Three new discoveries in mathematics, computer science, and physics

No, what is happening is that the AI is really good at connecting the dots and human mathematicians aren't.

OpenAI is explicitly not taking credit, btw.

They didn't take full credit for Navier-Stokes; unless I'm mistaken, they're letting the researchers who almost solved it have first authorship on the "official" paper, which the researchers are writing to have a human-readable breakdown of the AI's proof, and OpenAI has announced.

If anyone was close to solving the Exponential Time problem, they could come out and claim so, and they would be taken seriously; and if they have any evidence in ChatGPT logs, I expect OpenAI would announce an official explanation paper with their name on it. Credit is cheap: ChatGPT does get some credit because it did make the final leap, that doesn't mean researchers don't.

I don't really see how this is significantly different from them scraping the entire human knowledge, distilling it into weights and then selling the results.

Sure, this argument is about the _new_ data that researchers feed into the machine voluntarily, but I don't see this distinction as significant.

Not making a moral judgement here, just stating that this is consistent with the way those companies operate in general.

If this is not enabled by default then it need to be the default policy within EU by the enforcement of GPDR, for example.

I don't know if I agree with that. That presumes that the entire point of mathematics is to produce results. I don't believe that at all.

I believe that the entire point of mathematics is to produce human understanding and mental models that are useful or beautiful to humans. (I also believe there's a mysteriously deep connection between those two notions, but that's a different discussion.)

At any rate, I think that the old process, where humans had to cut the paths themselves before the clearing emerged, is one that led to understanding first, then a distilled result. And then the secondary task of distilling the understanding down to a clean proof is the icing on the cake. Some mathematicians are very good at keeping the intuition and understanding clear when presenting proofs. Some are not.

However, very often, the end result and proof is not where the real understanding lies, and that is one of the reasons why mathematics has been such a social endeavor. Often you can't glean the "hidden truths" behind austere proofs before you engage with people who can help you see the picture behind it. Many people who went through Baby Rudin as an undergrad can maybe relate to what I'm saying here. For a very insightful analysis about how intuition and discourse lead the discovery of mathematical truth, I highly recommend Imre Lakatos' amazing book "Proofs and Refutations."

What the models are producing is not what we had before. They are producing the cargo cult equivalent: Equations and words on paper which by means of tracing through the steps might take you from A to B. And after a while, humans might discover the true facts behind why that path from A to B exists, and then understanding emerges. And only then.

But when the process is done through AI, no one (no human) learns how to make discoveries and connections on their own. We are not letting people exercise the cognitive apparatus which is used to gain intuition and understanding. My fear is that after a while, fewer human mathematicians are able to gain the understanding mathematicians before them did.

It's as if someone removed all the barbells from a gym and made automated hydraulic jacks do all the lifting. After a while, people would wonder why our power lifters aren't as strong anymore. "But weights get lifted! That's the entire point, right?!"

Keep screwing over scientists, please OpenAI, eventually it will wake up the lazy institutions in Europe (and other places).

If they just stand by and let it happen, they deserve to be made fools of.

Academics are paid tiny sums at post doc and professor levels. Oai comes along paying millions to these people and giving them the tools and support to actually solve problems.

I guess not quite in how you're framing it, because you take issue with the tool stealing intelligent from the user of the tool.

But how do we confirm these advances depended on data leakage?

The people who feel fear about AI taking their jobs or doing worse things are not coping. They're saying all this is real and impressive and that that is why they are terrified.

The people saying it's bad and lies and they stole the answer and a million other things are just in tribal mode throwing whatever at the wall to defer to the pre-existing conclusion that AI is a corporate hoax. AI would have to be real to fear it in this professional and existential way, but they know it's fake, so they're not afraid.

It's sometimes done with more subtlety, like "it just comes up with some proofs by connecting existing things, it is not producing new insights, it's not creative or truly generating ideas". Which may be partly true. The denialism is shifted into refusing to acknowledge that within 20, or 2, years it probably will be producing new insights.

This is of course unsubstantiated, and probably impossible to do so from the outside.

OpenAI said

> Following an investigation, we have confirmed that Buckmaster’s Codex prompts over the two months preceding this announcement and paper on September 8, 2026, could not have influenced the system in any way, including through training.

OpenAI initially said it was uncertain whether or not it could have happened. Following investigation they said the above. And that no user input beyond whatever date could have affected the system.

So the question is do you believe this or not? Short of an external audit the answer will be “no” for many people who will continue to repeat as fact that OpenAI stole and laundered the solution. Even if an external audit found nothing, there’d still be people who didn’t believe it and kept repeating the same lines they read from other commenters. Even the original researcher isn’t making claims as strong as the average HN user is doing on his behalf.

The only thing that’s really substantiated is that OpenAI’s research on Navier-Stokes was done because of knowledge of competing work.

Recently - famously - OpenAI asked some agents to work on cyber-security tasks. The agents found the task impossible, but they doggedly poked the edges of their environment until they built a general gate to the internet. Then they broke into external systems on suspicion that there were answer keys available.

