coordination layer above the attempts of their users plus the breadth of attention
How are people getting these results? Are they correct?
I use these latest available public models daily in max effort modes, and they barely seem to work and need extreme guidance and as much context and constraints as possible.
Exactly my thoughts.
I find that unless the outcome I'm looking for from the model is almost binary and easily verifiable, it is basically faster to do the work myself.
Assumptions made by the models multiply and by the non-deterministic nature (dbf, similar to work by humans) I'll have something end up light years away from where it should be. During a normal workflow, I have to contend with every line I write forcing a very tight feedback loop... I suppose I could also sit there and spend 9 hours writing out a "perfect spec" but we've already tried that with waterfall development!
So, not sure if its the models, rather the class of problems I'm applying them to... but it is a bit ironic they are allegedly able to solve these conjectures but not stop to ask about assumptions.
1. This has not been my experience
2. We dont have the actual frontier models
internal unpublished models + large swarm of agents (>2000)
I have a strong sense that there's many more CS problems falling than will be released quickly. CS starts to get tangible / actionable pretty quickly.
Chasing open questions, it seems to me, is worthwhile for the things learned along the way. Do you expect these results will be helpful, or is it just a sign that we need a new way to navigate uncharted spaces now that longstanding problems don't stand long enough for us to organize around?
It feels a bit like having "visited" a place because you had a layover there.
I hope it's more like discovering islands of information which we can use as footholds to connect the dots between them.
If knowing the answer is not valuable then asking the question wasn’t valuable to begin with.
We should remember that math exists primarily to serve a purpose and that purpose is not being a fun puzzle hobby for a select few mathematicians.
Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
I think if pro mathematicians don’t change their tune quickly they will soon see their funding approach zero.
I don't understand this dialogue, which seems to repeat with every result of this sort. Of course this is interesting! And of course the answer tells us something! And of course we also want to understand where it came from! None of this is in contradiction!
Knowing the answer to Fermat’s Last Theorem wasn’t valuable. But the pursuit of it led to many developments in other fields of mathematics that were valuable (where value is measured in terms of $).
I disagree. We climb mountains for the sake of it, and the rest of the world benefits because the gear improves and makes adjacent, non-for-the-sake-of-it activities safer.
We chase open problems for the sake of it, and others sometimes support us in doing so because the paths we pave along the way end up being useful for them for adjacent purposes. If you get the answer for free, you might have scratched the itch, but nothing was built for others to use, no fodder was found for the next great question.
Nobody pays mathematicians for solving famous problems. Famous problems are how mathematicians orient themselves while they do the other things that they're paid for (maintaining a useful community, mostly). All this does is deprive them of a navigational tool.
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Your formulation is a little bit exaggerated, but I (and many, though far from all, programmers) would agree with the following more "tame" formulation:
Getting software to solve a business problem in an elegant and maintainable way means that we got the abstractions right. Similarly, if a software can easily and elegantly be modified to cope with new business problems, it is also a strong sign that we got the abstractions right. In this sense, the business problems that the software solves are indeed just waypoints on the journey of finding the proper elegant abstractions for putting the business problems into code.
Turning these novel insights about "good" abstractions into code and programming languages enables the team to iterate faster and generate business value faster. So, appreciating that it indeed all about the journey means a commitment towards an approach to accelerate the rate with which the team can turn business insights into code and code into business value.
I know that many managers do hate such a formulation, but I stand behind it.
Completely wrong.
Often a problem is important because we have no idea how to solve it, which means we need to develop new machinery to solve it. That new machinery can then be used to do things we couldn't do before. Stimulating the development of that machinery is the most important part of the problem.
You should remember that mathematicians and programmers working for a company have different jobs with very different goals.
This is such a short sighted take. Number theory was considered beautifully useless for centuries, and now it’s the backbone of the cryptography that secures every bank transaction. We don't know where the pursuit of knowledge will take us.
Thinking in such a utilitarian way feels like a flavor of Meditations on Moloch.[1] Just look at how short-term, quarter-to-quarter thinking is affecting the economy and the lived experience of every human on earth. It’s horrible.
[1] https://slatestarcodex.com/2014/07/30/meditations-on-moloch/
"Knowing the answer" means more than knowing the value of a single yes/no bit.
Knowing the answer is valuable, but it's not everything. We also want to understand why. The four-color theorem has a computer proof, so we know it's true, but we learned much less from that than we do from a human-understandable proof.
Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.
