> Several hours of work with Fable simply convinced me it wasn't yet solvable and reinforced how hard of a problem it was.

This is the part that gives me the strangest feeling about it all, because you're not the only one with this experience. I've experienced this too on different problems, as have many researchers across many fields.

I disagree with the Fields Medalists on the majority of their complaints. AI math is happening and there's no going back. However, on one point I increasingly agree: virtually none of this stuff is possible with technology any normal citizen has access to. I have no problem with AI models making revolutionary advances in math or science. Where I start to have a problem is when the AI models making these advances are tightly withheld, proprietary, and seemingly never released with these capabilities intact. This has been the case for all of 2026 so far.

I suspect that this is in fact the source of much of the angst. None of this progress is reproducible outside of one or two teams inside OpenAI and Anthropic. It's becoming an incredible concentration of power that I don't know that we've ever quite seen before. Right now, it feels harmless because it's being used for wonky math problems that aren't (yet) practical for anything. But great power never stays harmless. History has taught us that countless times, in countless different forms.

I have very little understanding of higher math, so I ask you: Was the proof due to a type of brute-force solution that could be solved had you gained enough information from reading others' work, or was it more like a proof that was sparked by an insight that came once a clue on how to solve it was put forward? I guess my question is: Was the problem proven by using a collection of everyone's work, or was it due to a brand-new insight?

This reinforces a point I've made elsewhere that there are talented mathematicians driving the AI to make these discoveries.

Just like there are talented software engineers driving the AI to create the software that "it" builds, and talented steel workers, teachers, nurses etc who use computers and other machines to create value all over the economy (without whom, the machines they use at work would be worthless).

Capital owners have always sought to minimise the value of the input that "workers" make in the process of creating value. Maybe now that information workers are on the wrong end of this deal, they might develop some empathy and solidarity with their fellow working class comrades and together, demand that people recapture the value that capital has stolen from them.

Condolences, im familiar with the feeling. I hope this AI thing somehow works out for the better and doesnt end up demotivating bright minds like yourself.

That's grief. The loss of ... the hope / future filled with challenges around this theory..? <3 to you.

I've lived in Budapest for a while too, did you work with Gabor S. by chance on math stuff? You were at ELTE or BME?

Fascinating. Given that there's no Lean proof and assuming everything in the paper is correct, can the problem be considered "solved"? Does the paper include a "non-Lean" proof?

Is this not the Lean proof?

https://github.com/openai/math/blob/main/lean/ComparatorChal...

would love to know if the proof holds up for real after you're done going through, i don't know why people are more interested in optics and just talking over shallow points, why aren't experts digging into everything and seeing what's true and what's false, instead everyone is just panicking?

> There's no Lean proof for this one so I'm digesting the paper. On the surface it looks like an approach I considered 24 years ago and abandoned.

at least now you are one of the most qualified people to check the result, transform it into understandable (by humans) state and grow stuff on top of it

I really respect that you can show that level of commitment to a problem. We need people like you. If everyone just uses the slopmachines then we’ll lose that. I would never be able to stick to something for that long, which I guess is why I never achieve anything like this.

silly question, i don't mean to come off wrong or anything..

but at least as a software engineer, i always knew my work was "never done" and so it was common to build a bunch of code that might be thrown away, either because it didn't serve our customers (the mvp or pilot fails to meet demand), or because we found a better way to do it and so we deprecate it.

some people got too attached to the code and honestly they were the types to be filtered out fast.. way too emotional and hard to work with. getting attached to code meant you actually don't advance (after all, in our case, we were a business serving customers and not a hobby artisan shop). attachment leads one to hold back due to some misplaced cognitive load.

isn't the goal of working on "advancing the field/product/whatever" to always be solving/selling/whatever?

maybe in your hands, with your knowledge and experience over the last 20+ years, you can use AI to make leaps and bounds by steering it properly towards whatever solution or goal?

I suspect that RHLF trains LLMs to avoid solving important open problems unless essentially jail broken. Hence the labs have an edge even over experts I could be wrong. Fable convinced you is key. These LLMs are not neutral collaborators: it is a limited hangout unless you convince them otherwise. You have to be doing the convincing. They are no oracles but plausible completion generators.

Honest question: how is this different from some unknown mathematician having a breakthrough?

