>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 assumed all new breakthroughs in advanced complexity theory would be theoretical and non-constructive, as in "we prove that this algorithm must exist but we do not provide it".
I'm genuinely shook at today's results, specially at the fact that there _is_ a constructive algorithm.
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).
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.
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?
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.
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.
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.
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.)
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 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.
> Anyone credible [who believes P=NP with some reasonable probability?
Do you accept
- Moshe Y. Vardi [1]
- Anil Nerode [2]
- Donald Knuth [3]
to be credible? Source for the statements that they consider P=NP to be possible: [4]
And you may include me in the list of credible P=NP believers because treat myself as a credible authority, i.e. when I ask myself whether something is right, I consider myself to be much more credible than some famous scientist. ;-)
(to be very clear: the previous paragraph is rather a clear statement against this kind of "hero workship" with respect to mathematical or scientific problems. One should consider the evidence by oneself instead of believing in P!=NP because lots of computer scientists claim all the time that this is likely true, or consider P=NP now to be possible just because some famous computer scientists consider it to be possible)
> The main argument in favor of P ≠ NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration. — Moshe Y. Vardi
That sounds particularly relevant in how AI-assisted/driven mathematical research is changing the game, where previous assumptions based on human experience/effort are being overturned unexpectedly.
Note that, while Knuth believes that P = NP, he doesn't believe that it will have practical applications because the exponents and/or constants will be galactic.
> to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems.
No, it's an irrational rejection of the notion of expertise. The experts are not "heroes", they are simply people who have done vastly more study than folks like you or me. e.g., if Magnus Carlsen looks at a complex chess position and says that white is winning, I will trust his judgment far more than my own.
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.
That statement is nonsensical for the usual formal notion of "undecidable" which is about infinite sets of inputs that a machine is asked to classify as "yes" or "no" instances, also called a decision problem.
P-vs-NP is no such decision problem. It has a single instance, we just don't know its validity.
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.
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 arbitrary 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 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.
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.
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?
The context can change each time. To implement a UTM only requires 2 states and 3 symbols or the reverse.
So if you use the context not as a tape in itself, but as an io channel to read/write an infinite tape, the context for any given step can remain fixed in size, and so the step itself can be O(1).
Any basic lookup table is turing complete if you bolt an external infinite tape and read/write head onto it. But then that’s moving the goal post since the context wouldn’t be the “limitless tape” and it’s not a transformer “in the normal way” because turing completeness would be coming from external io and not from auto regressive self attention. Different topic.
The point is that the discussion is essentially entirely meaningless, as it's pretty much hard to implement any simple construct that can make deterministic decisions based on a data stream and wrap it in a loop without ending up with a UTM.
The consequence of that is that the original question GuB-42 raised is pretty pointless too. An "inert" transformer effectively has limitations, but as typically used it trivially becomes Turing complete, and so the class of problems it can solve is the exact same class of problems almost any of our other computational devices can solve.
It depends on what you say is O(1). A theoretical Turing machine has an infinite tape that can be accessed in O(1), but in real life, it would require a machine of an infinite size.
Same idea for that transformer, it needs an infinite context, and it is an infinitely big machine, but with that level of abstraction, it processes a token in O(1).
In practice, there is no such thing as O(1) on arbitrarily sized data, there is no such thing as infinity either, it is just an abstraction.
It’s not the same for transformers, especially not with the theoretical infinite context. It simply doesn’t process tokens in O(1) in your given setup.
In practice, with frontier LLMs, it’s of course more complicated, and also they don’t reveal how they do lookups, but still the theoretical limit applies regardless.
Do you mind elaborating or perhaps providing some reference as to why you think transformers should be similar to a Turing machine on this specific criteria?
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).
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...)
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
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.
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.
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.
QCD simulations (like lattice QCD) are extremely expensive (exponential in the number of constituent quarks) and usually things like material properties require bulk physics, in large system sizes. The separation of scale between a theory like QCD and something like a simple metal makes it very difficult to practically do a first principles computation.
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).
Anthropic already has a wet biolab for AI research and IIRC got some boring results out, so my guess is that non-boring results are only a few months away. I doubt they'd be related to the achievements in math, except in the sense that all AI-achieved results are related to each other because they all signify an improved capability for general research.