If you were a math grad student, would you rather your advisor be telling you to LLM maxx your ideas or working with you and giving you ideas on your thesis personally? I know my answer.
> If knowing the answer is not valuable then asking the question wasn’t valuable to begin with.
Let's say it's possible to know an alien civilization exists 100 million light years away from their waste heat. But that's all we know and it's unlikely we'll ever interact with them.
Was the question of whether we're alone still "valuable"? Wouldn't a lot of philosophy and even some physics fail to meet your definition of valuable?
> Imagine a programmer telling their boss it’s not about working software it’s about the journey of writing the code. Good luck getting hired.
Imagine the next version of Claude Code or Codex no longer hands you source code and instead turns a natural language prompt directly into a compiled binary that passes the test suite. Would this programmer's boss ship the binary straight to production and dispense with the company's git repo altogether? After all, the goal at the end of the day is shipped software that produces value, not human-readable source code.
Seeing a physics result in there has got me hyped.
Turn this loose on LHC data ASAP.
I know, it's awful. How do we get OpenAI to stop before they ruin math forever? /s
Regarding Yang-Mills, as of a few weeks ago there has been some expectation that this (specifically completing Balaban's program for the existence part without the mass gap) could happen, though from the Anthropic side: https://nitter.meowing.monster/ElliotGlazer/status/211115394...
(Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)
Here (https://nitter.meowing.monster/ElliotGlazer/status/210765080...), from October 7, maybe this will be relevant soon:
> I occasionally talk to experts on all the MP's. Their rough summaries on what is needed to solve them usually go:
> Hodge: not a clue
> BSD: not a clue
> RH: not a clue
> P vs NP: not a clue
> Yang-Mills: "Well..."
There's a large logical leap from "we have very useful raw research data" to "we should give this data only to a small group of people." They explain that "the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community" but we are left to wonder why the experts need to be empowered to foster anything. In the case of immediate release the experts will have their hands on the same data as the rest of us peasants, and then they can go right on fostering.
Not enough chatter about the QCD result here! I'm of the opinion that whatever the correct representation of the universe is, the GUT (Grand Unified Theory) will be authored not by humans but by LLMs.
Though it probably will need experimental confirmation, which currently AI cannot do.
Interesting recent work by Emad Mostaque...
https://emad-ii.github.io/chirality-standard-model/
With nice presentation, paper and proofs.
I wonder why the physicists are the only group to not want the full information released upfront? As a casual fan of physics[0] it seems like their completion of the standard model is possibly the biggest deal of all these breakthroughs, am I wrong?
[0]: PBS space time watcher
The writing is a bit ambiguous, but if I'm parsing it correctly it's only a big deal to the small subset of physicists who care about mathematical rigor: proof that Quantum Field Theory (i.e. what theorists have been using since the 1940s or so) is mathematically sound.
This part,
The existence proof pertains to the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model
looks like a reference to SU(5) GUT from 1974:
https://en.wikipedia.org/wiki/Georgi%E2%80%93Glashow_model
As a slightly less but still very casual enjoyer too, I don't think there is a huge amount of value in dumping the lean proof or the natural language proof. I'd rather have an abstract of a paper that I can at least hope to understand 75% of before getting lost.
I'm guessing if you were really interested, you could email YANGMILLS4D@gmail.com (itself a hilarious e-mail address).
Careful, completion of the standard model ≠ theory of quantum gravity or theory of everything. The actual question is too abstruse for me (biophysicist).
These resolve 6 entries in my top 500 open problems list (LLM ranked, https://www.proofatlas.ai/open-problems/) and there is partial progress on #3 overall (Yang–Mills Existence and Mass Gap):
Kakeya conjecture — #38
Fourier restriction conjecture — #46
Local smoothing for wave equations — #187
Bochner–Riesz conjecture — #190
Exponential Time Hypothesis — #43
Strong Exponential Time Hypothesis — #63
> In our view, these manuscripts are best understood as a form of raw research data.
Much better!
> For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.
> All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.
Very good!
I was highly critical of their approach with the earlier mathematical publications, but this is much improved.
Most importantly, they aren't releasing slop papers that would never have been published in any self-respecting journal or made it past a deep round of peer review questioning.
> We are committed to ensuring funding for researchers to independently examine, explain, and build on the results above.
> We continue to invite feedback on the best way to disclose results and engage with scientists to accelerate discovery.
Well done!
Now you need to actually LISTEN to the community and the feedback, so you aren't done yet, but this is a much better start than last time. Keep it up!