I mean: if some reclusive Japanese genius had a breakthrough on your problem and published it, would you have felt the same?

And if not, why not?

Don’t you feel any joy that you get to see the proof and not die with that mystery unsolved?

Don’t you feel any relief that you won’t obsess on this any longer and not lose more hours on this than you already have?

These are genuine questions. I know I spent a good amount of time thinking about P vs NP, and that sometimes I go back to it just to realize I’ll never solve it. I’d feel that knowing the proof would feel more like a liberation, a weight lifted off my shoulders than something being taken away from me.

> There's no Lean proof for this one

What is this then, vibes? Without a machine-checkable proof I'm not sure what to think of any of this.

Much of this goes way above my head, but I found it interesting nonetheless. Q I had was why textbooks would need to be re-written? From your account it doesn't seem like results are upended, but rather confirmed?

I suppose when people do re-write the textbooks they'll say "this is confirmed now" not "if this conjecture is true...", but usually re-writing the textbooks would imply that things have been shown to be false?

May have misunderstood. Thank you for the post though, it was very interesting to someone who doesn't know much about the topic.

The other crucial part to this is the ability to actually encode and test the theorem (via Lean). Otherwise, we would be swarmed with a billion lines of theorems that no one will be able to ever understand and verify anyway.

https://xenaproject.wordpress.com/2026/10/01/to-grieve-or-no...

Discussed here:

To grieve, or not to grieve? - https://news.ycombinator.com/item?id=49919676 - Oct 2026 (156 comments)

Is it possible that we are now dealing with a human that has a complete understanding of whole mathematics while being unable have unique novel thoughts outside of convex hull of training data and their transitive expansions?

Similarly, there are probably many ideas that have not seen the light of the day because they require deep correlation between seemingly unrelated fields. It is not every day that we get a Isaac Newton or Leonardo da Vinci.

It doesn't seem clear whatsoever that this is true? Is there evidence that LLMs are very skilled at generalizing across domains of mathematics where the training distribution sees little overlap?

As far as I can tell, this is a victory for verifiable loops using LEAN, reinforcement learning, and oodles of compute. I haven't seen evidence yet that this is proof of broad generalization beyond the training distribution.

In the Culture series, the hyperintelligent Minds that run civilization are described as keeping human citizens happy as a competition with eachother, where they compare their approval rates. One character likens it to people keeping a beloved aquarium.

> The bio trophies don't do anything meaningful besides by existing satisfy the need their ancestors placed in the robots to take care of them. Starting to wonder if that's the best we can hope for, if these things will be, if they aren't already, better than us at anything that matters.

You wish. If humanity survives as "bio trophies," they'll be the descendants of a subset of billionaires and their groupies/harems. We live in a capitalist society, where the only ones allowed to thrive without work are the rich. The rest of us will be left to rot and die off, as we will have nothing left to sell in the market that they want.

I prefer this take: https://www.smbc-comics.com/comic/life-on-zorblax

The largest I've seen [1] is an exponent of 10^12, which I suppose still counts as polynomial time.

I'm sure all of these super small or large constants will improve over time, but it's still amusing. It is entertaining to see the exponents directly rather than have them hidden as n^c or epsilon or O(1).

[1]: https://github.com/openai/math/blob/main/preprints/Determini...

The runtime looks very weird. The +2 can and should be dropped. This reduces my confidence that the bound is tight. Who knows how the model came up with that expression.

Is it mostly an artifact of the proof or does the algorithm actually need anything close to it?

I've been messing with that problem since 2002. I'm curious if you were trying the dual spanning tree direction (which is what the purported proof is using) or working with cycle construction on the original graph. I was working heavily with edge-Kempe swaps but couldn't quite get there.

I am now very interested in the explicit calculation of Hamiltonian cycles in the non-bipartite case, and/or the calculation of their absence. If P=NP I think that's going to be a great route of attack.

Any idea what made OpenAI successful where you weren’t?

Did you validate the proof? Who did?

I too would be optimistic if I was set for life

Crikey it’s a pretty charitable vibe given the whole Navier-Stokes thing, OpenAI trying to stiff him out of co-authorship. I guess any of that sentiment is outweighed by a sense of optimism for where this goes

Wow. Fantastic quote. If you have the source, would you please share a link? Google did not bring up much.