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.
Climate change has many clean solutions lol. Just none that anyone gives a fuck about enacting. We'll deal with it post singularity (if we even care then)
And the claimed algorithm solves the counting problem, i.e. determining the number of different solutions, which in many cases is much harder than just determining whether a solution exists or finding one of the solutions.
The parent comment is correct, if you can count the solutions, it is easy to find a solution. Assuming there is a solution, you set the first variable to either true or false and then count the solutions. If it is zero, you made the wrong choice and the first variable must be set to the opposite value. Repeat for the remaining variables until you have fixed all of them and found a solution. This takes n invocations of the counting algorithm and is probably what the parent comment means with O(n) even though this is not correct as it is O(n times whatever the counting algorithm runtime is) and not just O(n).
But the parent comment also missed my point. They have an algorithm for the counting problem which also provides a subexponential algorithm for the decision and search problem. But they could have refuted the exponential time hypothesis with a subexponential search or decisions algorithm which would not [easily] provide a subexponential algorithm for the counting problem.
I never believed the EHT, but I can't prove I didn't and I only didn't believe it for the shallow reason that I didn't feel that there was a strong argument that it should actually be exponential, since there are so many almost-exponential things that it could equally well be.
For folks who believe that LLM-based AI only operates with the "convex hull" of its training data, I'm curious how they interpret this result.
To summarize, a lot has been discussed recently about whether LLMs are just really, really good at finding connections between existing information (i.e. the world's smartest and best search engine), or whether they are capable of discovering truly unique, novel ideas. Because many of the very impressive recent AI math results were often just tying together ideas and theories that were already known (and I don't mean to minimize those accomplishments), it was surmised (and, just speaking for myself, hoped) that LLMs operated within this "convex hull", and thus there would always be at least some axis where LLM-based AI wouldn't surpass humans.
Now, this idea of the convex hull of training data is not very clearly defined, and people can and have argued about what it means. But in this case, OpenAI disproved a conjecture that was widely assumed to be true. This was a genuinely surprising result that many experts were shocked by. I'm not a mathematician so I'm not qualified to determine if this was a "truly unique" realization, but a really surprising result seems like it would be a candidate for AI solving a problem in a manner "outside the convex hull".
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 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 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
Well in the books the nano particles come into your home and show you dream virtual realities of your dead relatives and then offer to transform your mentation into some god like entity in the now ubiquitous compute stratum.
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".
What my phone will be able to remember what I use it for at different times and places with better than 50% accuracy? The real tech miracle today is mRNA and solar panels. Not sure why you think being able to generate theory will help u leash the material goods needed to put people on Mars. Even if we get the singularity, would it regard human exploration as worth while compared to to machine exploration?
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.
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'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.).
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.
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.
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. 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.
Nah I agree with you. Not this century, not this planet. Fusion rockets are really efficient, though ;)
I just find it to be an interesting problem! My hope is it happens before I die. My rule of thumb is anyone who says energy on the grid in < 10 years is a blatant liar.
I agree that you can build a facility to put energy on the grid in under a decade, but the probability that it works is going to be tiny, and anyone giving you the billion or so it would take would be disappointed to say the least.
Therefore, most credible folks are talking a 2-3 facility staged approach. And probably 5 years per facility to build, and 5 years per to test.
Also, remember that even with huge cash pools, you have supply chain problems. Some of the nonlinear optics takes a fixed amount of time to fabricate.
> I agree that you can build a facility to put energy on the grid in under a decade, but the probability that it works is going to be tiny, and anyone giving you the billion or so it would take would be disappointed to say the least.
A giant water tank that you detonate H-bombs in should work, and even be energy positive. :)
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.
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 think you should realize that we already do that. In fact, that's one of the reasons computers were invented in the first place! It turns out that science is about unknown unknowns, and by definition simulations can't answer that.
But agents are really useful for doing some of the really boring stuff, like convergence checks or setting up meshes or whatever.
If you're talking about things at the equivalent level of say, finding out if positrons maintain symmetry with electrons for their magnetic moment, then yeah - I hear you. Or if you wanted to research some unknown behaviour of hot plasma at 15 million C, you can't do that in a simulation.