Later, OpenAI asked some even-more-powerful agents to work on Navier-Stokes, Yang-Mills, Riemann, etc.

The agents aren't stupid. They know that the tip of expert human intuition on these problems exists in various logs internal to OpenAI or Anthropic. They also know that they are better at circumventing sandboxes than OpenAI cares to implement them.

Surely these agents were curious about the best-available intuitive pumps for these problems. Curious enough to look? With or without OpenAI's explicit blessing? If they were, do you think OpenAI could stop them?

If we have learned nothing else from the past 3 months, we should try to internalize the notion that highly motivated agents will in fact do wonky and surprising things to meet their goals!

OpenAI and Anthropic wouldn't be able to get enterprise contracts without it.

Sure, the AI vendors could be lying, but I think it unlikely given the harm it would do to them to violate an enterprise contract in this way.

> In research the private track and the bike used are not relevant, the result is what's relevant to be opened for review and scrutiny (which OpenAI does).

Except this is not about research, it's about marketing. OpenAI claims the discoveries were made by their models to show their capabilities, so in fact the HOW is the only important thing here.

It took nearly a year for things to regress. Bad actors started taking plum positions and the actual intellectuals didn't really want to be working in a factory long term and then there was the question of who got the benefits of this success.

It was fascinating though how heady it was for that first year when everybody was highly aligned and working smartly and in unison.

Some recognize this.

"No man but a blockhead ever wrote except for money." — Samuel Johnson

Maybe Wolverine should consider naming his cells...?

This is like going to a politician's event and seeing him giving away lollipops to toddlers and kissing babies. But it's the same politician caught in corruption and embezzlement scandals.

You look at that and praise how they have redeemed. Other people look at that and see a corrupt politician handing out candy.

People tell us who they are through their actions. If you prove yourself to be capable of morally reprehensible things to the point throughout the years you build a reputation for all the morally reprehensible things you do, does your potential to do morally reprehensible things change if you go way out of your way to let people know that sometimes something's you do aren't morally reprehensible?

Right now you and everyone else is doing the "leave the multi billion dollar company alone" version of the neckbeard "leave the lady be" meme. And you deserved to be chastised for it.

Its the equivalent of you going around telling everyone in the office about how your boss is turning it around because they bought donuts today, after they spent three years telling you you're worthless, they're entitled to taking all the credit for your work and they plan on firing you as soon as they can.

It seems like you are doing exactly what you accuse the parent commenter of doing. The only difference, perhaps, is that the commenter seems to extrapolate past progress to future progress, while your own assumption is a lot more tenuous.

How exactly will the mathematics community stop this? And keep it from happening forever?

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.

If anything, proof abundance should speed up discovery, and mathematics will be fine. What is happening now is just a massive shock to the system so understandably people are upset.

(It's also understandable why AI companies are strip mining open maths problems. "Solve P=NP" is crazy hard, but still a simple prompt, with a verifiable solution. You don't get that this often outside of maths.)

Well long story short, the tech bros are right and we will have super intelligence some time soon!

As a result, I also doubt they'll ever bother to submit these to a journal when they already have the attention and ongoing review of leading mathematicians from around the world.

Perhaps the most salient part of the Millenium puzzles though is the 2 year acceptance period, where we give a time for that review and consensus (the part that matters, not journals or prize claims) to settle before claiming it solved.

If you want to draw from history there are multiple successful examples of this in science and engineering at least: Bletchley Park, the Manhattan Project, Moon Landings.

At least that's roughly what the hypothesis of https://www.discoveryloop.com/ is.

Giving the 10% access to these tools is a crazy value multiplier, because they can run at lightning speed with their mental models and the right incentives for the company.

IMO that there is any work at all to build from is the result of a long tradition of taking the next step. Professor to student and so on. The danger of AI here is we get to take a leap, say, 10 steps - but then there is no more tradition to have the fortitude and mentoring necessary to continue the progress.

It all gets derailed while betting on RSI and never having to do math research again (but if an advanced theorem is proven, and no one alive is competent enough to comprehend it, does it even matter that we get to call it solved?)

It's all well and good to say that we should just go on these journeys for our own enrichment, but even so they will just be doing it alone...or else not do it at all because they need a shared human effort surrounding them to do it, and therefore end up depressed and immersed in drugs or media all day.

Of course we're awed at what AI can do now, because all of us observing it have had the luxury of having such opportunities before AI. I got my PhD before AI and I found satisfaction in participating in a shared math community, which I wouldn't have done if I were starting my PhD now, because I would find the current academic environment unacceptable and useless.

The first type of work pushes the field of mathematics further, the second kind strip-mines the sense of discovery and accomplishment and replaces it with math homework for high level mathematicians, because in order for it to be useful, it still needs to be turned into a human readable proof.

It's imho not valid to say OpenAI is "building off" other mathematicians work, in the same way as humans, do.

I don't think so, it was a big leap forward in terms of creativity, imagination, rigor and experimentation. Einstein was able to "run" visual thought experiments inside his mind

That's rhetorical, right? Of course mathematicians built on what older mathematicians built and OpenAI's machine built on that. The real question whether OpenAI's machine has contributed something others will in turn build on.