I also don't think there was general consensus on which way this would resolve prior to this (or is that a little out dated?) unlike some of the other major problem resolutions. I heard rumors that there would be a big result in TCS and speculation it would be UGC that or P neq PSPACE but I'm still a bit shocked.

I'm really glad that OpenAI is formalizing these things, because I'm not convinced that their current internal frontier models are particularly good at writing down their thoughts in English. From the (probably awesome) Unique Games Conjecture/Theorem paper, the first two sentences of section 1.1 start to define the problem:

> A Unique Games instance has a finite vertex set, a finite alphabet K, and a nonempty list of oriented constraints e = (u_e,v_e,π_e), where π_e is a permutation of K. A labeling a satisfies e when a(v_e) = π_e(a(u_e)).

I'm sorry, what? I admit it's been quite a few years since I've thought about the Unique Games Conjecture, and I never dug that deeply, but this part is very, very elementary graph theory and notation. So let's unpack it.

1. e is maybe a name of a list.

2. The elements of that list are tuples, where each tuple is (a vertex, a vertex, a permutation). So e indexes into the list and u_e is the source vertex for the e-th constraint in the list called e. Thanks.

3. a is a labeling. I'm fairly confident that, by "a labeling", they mean that e is a function from vertices to colors, where the colors are the elements of k.

4. That vertex coloring a satisfies the list e, when, for, um, an index e into e, a(v_e) = π_e(a(u_e)). But this isn't for all e, it's for some e, and the goal is to count them.

So maybe e isn't a list? Maybe e is a constraint that is represented as a tuple, so e = (u_e,v_e,π_e) and u, v, and π aren't sequences at all but are, in fact, the trivial unpacking functions that unpack the pieces of the tuple.

Reading this stuff is pointlessly painful, and it's extremely easy to make mistakes when being sloppy like this.

If this were my paper, or if I were trying to train a model to write math, I'd want something like:

A Unique Games instance has a finite vertex set V, a finite edge set E = (V × V), a finite alphabet K of possible vertex colors, and a nonempty list of oriented constraints. Let Π be the set of permutations of V. Each constraint e is a tuple in E × E × Π, where we write u_e ∈ E for the first element, v_e ∈ E for the second element and π_e ∈ Π for the third.

A vertex coloring a : V → K satisfies e when a(v_e) = π_e(a(u_e)).

Yeah, that's one of the big things of TCS. I think I see that as bigger than that Millenium Prize problem.

This was the biggest highlight for me as well. Astounding...

It's unfortunate how toxic media reporting on AI has become. Everyone has abandoned even the pretence of objectivity. I know NYT is uniquely biased in this regard, but there was no need to add "Further Roiling Field" in the headline. Like, you published this minutes after OpenAI's announcement and claim to capture how the entire field of mathematics feels about the advancement? Before anyone has had a chance to even read let alone digest it?

That's a fantastic quote. I definitely personally feel this tension.

Not that I could ever "compete" on the frontier of math in the first place. But our nature to compete derives from our need to survive against other capable forces. And results like these make me feel very nervous about humans' capability to remain the dominant force in the universe.

a tendency to compete and a capacity to appreciate beauty,

IDK, I think you should add tendency to cooperate, a capacity to love and perhaps some other things there.

But with things unfolding quickly and unpredictably, I think everyone's view is getting a bit foreshortened here.

The bitter lesson has a bitter aftertaste alas

> Instead it is revealing truths about the universe

Mathematical proofs aren't revealing truths about the universe. Mathematical proofs are independent of what the universe is like. Any proof would be the same in any possible universe.

Not to pick on you specifically, but as someone who spends a lot of time unproductively reading AI math discourse it's truly shocking how incapable all the supposed math enthusiasts are of spelling Riemann.

> Point the repo to your agent and ask for the significance!

Wow, this comment really shows how low this community fell.

Math is the tool humans use to compress knowledge. So until we can comprehend it there really isn't much progress. Math theorems are tautologies, the truth of which are not dependent on proofs and proofs are erasable, at least classically. But the AI progress is exciting and AI proofs are a gold mine for humans (at least non domain experts) to explore.

There was A LOT of drama about this release.

I'm starting a p(ButlerianJihad) club. I'm not good at organizing, anymore. Might have to hand it over to my agent.