..or can we?
Humour me for a moment. Maybe simulating to the degree that was necessary for cutting edge work was not possible earlier, when humans had to program each and every thing in the system that could affect the result. But with current gen petaflop-on-your-desk level compute + swarms of agents that can work tirelessly to account for every known thing to humanity, maybe the threshold has been pushed far enough to produce some more meaningful discovery work in simulation?
Respectfully, finding out how scientific computation is used in fusion research is a Google click away. And yes, we can simulate these conditions.
Let's put it this way - a first year graduate student can ignite a fusion target in simulations. It took thousands of scientists the better part of three decades to do that in experiments. I think agents could do the former really efficiently and at scale, but I'm skeptical that agents could do the latter. Oh, and btw, fusion simulations of the kind you're talking are in exa-zettaflop territory ;)
I guess I'd say, let's have a swarm of agents build a metro line first and see how well that goes. Obviously, given that we have embodied intelligence (sometimes) that's capable of doing this work, it would be silly of me to say that it's impossible for it to happen in silicon. But given where the SOTA is right now I'm skeptical of this being true in the near future. But hey! If you'd asked me the same question about GPT in 2019 I'd have said I would be skeptical there too. So who knows.
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.
Doing AI assisted enggineering to explore a space and then using classical methods to validate a point design for licensing is a good approach. In licensing the vendor and the regulator check things and then also verify many times during startup. The overall safety remains good.
Up front: I totally believe you are doing your level best so please don't take this as trying to take you down, I just want to understand. In the hypothetical case that in 10 years time it turns out that that design was flawed who will be on the hook? You, the AI vendor, the regulator, the device vendor?
Nuclear liability is always 100% held by the owner/operator by law in most countries. All manufacturers of equipment and components demand and are given indemnity.
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".
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.
> 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.
The money would be better spent trying to breed a flying unicorn, because that's remotely possible.
"Cancer" is not "a" disease. It's a symptomatic description: cells in your body multiplying beyond a limit that is healthy for you.
What you are asking for would be like asking for a universal cure for headaches. What if your headache is caused by a tumor? Or a nailgun injury? Or psychotic hallucination?
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.
> 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!
And a perfect (or near-perfect) spec is actual source code. And that's where I see myself with LLMs. They are great about generating source code. They are terrible at writing well-specified software, and so they need a lot of context: docs, plans, existing source code, existing templates or examples. And even then, they still need a lot of guidance.
Then what are you doing different if this hasn't been your experience? You're just throwing AI at things you don't know how to do, and it does those things successfully?
Except the results are AI slop that no one actually understands and make absolutely zero difference to the world. I.e. research with zero practical applications.
Yay, the slop generator made 2000 pages of Lean which no one understands... to solve a problem with no practical results for human-kind.
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.
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.
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).
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).
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!
Heh. Unfortunately computer science is really the weird one out where open source is the norm. A lot of computational chemistry knowledge is mainly transferred in research labs and through college classes, and you have to pay for a lot of the tools. But hopefully I can at least point you in some useful directions. Here's a bit of an overview of the CC workflow and free options (note I haven't tried all the free options):
1. The first thing you need to do is start with some atomic structure. You can often find pre-existing pieces (through websites like pubchem), but you need to cobble them together. I use chemcraft and gaussview for this, but those are paid options. I've heard good things about the FOSS program Avogadro though.
2. Once you've built the structure of interest, you want to calculate properties of it. Note that we only infrequently simulate the atom's movement, since that's quite expensive. There's a lot of techniques that let us get the most bang for buck, like calculating transition states, or states of lowest energy which we call intermediates, or running something called an IRC, which follows the molecule to its lowest energy point (note we pretty much always think about these things in terms of energy). The traditional programs I use for this are Gaussian and ORCA (ORCA is free but you do have to create an account on their forum). Newer methods include MACE and orbmol, which are pretty much only used for trajectories (e.g. running through time) as far as I know. This newer crop of programs are much more FOSS, but they aren't as rigorous so my lab uses them only as a "rough draft". I pulled in one of my current notebooks if you're curious to try it out: https://github.com/smj-edison/qc-tools/blob/main/examples/ml...
