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.
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.
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.
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.
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.
Enough! I think it's a reasonably fair test of the models. Let's put it this way, I use the same process to resolve problems outside my peak expertise, and I find it works well.
I think the solution is easy enough that any competent colleague would come up with it. Some variant on it has probably even been published (hence why I'm not chomping at the bit to publish). I think it comes down to the models being extremely spiky, and the problems they're trying to solve being easily verifiable in code.
With my personal little test problem, that never made a difference.
However current frontier models got good enough at websearch and thinking through the problem, that they can finally piece together the answer from my own public writing (eg on stackoverflow). And recently someone wrote up an answer to the problem and put it on Arxiv.
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.
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.
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.
Hah. Fusion scammers are a dime a dozen. Unlike SaaS apps or finance, though, you can't sweet talk a neutron into existence, so they fizzle out pretty fast. I'm more worried about fusion dying because nobody comes up with a good enough idea than someone stealing my ideas.
Going on a tangent: I think fusion is pretty much useless this century. We already have energy generation technology that has high capital costs and low running costs: it's called nuclear fission and works fairly well.
Fission hasn't exactly taken the world by storm.
There are a few nice advantages fusion has, but they are more important once we want to reach the stars as far as I can tell; not this century on this planet.
But I'm just an armchair analyst. You probably have a much better understanding.
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'm seeing estimates of a %95-91 failure rate on attempts in generally (or, conversely, a max of ~10% success, but likely lower, since we can only go based on how many attempts they are willing to mention in their statements). E.g., from analysis in https://medium.com/@mehmet.ozel2701/openai-tried-about-4-000...
I wonder how many of those founder at the "paper" to formalization step, or at earlier or later steps (e.g., proof doesn't turn out to prove what the paper said, hits a bug in the kernel, etc.).
based on how bad these models are at one-shotting complex real-world software engineering problems -- it always has to be a conversation -- there is absolutely no way they're one-shotting complex math proofs.
Also, goes without saying, but put Sam Altman in jail, obviously.
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.
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 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.
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 $).
> 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.
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.
To say that we must have people that know how to X, because otherwise no one will be able to teach how to X, is kind of circular reasoning. Why is X important to begin with? What’s its inherent value to society?
I could address your situation in several ways and it would be a long discussion that a quick soundbite wouldn’t really satisfy. But at the very least, your argument is one of the arguments I would use to question the privatization of pure math by LLM companies. Pure math has little value to society so AI companies should consider not getting involved when they aren’t really making life better for anyone and are possibly harming young academics.
But to give a brief answer to your question directly: In my view, and I think many if they really thought about it, the point of life is to overcome. If there is an AI that spits out the answer to questions that don’t directly impact human wellbeing instead of letting humans explore the answer at our own pace, we should question whether that AI is worth keeping around or at least relying on too much.
Obviously the question is more nuanced when it comes to math, because even pure math can have applications to understanding reality and improving human wellbeing.
So the question then becomes what do we do with the AI when it can both improve our wellbeing and remove meaning from our lives? Is a life without struggle worth a life without meaning? Personally I don’t think so, so I think we have to optimize all of the benefits of AI while essentially taking a harsh stance against all the harms. Currently the main harm is to education, and that’s why I think we should question and probably even disallow its use by students still working to demonstrate their skills.
the assumption is that training llms is greater than training individuals. llms may lead to their own "breakthroughs" but so have many people without the use of llms. is the thinking that we'd be dependent on llms and so we are now forced to be only able to discover what llms find discoverable? sounds like a loss
> 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.
> 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?
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.
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.
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.
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.
That is also not how it works. The theorem says that there exist some true statements for which there is no proof. This is assuming the axioms are consistent.
Can something be both true and unprovable without being a self-referential Godel sentence? I assume most statements that aren't self-referential are independent, like the continuum hypothesis with ZFC.
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.
> 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
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.
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).
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.
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.
>We give a deterministic algorithm for exactly counting the satisfying assignments of
an arbitrary k-CNF formula [in superpolynomial but subexponential time]
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?
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).
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.
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...)
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.
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.
> 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!
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).
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.
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):
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 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.
> 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.
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.
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?
We already do have a very good understanding of QCD, it’s a very successful model with 70 years of experimental results behind it. This proof may offer new insight but most likely will just be confirming what we already knew.
Why haven't these theories had more impact on materials science? We use DFT and other greatly simplified approaches (with tremendous results) but still I suspect we are massively underperforming in terms of producing materials with useful properties.
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...)
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.
[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...
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.
Also glad to see that one of the authors of the paper stating it is also in the loop here.
2026 is weird.