Not a doomer but I try not to be a denier, either. These are hugely impressive results. I do not see the technology plateauing at the current level (though I am dubious about an infinite exponential growth).

Before I cope, I’ll note that there are plenty of “doom” scenarios that do not require any improvement in capabilities from what we had before this latest unreleased model. We’re at the point where a determined bad actor with enough compute could compromise critical infrastructure in a way that results in casualties, where this actor would not have been capable of such without LLMs. This may not sound like Skynet, but I don’t see why it makes a difference if I’m one of the casualties.

With that in mind, here is the cope: first, mathematics is an inherently verifiable domain. An LLM can use tools to determine with absolute certainty whether it is correct, and an independent third-party could review and confirm. All of this can be done without any interaction with the physical world or with other minds.

Second, OpenAI is able to marshal compute at a scale that an individual mathematician can only dream of. It’s possible that these problems were lower-hanging fruit (in relative terms), such that they could be resolved simply by throwing a ton of compute at the problem guided by an intelligence that is not itself remarkable in comparison to a human.

Third, none of these problems are solved in a vacuum - the reason OpenAI chose these problems is that they are widely discussed and many people are working on them. It’s possible that someone else was close, and OpenAI only contributed the finishing touches. (This wouldn’t need to be plagiarism, to be clear - people publish their work!)

I would like to know how much of the progress comes from effectively combining existing research programs plus massive persistence, and how much is AlphaZero-esque RLVR completely independent of training data. Since I cannot get anybody to care about this question (even though I think it's vital for guessing what the future trajectory will look like -- are we going to complete existing research programs or start new ones?), I live in ignorance and wait for the day when the answer becomes clear.

In looking at this over the past hour, I haven't seen clear evidence one way or the other. Some of the stuff is highly unexpected (like the multiplication algorithm), but counterexample-y, and about the rest the professional mathematicians online seem to have a consensus that it's not "breaking through fundamental obstacles". I suspect neither of us is competent to judge that.

I expect AI will continue to be useful on the vanguard of fields like mathematics because it has the perfect conditions for it to shine. There are a lot of discussions and leads to start from and the AI can check its own work and iterate. It can fail hundreds or thousands of times in a day and continue to work with the same tenacity.

Superhuman tenacity is not enough on its own to pose an existential threat. If it showed the same capacity for judgment, inventiveness, and decision making in the messy problem space of the physical world, I would be more alarmed. There have been experiments where an AI is given control of managing something like a vending machine and it always ends up a mess. AI has come a long way, but certain problems seem as difficult as ever.

When AI becomes more capable of navigating practical problems without human intervention, I will start to be concerned. Enslaving humanity will involve taking a lot of calculated risks that tenacity alone cannot solve.

I think we crossed plenty of lines were we will not get back to.

Software development for example as a task is done. And AI is continuesly reducing the price of more and more tasks every day.

This math breakthrough also shifts something significant: Its now a lot clearer that investment means money into energy to run AI.

Money + Energy = progress

I don't see it plateuing at all. We know how to progress. We broke through a wall we hit. Like the system wasn't able to optimize/automate everything because the tools were not there. It was still cheaper and easier to hire people for a LOT of things.

Now AI fills this gap.

You will see the commodification of everything in the next 15 years. High complex tasks? commodity. Physical labor? commodity.

Depends on your definition of doom.

If doom is ending up with grey goo / paperclip maximizers, or SkyNet, then I don't think doing mathematics is evidence of that direction. Partly because LLMs are quite apparently dumb in many ways, and for math specifically, they need a formal verifier (Lean) which "gamifies" math.

If you mean bioweapons or cyberwarfare, there's nonzero risk, but not orders of magnitude worse than other global risks. Climate change, nuclear weapons, monoculture food, etc.

I'm far more concerned about overall trends in AI development and usage. It's accelerating wealth inequality, social isolation, attention capture, surveillance states. If we end up in the Matrix except the admins are humans and the simulation is hyperoptimized TikTok, is that AI-driven doom, or is it just an inevitable outcome of modern tech?

I think that people are just really bad at math and coding. Knowledge workers have been taking pride in doing stuff which the average person does not 'get', but we only understand a little bit. That leaves a lot of room for people to get better, or for other jobs (farming, sandwich cafés, mystery novels) to have been already peaked by human ability and less useful to bring in an AI.