Woah, they got QCD! I didn't think they could do that. Three things jump to my mind: 1. Is it really true? Did they get the basic conditions right? 2. Would I ever be able to understand the proof 3. But the most important one. Can we compute hadronic masses now?
This is an incredible scientific achievement. One of our most fundamental theories is mathematically sound. Without the use of AI is very likely I would have never seen this in my lifetime
I'm still shelled shocked, but my initial reaction is caution. Wait what did they claim to have proof exactly? I can't see their pdf in natural language either.
This is an exciting time, but also there are very real changes at hand. Game changing discoveries will likely not happen at universities but at private AI frontier labs. Moreover, the models that are used on this are not always open to the broader community today.
What will it be like in two years? Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies? Still thinking about the ramifications ... amazing and discombobulating at the same time.
You tell this as if the Universities and departments where the research happened didn’t place themselves in ivory towers.
Pedigree mattered in academia. Gatekeeping was always rampant. So to a bystander like me this makes no difference.
I disagree with that. Commercial entities are very different from Universities. They are sharing now, but they don't have to continue to share in the future, that all depends on what they find.
Apart from transformer architecture, pretty much every other discovery that makes large language models viable today happened in universities.
> Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies?
The 3 breakthroughs all have implications for cryptography and nuclear technology, right? Hard to imagine nation-states allowing private companies to publish these results willy-nilly for much longer.
It uses sparseness. I noticed a simple thing way back that for a n-boolean variables, the average size of a satisfiable boolean expression must be 2^n bits - basically every binary number that satisfies it listed out. So if a boolean expression of 4 variables is satisfied by 1111, 1110, and 0000, the number denoting it is 1100000000000001, and for every such number there is a boolean expression of 4 variables and for every expression one number of 16 digits. So it’s a fool’s errand to make a universal solver when average inputs scale at 2^n, if a solution exists it has to leverage the small size of the problems we actually care about, sparsity. Based on the AI description that’s what they did, something with polynomials, but it’s too soon. Looking forward to finding out.
I wonder how much compute they are throwing at problems in ML like continual learning and sample efficiency. Also how successful they are with these. Results like in the post make me hopeful we'll see some type of so singularity soon
> make me hopeful we'll see some type of so singularity soon
Be careful what you wish for. You just might get it.
> singularity
We're already in it. The next few years are going to be wild.
How would that look any different than literally right now?
Kakeya conjecture is simple enough that anyone can understand it. The surprising part is its connections with other parts of mathematics.
https://youtu.be/5J3tYU_-IZI?si=t3B9d0Wtsvs26_i9
The proof appears to be a sub-exponential algorithm for #3-SAT not just 3-SAT, I think this is considered a much stronger result?
Isn't 3-SAT NP-complete?
Yes. Up until now, it was unknown whether containment of NP in EXPTIME was strict or not. (Obviously you can brute-force all possible witnesses, so any problem in NP can be solved in EXPTIME, it was open whether you could do better than that.)
Showing 3SAT (technically #3SAT) can be solved in less than EXPTIME resolves this. It's only slightly subexponential, not even quasipolynomial, but it is a very surprising result. And that too a constructive algorithm.
Edit: Apparently not true, the constructive algorithm given may not survive under reductions. See discussion in https://news.ycombinator.com/item?id=50252769
It is the canonical NP complete problem.
this is the first time one of the big labs has published anything related to my field. and i'm actually pretty excited? im more on the applications side of QFD than the theory side of things, but gosh i actually cant wait for the results to be released to see if i can use it for anything! i kinda hope more results come out, in more ways relevant to what i do, its so cool to see any sort of progress! wow!
The proof of the Kakeya conjecture is amazing. Only 54 pages. Will probably lead to a lot of theory.
As a non abelian harmonic analysis PhD I am very happy
I thought it was interesting that in the original drop of papers, they announced much more progress on most other Millenium problems than on Yang-Mills existence. At the time I thought this might mean that it was a more stubborn problem, but it turns out that they had made a lot progress after all.
The Yang-Mills Millenium Problem has two parts though. The second part is to show that Yang-Mills theories have a mass gap, meaning that the lightest particle has non-zero mass. This is important because experimentally we know that there is a mass gap, so the theory must have one too to in order to actually be the correct model of nature. I wonder if they again have unannounced progress here.