3. Interpreting results/debugging problems. This is currently done in my workflow using a python script to extract values from the job runfiles (oh yeah, did I mention the terrible DX? A lot of this stuff is done by hand-writing a job text file and then using regex to parse the logfile). We do a lot of analysis based on relative energies (this is something called transition state theory of reactions).
I definitely glossed over a lot of details, but this is long enough as is. Definitely let me know if you have questions! If you do want a "hello world" though, I'd definitely check out the notebook I linked to. You'll have to create a file called "start.xyz" with some sort of chemistry program (or use AI to pull one for you), and you can run the system through time.
To add to this I think there are “reductionist sciences” and “systems sciences”. Reductionist sciences are ones, like physics where there are lots of copies of a thing with near identical properties at different scales (the periodic table would be considered an incredibly short list and that seems to be one of the longest list of “different things” in physics).
Systems sciences are ones where each scale has lots of things with unique properties. If you think of “biological things between 2-4m” the amount of unique objects with unique properties and unique interactions is absolutely mind boggling. These sciences generally do not lend themselves to clean highly parsimonious models. The best you can do is coarse-grain a thing as a system (some very roughly defined set of interactions between different things) with certain behavior while accepting that you are losing enormous amounts of predictive power.
I suspect AI will remain worse at systems sciences for a long time because the “possible space” of models for a particular system is much larger than in the reductionist sciences and navigating it takes a human taste and instinct that LLMs don’t seem to have (yet).
Riffing on your idea and the last paragraph.. That is quite the load-bearing "yet", it seems a lot of future potential depends on LLMs acquiring taste and instinct to cut through an immense possibility space to hone in on "interesting" areas of problems, theories, questions and solutions.
When I imagine how a musician acquires good taste, it often involves a master-apprentice relationship, or emulating role models at first, so it might require finding human beings at the top of their field getting personally involved.
Good taste also is built up from consuming/enjoying tons of material (art, music, literature, etc) - but not just a large quantity of any quality, but curated selections of high quality soure material, which again would likely require human beings who already have some understanding of quality to provide it.
Another aspect of developing taste is lots of time spent alone, just the musician with the instrument, not only practicing exercises but playing, exploring, getting to know the instrument intimately, the full range of sounds it can produce, composing, listening, and expressing oneself. Surely mathematicians go through similar regular meditations and introspective time with the medium, discovering what one likes and enjoys. I imagine language models can be taught to spend time alone and "train itself" without any given goals or guidance, perhaps deciding on its own reward functions.
One other thought, I think another aspect of biological sciences is just how much effort it takes to do research. Hypotheses are a dime a dozen, but actually verifying these is quite difficult. It takes a lot of patience, time, attention to details, and even then it often fails to replicate.
I don't think LLMs will just magically be able to connect all the dots, because "garbage in, garbage out" still applies. And there's just too many invisible intermediate states, there is literally just not enough information to fully constrain a model. Hence my interest in scaling computational chemistry to get these intermediates :)
Your "reductionist sciences" I would instead phrase as "exact sciences", where the input variations are known (even the most generous particle theory only has a 'few' fundamental particle types, such as up-quarks and photons) and their interactions are either well-known (Fermi Principle is apparently inviolable), or reasonably expected to be known someday (exact relationship of gravitons, IF they exist, with other particles).
The "inexact sciences" use either truly infinite, or practically uncountable, inputs - such as the possible shapes of a single protein, which affects the outputs.
You could criticize this grouping by saying the number of particles in the Universe is practically uncountable, but sufficiently accurate statistical models can be built (a galaxy exerts <this much> gravitational force). We can't do the same with the "inexact sciences": we can't predict how a mole of protein will react with a mole of enzymes, without physical testing. OK, maybe for some very low levels of complication this could be reliably predicted, but in general it may be a !P problem.
> 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
the NS problem is not physics and not predictive, it happens at scales smaller than molecules, so it doesn't occur in real fluids
also the "NS millennium prize problem" hasn't been solved until OpenAI actually gets it published in a Math journal, and I would say it has been ruined until they do
> the NS problem is not physics and not predictive, it happens at scales smaller than molecules, so it doesn't occur in real fluids
1. How is it not physics? It's goal (as laymen me crudely understands it) is to determine if a singularity could be physically realized in a perfect liquid.