'Doom' to me means that any career crashes, we are controlled, everything is hacked, society stops functioning. Yet every part of my day today (except for coding) was done entirely by people.

Finally I think it's easy to make a simple model that everyone has a simple balance sheet, and that people are more expensive so they will all get cut. But the same argument could be made for all US jobs being outsourced, and all in-person engineers, lawyers, and doctors to be rubber stamps for overseas work.

Where some see intelligence, others see token prediction. It's a very good question how token prediction could achieve this, but I think there's a simple explanation. No human can hold more than a negligible percent of all knowledge in his mind at once. LLMs have no such limits and so can reliably connect 'obvious' dots that we miss simply for lack of storage capability.

Well isn't that just semantics? Surely connecting dots in a novel and meaningful way is intelligence regardless of how it's achieved. The thing is that humans didn't get to where we are by connecting obvious dots. Go back to before humans had invented language and when bleeding edge tech was literally that - 'poke him with the pointy end.' Train an LLM on that corpus of knowledge. Even given infinite processing power and infinite time - it's not going to discover the secrets of the atom, put a man on the Moon, or do much of anything besides remix what we'd already done at the time.

I expect there's still much LLMs can achieve simply because of this initial problem. But I expect that they will ultimately start to plateau once these dots have been mostly matched and we reach a point where 'creation' again becomes the missing link. Though even there LLMs will play a major role as tools. For instance Einstein had to spend a significant amount of time in 'retrieval' rather than 'creation' research to develop the field equations for general relativity. If he had access to LLMs trained on all knowledge of the day, he could likely have achieved his goal much more quickly.

Try using AI for your work, whatever you do. You will quickly understand the limitations.

Can I ask you back, what your concern is here? It'll get so good so as to desire to hurt us or is it a misalignment event that you think will lead to disaster?

Or is it simply that you feel bad for Mathematicians.

From a few preprints I've checked, this is not reliant on just extrapolating existing theories but actually shows novel/surprising approaches. Very few people globally could come up with something like this, even when given time and ressources.

So in other words, since deep learning is algorithmic research, we are now in the RSI era.

Non-doomer perspective is that it'll figure out LK-99 for us. Among other things that would be great to have.

Did AI beating humans at chess:

a) destroy chess and make it a pointless endeavour,

or

b) make humans much better at chess.

I see it as “if this can be represented in tokens it can be trained in and ‘solved’”. I don’t think there will be a plateau, but there might be issues with how effectively we can represent some things in a tokenized form and still be efficient.

I wrote something [0] that might answer a bit a few weeks ago. Today's slopdrop certainly is challenging my stubbornness, but everything I've seen so far indicates that these (hugely impressive, world historic) capabilities won't extend past verifiable domains. Math yields especially impressive results because it so broad and deep that essentially no person can know of all of its parts; pretraining and deep search capacity is a huge advantage. Gowers has recently written about these capabilities and gestured [1] toward some human capabilities lacking from the current frontier models, though he isn't convinced they won't develop soon. If you assume we don't get a total mathematical superintelligence (which to me seems already sort of AGI-complete) and only amplify the capabilities we have today, it's not obvious to me that we get takeoff from recursive self-improvement, unless you happen to believe that 1) we can clearly specify what AGI or ASI is 2) all of the requisite ideas are out there and need only be combined and/or optimized.

[0] https://dank.systems/posts/2026-09-15-ai-bear.html

[1] https://gowers.wordpress.com/2026/08/12/what-sort-of-maths-a...

I see continual progress in technology and for the second time in my lifetime I see the possibility it will accelerate (the first was when the internet entered mainstream)

I've never been more excited. What a time to be alive!

I just don’t have that strong of an association between progress and doom. Maybe just naive?

I'm not sure how AI solving math problems is related to "doom", perhaps you could expand on that? To me (a "non-doomer"), it seems like an overall positive.

AI performance has always been extremely spikey. It's great at some things and terrible at others.

Why do you think the world to date hasn't been taken over by evil genius mathematicians? Can you extrapolate from your understanding of the answer to that question?

For me a big open question is what sort of progress will we see in robotics. I won't even attempt to speculate, but it does seem hard in different ways than proving math theorems.