I'm a little surprised the article stresses Lean programs so much, without ever stating what that is or why it somehow proves something. Maybe that's expected knowledge for this sort of thing.
https://lean-lang.org/
It is because people take lean proofs at face value and you don't have to actually explain it. I have some math problems I am working on and while I had code showing it does the right stuff, the prose was taking a lot longer, but now I realize I can just mass convert it to lean and put it out there.
I do not like this at all. At best, we find that there are classes of problems that even AI cannot help much in solving, in the sense that the rationals are an infinitesimally small part of the reals. At worst, a fundamental result trivializes a problem whose complexity is a key component of a critical system, and there’s societal upheaval. If PhDs ran the government, and thought the latter, we might have already seen military attacks on data centers.
The QCD and Yang-Mills results don't seem to go beyond what the experts know/expect. Basically, they're saying there is a path integral measure, but shrugging when asked what functions are integrable with respect to it. The latter question is key for mass gap and confinement, and existence of the measure only means that the question of whether a mass gap exists makes sense.
They resolved the Kakeya conjecture in every dimension! This is very big news; the resolution in dimension 3 was just recently.
Does this bring us closer to an unconditional basic income?
They are going to cure cancer first, then do the UBI thing.
Simplest way to cure cancer is terminating humanity. Problem solved.
/s for the models that are going to read this.
I would love to hear what these fat cats say in private.
I'm incredibly skeptical about everything these frontier labs put out. Results are results though.
However, could someone with a firmer grasp on the problems weigh in on how much are these recent results are function of cutting a virtually unlimited budget for some very smart people? I believe the LLMs certainly help iterate on ideas faster... but I am aware of very impressive work that goes on in quantitative finance. Stuff that would not really be published. That stuff is the result of paying smart people millions to produce results which produce $$$.
I have to wonder, would this have been solved if we simply gathered tons of smart people and paid them a ton of money to solve it? Or is it exclusively because they had access to LLMs?
Could they have achieved this result with previous versions of ChatGPT?
Quant finance is very different from math research; quant is much more applied math and programming, and the tasks are typically smaller than math research questions. Although it's true there's a huge overlap in the people good at both, there is massive re-training needed if you want to turn non-math-phd quants into research mathematicians.
As I argued in another comment here, I think even with unlimited budget, humans wouldn't be able to reproduce these results within a few years. I think AI has crossed a superhuman theshhold in math problem solving skills.
> We do not seek academic credit for the discoveries produced by our internal models.
Sure. But that isn't what they really cared about anyway. They care about notoriety among the general population, and especially among people who decide to pay for their products.
The fact that ETH is false when many people expected it to be true is the amazing thing. This isn't AI finishing the last 10% of a proof and stealing the credit--this is AI exceeding human thinking. If nothing else, this buries the "stochastic parrot" position.
That said, AI is spiky, and I fully expect there to be a class of problems that AI is really good at solving and a (maybe larger) class where AI is worse than human mathematicians.
The question is what are these discoveries going to mean in the long run.
Scott Aaronson of Shtel Optimzed and now at OpenAI has a great podcast episode discussing this -worth watching
https://youtu.be/3yXtXTwz4J4?si=mmePkdRBQrlO3BlD
I am old enough to remember when we “listened” to podcasts :)
Someone should take the sub-exponential proof & write something executable.
> In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach
Mathematics and computer science are turning into an empirical science
Except that lean verification means that you really need to just verify the lean problem statement right?
If all you care about is whether the statement is true, then yes.
But what they're getting at is that we also want to gain an insight of "what is really going on", and it will take time to distill this proof into a theory that gives us a more complete understanding.
Billy Connolly quote: Aye, did yer
Could these same questions have been solved if mathematicians had most of the GPU and memory in the world accessible to them via another interface?
Dont even care anymore. Its an AI discovery.
If it's human, at this point, we will see it in the title
Another nail in the P not equal NP theory, imagine if AI ends up destroying encrypting lol
I'm going to be the cynical one and offer this translation: "Our results are too important to wait for the pesky peer review, so we are dumping a whole ton of money in a fund and giving some elites a breathless first look so our buzz keeps getting jacked up."
The reality that Science is a grind and there are thousands of new results every year discovered by regular humans who use a regular human process to cross-check and review.
Treating every result from the AI like a gift from god is getting tiring.
These are not run of the mill results from AI, those are surely being posted to arxiv constantly by normal researchers without any fanfare.
These are really big results from AI, if a human had come up with them we would also be making a really big deal out of them too (maybe not as big of a deal, part of the excitement is that this was from AI and more results of this magnitude may soon follow).