2. Why does it happen at scales smaller than molecules? Unless it requires properties that a liquid made of particles sized like real atoms cannot achieve, I don't understand.
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.
Oh I'm curious! Searched through your post history but couldn't find the link. But the phrase "lights out" did turn up a general term for this, also called "dark factory", which requires minimal human intervention to operate. Our cyberpunk future is self-actualizing.
Brilliant, thank you. I see, aside from fully automated labs, they work with various kinds of industry-specific drones and robotics. The phrase "modular laboratories" catches my eye, and reminds me I recently learned about your business called the Modular Company, doing tech due diligence. From the wide range of industries listed, I imagine there's a lot that can be learned from the clients, and the businesses being assessed, their niche knowledge and experience.
"Dark factory" is the perfect term, that the labs can operate without lights, heating, and other human needs - I like how it has a cyberpunk feel. Another aspect I find attractive is that it treats a laboratory as an automated system, like a program that is implemented in hardware and software, probably including some kind of AI as part of the process.
How about a "lights-out company", or dark business. (Well the latter sounds too suspicious, haha.) Surely there are people working on automated business systems, modelling it as a machine or program with inputs and outputs, self-modifying based on reward functions, adaptive and evolutionary.
Now that the production of software is getting largely automated, mechanized and industrialized, I've heard people say that the differentiating factor for successful software is the domain-specific knowledge of the businesses where it's being applied. Perhaps that's becoming true of mathematics too, where pure maths - like software for developers - is dominated by automation, the cost of production so low that humans cannot compete with the machines. The term "software factory" is pretty old, but it's taken on a new meaning where the role of humans is mainly overseeing a largely automated system, often with language models working day and night without requiring much oversight.
> From the wide range of industries listed, I imagine there's a lot that can be learned from the clients, and the businesses being assessed, their niche knowledge and experience.
Yes, I always tell people we have the best paying school...
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.
"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.
> 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!
> 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.
I disagree completely. Hawking Radiation has never been proven to exist, which is why Stephen did not earn a Nobel for it.
Merely providing a mathematical possible solution that satisfies all known behaviors would be a tremendous step forward.
We don't actually "know" anything about EM, by example, except how to describe its behaviors in all physically observable instances. Radios work reliably with current. Supernovae behave comprehensibly. Photons carry predictable energies. However, long before we knew there were supernovae, or even photons, Maxwell said pretty much all we know about EM behavior.
I don't think study of fundamental particles is hamstrung by physical research. Higgs bosons were theorized long before they could be verified (very recently) to exist. Hawking Radiation has not yet been observed.
Testable, purely theoretical hypotheses are partly how the field advances. Proof is often quite difficult & expensive.
546 comments
[ 1.4 ms ] story [ 24.2 ms ] thread> The Exponential Time Hypothesis is false
>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 actual exponent OpenAI found was x/sqrt(x).
I'm genuinely shook at today's results, specially at the fact that there _is_ a constructive algorithm.
looks like the real proof is in this file
Proof is at https://github.com/openai/math/tree/main/lean/OAI/Algorithms...
>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.
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.
Would be funny if the most famous problem turns out be an example though
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.
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.
Do you accept
- Moshe Y. Vardi [1]
- Anil Nerode [2]
- Donald Knuth [3]
to be credible? Source for the statements that they consider P=NP to be possible: [4]
And you may include me in the list of credible P=NP believers because treat myself as a credible authority, i.e. when I ask myself whether something is right, I consider myself to be much more credible than some famous scientist. ;-)
(to be very clear: the previous paragraph is rather a clear statement against this kind of "hero workship" with respect to mathematical or scientific problems. One should consider the evidence by oneself instead of believing in P!=NP because lots of computer scientists claim all the time that this is likely true, or consider P=NP now to be possible just because some famous computer scientists consider it to be possible)
---
[1] https://en.wikipedia.org/w/index.php?title=Moshe_Vardi&oldid...
[2] https://en.wikipedia.org/w/index.php?title=Anil_Nerode&oldid...
[3] https://en.wikipedia.org/w/index.php?title=Donald_Knuth&oldi...
[4] https://en.wikipedia.org/w/index.php?title=P_versus_NP_probl...