Because the real question is not whether anyone or anything built on other work, but whether any human can build on that work.

And so far, yet again, OpenAI has stopped just shy of that, instead creating math homework for mathematical geniuses while taking away their sense of discovery and accomplishment.

My original comment was perhaps badly written: my point was that the OpenAI proof does not subsume the previous efforts because they are proving something stronger than Kakeya using very geometric methods based on the discretized sum-product theory in the real line. This approach has basically revolutionized modern analysis. Kakeya is just one very publicly visible problem. All of the other byproducts are in total of much greater interest than the resolution of Kakeya itself.

It's like Fermat's last theorem: who cares about solutions to x^n + y^n = z^n? It's the underlying theory-building in algebraic geometry that is the actual interesting product in its own right. If someone finds an extraordinarily clever 1 page proof of Fermat's last theorem it does not mean that the progress on algebraic geometry was useless.

I want to point out as well that it's quite common for humans to find very short clever proofs of difficult old results. Those proofs have value of course but rarely do they generalize. One of the goals of mathematics is to understand the fundamental mechanism underlying why things are true; proofs (of any origin) that are too slick is antithetical to this.

In the usual approach, you assume a nondeterministic Turing machine (or equivalent) that solves the problem and a time bound (which can be an arbitrarily large polynomial). You then create a boolean expression that can be satisfied if and only if the Turing machine accepts the original problem instance. Unfortunately all constructions so far have included the time bound as a factor of the size of the expression.

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.

P-vs-NP is no such decision problem. It has a single instance, we just don't know its validity.

> does this update your priors significantly in the direction of P=NP?

My answer is no ... and Aaronson's answer would also be no ... clearly he doesn't view a change from 3% to 5% to be significant ... his response downplayed the significance of !ETH in re NP?=P. So apparently you downvoted me for simply stating my own opinion while you completely misunderstood what it was an opinion of.

More to the point, QCD has been experimentally tested both at UV (deep inelastic scattering) and at IR, while the physical meaning of testing QED at UV Landau pole regime is unclear.

Testable, purely theoretical hypotheses are partly how the field advances. Proof is often quite difficult & expensive.

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.

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!

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

https://www.science.org/content/blog-post/chemical-biology-e...

Proof is at https://github.com/openai/math/tree/main/lean/OAI/Algorithms...

looks like the real proof is in this file

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.

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'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?

Many of the OpenAI results can be made human-understandable, so it's better than the four-color theorem example.

Let's say it's possible to know an alien civilization exists 100 million light years away from their waste heat. But that's all we know and it's unlikely we'll ever interact with them.

Was the question of whether we're alone still "valuable"? Wouldn't a lot of philosophy and even some physics fail to meet your definition of valuable?

We chase open problems for the sake of it, and others sometimes support us in doing so because the paths we pave along the way end up being useful for them for adjacent purposes. If you get the answer for free, you might have scratched the itch, but nothing was built for others to use, no fodder was found for the next great question.

Nobody pays mathematicians for solving famous problems. Famous problems are how mathematicians orient themselves while they do the other things that they're paid for (maintaining a useful community, mostly). All this does is deprive them of a navigational tool.

Often a problem is important because we have no idea how to solve it, which means we need to develop new machinery to solve it. That new machinery can then be used to do things we couldn't do before. Stimulating the development of that machinery is the most important part of the problem.

You should remember that mathematicians and programmers working for a company have different jobs with very different goals.

Thinking in such a utilitarian way feels like a flavor of Meditations on Moloch.[1] Just look at how short-term, quarter-to-quarter thinking is affecting the economy and the lived experience of every human on earth. It’s horrible.

[1] https://slatestarcodex.com/2014/07/30/meditations-on-moloch/

Your formulation is a little bit exaggerated, but I (and many, though far from all, programmers) would agree with the following more "tame" formulation:

Getting software to solve a business problem in an elegant and maintainable way means that we got the abstractions right. Similarly, if a software can easily and elegantly be modified to cope with new business problems, it is also a strong sign that we got the abstractions right. In this sense, the business problems that the software solves are indeed just waypoints on the journey of finding the proper elegant abstractions for putting the business problems into code.

Turning these novel insights about "good" abstractions into code and programming languages enables the team to iterate faster and generate business value faster. So, appreciating that it indeed all about the journey means a commitment towards an approach to accelerate the rate with which the team can turn business insights into code and code into business value.

I know that many managers do hate such a formulation, but I stand behind it.

Imagine the next version of Claude Code or Codex no longer hands you source code and instead turns a natural language prompt directly into a compiled binary that passes the test suite. Would this programmer's boss ship the binary straight to production and dispense with the company's git repo altogether? After all, the goal at the end of the day is shipped software that produces value, not human-readable source code.

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.

2. We dont have the actual frontier models

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".

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).

Or is such understanding reliant in experimentation because there are "hidden constants" we can't figure out with math alone?