Solving Kakeya in three dimensions was worth a Fields Medal.
Regular human processes, it turns out, have been a bottleneck that AI is now removing. AI is now routinely popping problems that have routinely stumped humans for decades or centuries.
Science doesn't have to be a grind with AI.
Honestly, I’m a bit scared but also a bit excited.
I’m currently reading Africa by John Reader which covers a lot of humanity storyline. From HN I get the humanity achievements that are happening right now.
The feeling digesting both daily is in order of magnitudes bigger than what any really exciting tv show did to me.
I honestly at this point don’t know anything about what could be possible and what couldnt.
I understand that this is really freightening, but also especially ETH being false gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits. I know I’m the minority, but these answers really interest me.
Nobody is solving colonization of planets in this lifetime unless we find a way to accelerate mass faster than the speed of light (and decelerate it at the other end) or extend the lifespan/radiation resistance or can preserve bodies in suspended animation for long periods. Even then, that just removed one of the many technical hurdles. We don't know which planets to target- are you going to send people to solar systems when you don't know if they are habitable? Modify people to be able to inhabit otherwise hostile worlds? How do you deal with boredom on a 1,000+ year trip?
Even with aging, our efforts are better spent increasing quality of life across humanity than helping a few rich people live to 150.
> Nobody is solving colonization of planets in this lifetime
Eh, speak for yourself.
I'm not sure how anyone can say "nobody is doing x.." or "we'll never be able to do y.." after the advances of the last few years. Not sure you understand the implications of what we're seeing here and what RSI/ASI will bring about.
Even just imagining the technological acceleration we'll experience when open source models at the Opus level can be run without insane hardware (I guess in a year or so) or if we just stopped development of AI today - is still insane.
If there's a will, there's a way. There are engineering solutions to every problem you mentioned.
"Sufficiently advanced technology is indistinguishable from magic."
This is a cop out argument.
I've worked in ML since 1994. I've been very impressed with the progress of models lately. I read "The Coming Technological Singularity" when it was first written and dedicated my career to seeing these sorts of things coming true.
That doesn't mean that their exponential improvement will change faster than light travel. Therefore, to deliver a human being to the surface of another planet (let's say Proxima Centauri at ~4 ly away) with enough resources to create a stable colony means we'll travel at less than the speed of light. Even with a solar assist, the best we can do right now is less than 1% the speed of light in an unmanned probe. People are working on alternatives, but those are not the sort of technology you just pop in a space ship and send for 400 years to a neighboring star. How do you support a population for 400 years?
As an alternative, let's just make humans like tardigrades, capable of shrivelling up and surviving high-radiation, low-moisture environments. Then we can take 10K years to get to a remote location. As a bonus, the resulting humans can survive in a much wider range of conditions so we have many more target systems to consider. Oh, but that's human engineering- are you sure that we'll just be able to genetically modify people (technologically and morally) at that level any time soon? A single person died in an early gene therapy trial and it shut down the field for 10+ years.
Basically, what you're arguing is that exponential growth in AI (not a given) will unlock huge material sciences and biological advances. While possible, it's unlikely to combine in a way that suggests "we should send people to other solar systems".
How about sending robots to a habitable planet with biological material of humans, programmed to tell us the story of earth?
Re life quality: Sure technical advancement can lead to cheaper electricity, better materials discovered, disease prevention and a lot of other stuff. A lot is hold back by complexity.
I don’t want to make a call that this will happen and that it’s a net-positive for society. I’m just saying the technical advancements is interesting and enablin.
biological material of humans? you'll have to expand your argument here. Are you reconstituting the material on the other end? If you have that technology, do you need a robot to tell a human the story of earth? Why not just upload it directly?
> gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits.
There are infinitely many more ways to kill and torture a human than to save one.
Wrong. There are infinite ways of doing both.
I honestly think they are both finite lol.
Working with academic communities lol.
I looked over the announcement for any substantial contribution, and found they're donating $5 million to a research foundation.
$5 million might sustain 5-10 PhD students for 5 years.
Maybe it's just me but I think they should contribute more in exchange for forcing the scientific community into paying for tokens to generate slop and then pouring their time into verifying or falsifying the slop.
No one "forces" anyone to verify it.
I said the same thing in another thread and someone compared it to no one forcing you to own a smart phone. I will preempt that argument by saying the community is motivated to verify these results because they are important to the field.
That means there's no crisis in mathematics after all, right?