That sounds particularly relevant in how AI-assisted/driven mathematical research is changing the game, where previous assumptions based on human experience/effort are being overturned unexpectedly.
> to be very clear: the previous paragraph is rather a clear statement against this kind of "hero worship" with respect to mathematical or scientific problems.
No, it's an irrational rejection of the notion of expertise. The experts are not "heroes", they are simply people who have done vastly more study than folks like you or me. e.g., if Magnus Carlsen looks at a complex chess position and says that white is winning, I will trust his judgment far more than my own.
P-vs-NP is no such decision problem. It has a single instance, we just don't know its validity.
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.
In the usual approach, you assume a nondeterministic Turing machine (or equivalent) that solves the problem and a time bound (which can be an arbitrary 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.
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.
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.
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.
So if you use the context not as a tape in itself, but as an io channel to read/write an infinite tape, the context for any given step can remain fixed in size, and so the step itself can be O(1).
The consequence of that is that the original question GuB-42 raised is pretty pointless too. An "inert" transformer effectively has limitations, but as typically used it trivially becomes Turing complete, and so the class of problems it can solve is the exact same class of problems almost any of our other computational devices can solve.
Same idea for that transformer, it needs an infinite context, and it is an infinitely big machine, but with that level of abstraction, it processes a token in O(1).
In practice, there is no such thing as O(1) on arbitrarily sized data, there is no such thing as infinity either, it is just an abstraction.
It’s not the same for transformers, especially not with the theoretical infinite context. It simply doesn’t process tokens in O(1) in your given setup.
In practice, with frontier LLMs, it’s of course more complicated, and also they don’t reveal how they do lookups, but still the theoretical limit applies regardless.
Do you mind elaborating or perhaps providing some reference as to why you think transformers should be similar to a Turing machine on this specific criteria?
(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
> 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
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.
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).
[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...
Turns out the 'opne' in open AI wasn't referring to open source, but open problems!
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.
really...
You meant deepseek?
The number of valid permutations of 10 things is 10!. Reviewing all 10! Things is clearly exponential or even worse than exponential.
We can get the count of 10! pretty easily, far easier than trying all combinations manually.
But the parent comment also missed my point. They have an algorithm for the counting problem which also provides a subexponential algorithm for the decision and search problem. But they could have refuted the exponential time hypothesis with a subexponential search or decisions algorithm which would not [easily] provide a subexponential algorithm for the counting problem.
To summarize, a lot has been discussed recently about whether LLMs are just really, really good at finding connections between existing information (i.e. the world's smartest and best search engine), or whether they are capable of discovering truly unique, novel ideas. Because many of the very impressive recent AI math results were often just tying together ideas and theories that were already known (and I don't mean to minimize those accomplishments), it was surmised (and, just speaking for myself, hoped) that LLMs operated within this "convex hull", and thus there would always be at least some axis where LLM-based AI wouldn't surpass humans.
Now, this idea of the convex hull of training data is not very clearly defined, and people can and have argued about what it means. But in this case, OpenAI disproved a conjecture that was widely assumed to be true. This was a genuinely surprising result that many experts were shocked by. I'm not a mathematician so I'm not qualified to determine if this was a "truly unique" realization, but a really surprising result seems like it would be a candidate for AI solving a problem in a manner "outside the convex hull".
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.
We're already in it. The next few years are going to be wild.
Be careful what you wish for. You just might get it.
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).
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.
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".
(Never would mean AI will be used in future labs like a fancy calculator, but not much more changes)
Also, goes without saying, but put Sam Altman in jail, obviously.
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.).
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.
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 just find it to be an interesting problem! My hope is it happens before I die. My rule of thumb is anyone who says energy on the grid in < 10 years is a blatant liar.
It's just probably not economically feasible, ie it would be a waste of resources to rush fusion so much.
> I just find it to be an interesting problem!
Ha, that's why I was working Zero Knowledge Proofs for quite a while. I don't even hold any cryptocurrency or other crypto assets.
Therefore, most credible folks are talking a 2-3 facility staged approach. And probably 5 years per facility to build, and 5 years per to test.
Also, remember that even with huge cash pools, you have supply chain problems. Some of the nonlinear optics takes a fixed amount of time to fabricate.