(Never would mean AI will be used in future labs like a fancy calculator, but not much more changes)
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.
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.
I think the solution is easy enough that any competent colleague would come up with it. Some variant on it has probably even been published (hence why I'm not chomping at the bit to publish). I think it comes down to the models being extremely spiky, and the problems they're trying to solve being easily verifiable in code.
However current frontier models got good enough at websearch and thinking through the problem, that they can finally piece together the answer from my own public writing (eg on stackoverflow). And recently someone wrote up an answer to the problem and put it on Arxiv.
Fission hasn't exactly taken the world by storm.
There are a few nice advantages fusion has, but they are more important once we want to reach the stars as far as I can tell; not this century on this planet.
But I'm just an armchair analyst. You probably have a much better understanding.
I wonder how many of those founder at the "paper" to formalization step, or at earlier or later steps (e.g., proof doesn't turn out to prove what the paper said, hits a bug in the kernel, etc.).
Also, goes without saying, but put Sam Altman in jail, obviously.
It feels a bit like having "visited" a place because you had a layover there.
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.
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.
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.
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/
Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.
But to give a brief answer to your question directly: In my view, and I think many if they really thought about it, the point of life is to overcome. If there is an AI that spits out the answer to questions that don’t directly impact human wellbeing instead of letting humans explore the answer at our own pace, we should question whether that AI is worth keeping around or at least relying on too much.
Obviously the question is more nuanced when it comes to math, because even pure math can have applications to understanding reality and improving human wellbeing.
So the question then becomes what do we do with the AI when it can both improve our wellbeing and remove meaning from our lives? Is a life without struggle worth a life without meaning? Personally I don’t think so, so I think we have to optimize all of the benefits of AI while essentially taking a harsh stance against all the harms. Currently the main harm is to education, and that’s why I think we should question and probably even disallow its use by students still working to demonstrate their skills.
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.
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?
Turn this loose on LHC data ASAP.
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.
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.
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.
Would be funny if the most famous problem turns out be an example though
EDIT: I am wrong (it's been a while) but there is nuance.
I said any computer program fails to prove some true statements. In fact, some 'true' statements cannot be proven by any program, as others have said.
But for those statements G, you can prove 'G is true if our system is consistent', just not 'G is true'.
So when we say G is 'true', we are really just asserting the consistency axiom that our computer program is denied.
I'm saying it'd be funny if Riemann were true but not proveable.
Though I guess it's actually funnier. If Riemann were false then there would be a counterexample, so it's clearly within existing axioms.
So if one can prove that Riemann isn't proveable in our current axioms then it has to be true.
G is independent of your original system T (cannot be proven true or false).
If you create T+ with added consistency axiom, then G is true and provable.
But G is never true and unprovable in the same system.
It's a lot like is/ought or theology.
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.
absolutely insane
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.
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.
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!
Is there anything like a getting-started for working with MACE or orbmol, a 'hello world'? Like maybe simulating Allolactose binding to lac1 ?
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.
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.
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]
I've got some reading to do it seems...
> The Exponential Time Hypothesis is false
Proof is at https://github.com/openai/math/tree/main/lean/OAI/Algorithms...
looks like the real proof is in this file
>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.
(Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)
> 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..."
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!
[0]: PBS space time watcher
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
I'm guessing if you were really interested, you could email YANGMILLS4D@gmail.com (itself a hilarious e-mail address).
https://lean-lang.org/
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
We're already in it. The next few years are going to be wild.
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
As a non abelian harmonic analysis PhD I am very happy
Mathematics and computer science are turning into an empirical science
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.
https://youtu.be/3yXtXTwz4J4?si=mmePkdRBQrlO3BlD
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 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).
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.
Even with aging, our efforts are better spent increasing quality of life across humanity than helping a few rich people live to 150.
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".
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.
There are infinitely many more ways to kill and torture a human than to save one.
Depends on what counts as a different way I guess.
And on if you're applying some kind of discrete physical model of the world or not...
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.
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!
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.
On the positive side, suddenly there's a whole bunch of open problems!
Turning the question arouhnd: what would be the most impactful thing for OpenAI or Anthropic to do to show large breakthroughs?
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?
Or is such understanding reliant in experimentation because there are "hidden constants" we can't figure out with math alone?
Why haven't these theories had more impact on materials science? We use DFT and other greatly simplified approaches (with tremendous results) but still I suspect we are massively underperforming in terms of producing materials with useful properties.
> 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
(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...)
https://en.wikipedia.org/wiki/The_Scorpion_and_the_Frog
Let's throw in more of that stuff, so they wouldn't look at the financials.
Yay, shills are here with downvotes!
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.