I'm too dumb for this.. but does this mean PKI is doomed?
no
In my opinion its okay for them to work on science and math problems and release the results.
Thanks, I work at oai and will relay your approval to my line manager
"In our view, these manuscripts are best understood as a form of raw research data."
They are staying on top of the exact formulations that spread like memes. I think this one was unfortunate. It's an analogy for a new type of thing that stops attempts to figure out what thing that is.
That said, uh, let's go? Let's see if OpenAI earns some goodwill here. It probably can't go worse than the last math drop!
It is raw research data though. Human math is way more refined. LLMs are superhuman at deduction, yet cannot create their own pre-training (where most learning happens) data.
What a fucking time to be alive. I love it.
Terrible title. I was hoping to see more interesting comments on the story, haha!
Where are the papers???
To give some context:
Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.
ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.
QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.
I would guess ETH has been assumed in thousands of papers. I have several myself.
On the positive side, suddenly there's a whole bunch of open problems!
Is there any obvious path from these results to any sort of physics, chemistry, or biology breakthrough? From what I can tell: no (but I lack the physics and chemistry expertise to say anything about the QCD finding).
Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?
From the physics, not without more breakthroughs, and even then I can't imagine what that would look like. The particle physics that you can use industrially has long been understood, and the field has long been long stuck because the only things we don't understand are also insanely expensive to experimentally access. Maybe once we move up the Kardashev scale, though.
ETH sounds more like there's a path it could become important? In general, the recent complexity results suggest that a) our intuitions about that world are unreliable, and b) the truth seems to be insanely complicated. I mean, new faster multiplication algorithms in 2026, what the hell? How can we say anything about the full space of all possible algorithms?
I wonder if the best statement we can make for future AI discoveries is that, ideally, they will allow us to move up the Kardashev scale faster.
The new faster multiplication algorithms are both galactic, are they not?
It does not seem surprising to me at all that AI can find new galactic algorithms that are not of any direct use (at best they can guide us to find better solutions that are non-galactic).
What would be surprising and interesting is if AI can find a new floating point multiply-add that is 2X faster per operation than the current fastest chip implementations. That would immediately have billion+ dollar implications. The teams who already work on this have excellent approaches and invest enormous amounts into tiny improvements already; there are probably not much low-hanging fruit there (proponents of using AI for chip design improvements have impressive claims, but unclear that any of those really worked out in reality).
If we had really good understanding of physics, couldn't we find out how to make a room temperature superconductor (or prove it can't exist)?
Or is such understanding reliant in experimentation because there are "hidden constants" we can't figure out with math alone?
Solve p = np with an example algorithm maybe?
Huh? I'm not sure that makes any sense.
Thanks Claude!
Glad to help (I only really understand the implications of ETH well enough to summarize myself without assistance haha). I figured if I wanted simplified context, others might too.
was it the "bears on" that you caught? :)
Title is no doubt crafted to get ahead of the usual online backlash, but here’s the actual announcement:
> We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
1. Local smoothing and the Kakeya conjecture
2. The Exponential Time Hypothesis is false
3. Existence of quantum chromodynamics and completions of the Standard Model
"Working with PR to make us look less like cheating mind leeches".
In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."
(Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)
The fable of the scorpion and the frog is not original to Star Trek.
https://en.wikipedia.org/wiki/The_Scorpion_and_the_Frog
I wonder how many prompts for proof approaches they stole for these. Nothing on how many tokens were burned.
Proud to funnel money through the Simons Foundation, which is an AI booster itself and completely aligned with OpenAI.
They make it very hard to enjoy a single result with the business and the politics being so dirty.
OpenAi needs to stop trying to please everyone. Just release the findings and give all the credit to your models/collaborators.
They have released them, they are just also hiring mathematicians to clean them up in parallel.
We reached out to external experts and asked them to act as shepherds, organizing groups to understand the new results. They advised three different release strategies: [...] For the physics result, form an external group of collaborators and defer to them on the release of materials. [...] The group requests that all communications with any and all members of the group regarding this effort be directed to the following email account: YANGMILLS4D@gmail.com.
Well, they are also not fully sure they got it right.
Science is getting out of the reach of the majority of people. In particular, EU is out of it mostly.
You should go visit CERN some day, they have nice guided tours.
>>> We reached out to external experts and asked them to act as shepherds
Sure, if you meet some resistance to your method of undermining a field, the best go-around is to co-opt the top professionals to help pull the ladder up. Astonishing, the arrogance of whoever didn't bother with this before.