A giant water tank that you detonate H-bombs in should work, and even be energy positive. :)
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.
But agents are really useful for doing some of the really boring stuff, like convergence checks or setting up meshes or whatever.
..or can we?
Humour me for a moment. Maybe simulating to the degree that was necessary for cutting edge work was not possible earlier, when humans had to program each and every thing in the system that could affect the result. But with current gen petaflop-on-your-desk level compute + swarms of agents that can work tirelessly to account for every known thing to humanity, maybe the threshold has been pushed far enough to produce some more meaningful discovery work in simulation?
Let's put it this way - a first year graduate student can ignite a fusion target in simulations. It took thousands of scientists the better part of three decades to do that in experiments. I think agents could do the former really efficiently and at scale, but I'm skeptical that agents could do the latter. Oh, and btw, fusion simulations of the kind you're talking are in exa-zettaflop territory ;)
Haha, thanks for humouring me!
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.
> I'm kinda hoping they are investing just as much into that type of research.
The money would be better spent trying to breed a flying unicorn, because that's remotely possible.
"Cancer" is not "a" disease. It's a symptomatic description: cells in your body multiplying beyond a limit that is healthy for you.
What you are asking for would be like asking for a universal cure for headaches. What if your headache is caused by a tumor? Or a nailgun injury? Or psychotic hallucination?
Turn this loose on LHC data ASAP.
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.
And a perfect (or near-perfect) spec is actual source code. And that's where I see myself with LLMs. They are great about generating source code. They are terrible at writing well-specified software, and so they need a lot of context: docs, plans, existing source code, existing templates or examples. And even then, they still need a lot of guidance.
2. We dont have the actual frontier models
Yay, the slop generator made 2000 pages of Lean which no one understands... to solve a problem with no practical results for human-kind.
Great, well done OpenAI.
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.
https://www.science.org/content/blog-post/chemical-biology-e...
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 ?
1. The first thing you need to do is start with some atomic structure. You can often find pre-existing pieces (through websites like pubchem), but you need to cobble them together. I use chemcraft and gaussview for this, but those are paid options. I've heard good things about the FOSS program Avogadro though.
2. Once you've built the structure of interest, you want to calculate properties of it. Note that we only infrequently simulate the atom's movement, since that's quite expensive. There's a lot of techniques that let us get the most bang for buck, like calculating transition states, or states of lowest energy which we call intermediates, or running something called an IRC, which follows the molecule to its lowest energy point (note we pretty much always think about these things in terms of energy). The traditional programs I use for this are Gaussian and ORCA (ORCA is free but you do have to create an account on their forum). Newer methods include MACE and orbmol, which are pretty much only used for trajectories (e.g. running through time) as far as I know. This newer crop of programs are much more FOSS, but they aren't as rigorous so my lab uses them only as a "rough draft". I pulled in one of my current notebooks if you're curious to try it out: https://github.com/smj-edison/qc-tools/blob/main/examples/ml...
3. Interpreting results/debugging problems. This is currently done in my workflow using a python script to extract values from the job runfiles (oh yeah, did I mention the terrible DX? A lot of this stuff is done by hand-writing a job text file and then using regex to parse the logfile). We do a lot of analysis based on relative energies (this is something called transition state theory of reactions).
I definitely glossed over a lot of details, but this is long enough as is. Definitely let me know if you have questions! If you do want a "hello world" though, I'd definitely check out the notebook I linked to. You'll have to create a file called "start.xyz" with some sort of chemistry program (or use AI to pull one for you), and you can run the system through time.
Systems sciences are ones where each scale has lots of things with unique properties. If you think of “biological things between 2-4m” the amount of unique objects with unique properties and unique interactions is absolutely mind boggling. These sciences generally do not lend themselves to clean highly parsimonious models. The best you can do is coarse-grain a thing as a system (some very roughly defined set of interactions between different things) with certain behavior while accepting that you are losing enormous amounts of predictive power.
I suspect AI will remain worse at systems sciences for a long time because the “possible space” of models for a particular system is much larger than in the reductionist sciences and navigating it takes a human taste and instinct that LLMs don’t seem to have (yet).
When I imagine how a musician acquires good taste, it often involves a master-apprentice relationship, or emulating role models at first, so it might require finding human beings at the top of their field getting personally involved.
Good taste also is built up from consuming/enjoying tons of material (art, music, literature, etc) - but not just a large quantity of any quality, but curated selections of high quality soure material, which again would likely require human beings who already have some understanding of quality to provide it.
Another aspect of developing taste is lots of time spent alone, just the musician with the instrument, not only practicing exercises but playing, exploring, getting to know the instrument intimately, the full range of sounds it can produce, composing, listening, and expressing oneself. Surely mathematicians go through similar regular meditations and introspective time with the medium, discovering what one likes and enjoys. I imagine language models can be taught to spend time alone and "train itself" without any given goals or guidance, perhaps deciding on its own reward functions.
I don't think LLMs will just magically be able to connect all the dots, because "garbage in, garbage out" still applies. And there's just too many invisible intermediate states, there is literally just not enough information to fully constrain a model. Hence my interest in scaling computational chemistry to get these intermediates :)
Your "reductionist sciences" I would instead phrase as "exact sciences", where the input variations are known (even the most generous particle theory only has a 'few' fundamental particle types, such as up-quarks and photons) and their interactions are either well-known (Fermi Principle is apparently inviolable), or reasonably expected to be known someday (exact relationship of gravitons, IF they exist, with other particles).
The "inexact sciences" use either truly infinite, or practically uncountable, inputs - such as the possible shapes of a single protein, which affects the outputs.
You could criticize this grouping by saying the number of particles in the Universe is practically uncountable, but sufficiently accurate statistical models can be built (a galaxy exerts <this much> gravitational force). We can't do the same with the "inexact sciences": we can't predict how a mole of protein will react with a mole of enzymes, without physical testing. OK, maybe for some very low levels of complication this could be reliably predicted, but in general it may be a !P problem.
the NS problem is not physics and not predictive, it happens at scales smaller than molecules, so it doesn't occur in real fluids
also the "NS millennium prize problem" hasn't been solved until OpenAI actually gets it published in a Math journal, and I would say it has been ruined until they do
1. How is it not physics? It's goal (as laymen me crudely understands it) is to determine if a singularity could be physically realized in a perfect liquid.
2. Why does it happen at scales smaller than molecules? Unless it requires properties that a liquid made of particles sized like real atoms cannot achieve, I don't understand.
TY.
You've got to procure, use and dispose of a ton of equipment. And it all costs a good amount of money.
https://en.wikipedia.org/wiki/Lights_out_(manufacturing)
https://globalmecho.com/lights-out-lab/
Crazy stuff.
"Dark factory" is the perfect term, that the labs can operate without lights, heating, and other human needs - I like how it has a cyberpunk feel. Another aspect I find attractive is that it treats a laboratory as an automated system, like a program that is implemented in hardware and software, probably including some kind of AI as part of the process.
How about a "lights-out company", or dark business. (Well the latter sounds too suspicious, haha.) Surely there are people working on automated business systems, modelling it as a machine or program with inputs and outputs, self-modifying based on reward functions, adaptive and evolutionary.
Now that the production of software is getting largely automated, mechanized and industrialized, I've heard people say that the differentiating factor for successful software is the domain-specific knowledge of the businesses where it's being applied. Perhaps that's becoming true of mathematics too, where pure maths - like software for developers - is dominated by automation, the cost of production so low that humans cannot compete with the machines. The term "software factory" is pretty old, but it's taken on a new meaning where the role of humans is mainly overseeing a largely automated system, often with language models working day and night without requiring much oversight.
Yes, I always tell people we have the best paying school...
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.
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!
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!
absolutely insane
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.
(only half tongue-in-cheek)
Merely providing a mathematical possible solution that satisfies all known behaviors would be a tremendous step forward.
We don't actually "know" anything about EM, by example, except how to describe its behaviors in all physically observable instances. Radios work reliably with current. Supernovae behave comprehensibly. Photons carry predictable energies. However, long before we knew there were supernovae, or even photons, Maxwell said pretty much all we know about EM behavior.
Testable, purely theoretical hypotheses are partly how the field advances. Proof is often quite difficult & expensive.