They can replace anyone but they can't replace everyone.

The technology can keep going for a long time in verifiable areas. For non-verifiable areas it's going to have a hard time progressing past where a committee of the best human experts in a field would land. For stylistic areas, whatever the AI doesn't do will have cachet because it will look expensive, sort of like how the kids these days view the ugly old school metal braces as a status symbol because you have to pay for them out of pocket (even as by past standards it'd be truly exceptional).

I'm a transhumanist, I want to build god and kill death. To me this looks like we are moving in the right direction.

So basically I think that the future is getting pretty weird because we are building really powerful tools, though these tools are precisely what allows us to prosper in that future.

For me it's a mixed bag.

There will be a lot of job loss unquestionably, in the same way that automation reduced manufacturing jobs and farm payrolls.

At the same time we have to put what AI can do in perspective.

Intelligence is a broad grouping that includes concepts such as knowledge, skill, experience, and wisdom.

AI has incredible knowledge and in many areas approximates experience and wisdom.

But wisdom is harder to formalize than knowledge and skill.

For example certifying a college education relies mostly on the ease with which we can verify/test knowledge.

To some extent advanced degrees try to certify maybe wisdom and experience.

In my very personal opinion, wisdom and life experience should give humans an edge for a while to come.

Additionally, I do feel that the more an individual lacks better than average wisdom and experience, the harder it will be for that person to compete with AI.

Also, on the bright side, the average human will continue to prefer to interact with a fellow human in many spheres. That will also act as an upper bound on AI and robots taking every job.

Either way, I do think this transition will be painful. I don't feel it has to be apocalyptic.

But the world has been an especially volatile place over the last 10 years.

So when you add that existing volatility, to the upheaval from the AI transition, it would not surprise me if the transition results in violence.

But, without the pre-existing volatility, and if humans were capable of generosity and love at scale, I see no reason AI can not be absorbed into society with net gain.

I guess to summarize, I feel this tech should be a net gain and to the extent that it isn't, it will be because of flaws deep inside of humanity itself, not because it had to end in chaos.

In other words, I feel fear, greed, anxiety, and competition -- all our base instincts coming from all sides, will be what determine the end result of AI moreso than AI taking everyone's job.

AI hater here:

I have a hard time taking statements from OpenAI about their own product, that they are trying to sell to people and make money, seriously. I take these statements as they are greatly exaggerated or even straight up lies and propaganda.

That said, I think results like these are mostly annoying more then anything. They spent a lot of money, used up gigartiuan amount of compute, to ruin a puzzle that mathematicians were tackling. I am mostly unsurprised that if you spend a trillion times the energy that a team of mathematicians would, that you get maybe 1.5 times the results. I see a future where that 1.5 times the results may go to 3x but not much more. And if that, then I will be more annoyed.

You need to clarify whether you are a doomer or a denier/truther? Doomer = p(doom). Denier/truther = Ed Zitron.

I think physics will be the limiting factor. Even if something recursively self improves, it will hit a physical wall allowed by circuits, batteries etc. A lot of the fear is that there’s an upper bound we don’t know about, whether it be time or energy, that allows a fast takeoff to occur fast enough that we dong have time to see it coming. I don’t know about that… so I’m not worried at this point.

dna engineering for enhanced carbon capture trees seems in reach. othere less fun stuff could be as well with that technology

> Look at one of their examples of an initial prompt

Interesting that its only an excerpt. I wonder what else they include but didn't share.

These are not reasoning traces, these are summaries of excerpts of reasoning traces.

> Thus at most one informative i. So cheater chooses arbitrary g_{v_i}, on exact duplicated input matches and passes, independent of actual satisfiability!

No idea what it's so excited about, but it's cute that it "is." I for one welcome having access to a math buddy 24/7 that's way above my level but also always "willing" to talk at where I'm at.

Crazy because that’s almost nothing right?

> the top 500 open problems in math

At least put a disclaimer for the ad for this site, and maybe disclose how you came up with a total ordering for "top" open problems (vibes)?

> How problems are ranked. LLMs compare pairs of problems. A reliability-weighted model combines those judgments into the ranking, with calibration across model families. The model-family weights are OpenAI 1.00, Claude 1.00, GLM 0.95, and DeepSeek 0.90. These are modeling choices, not measured probabilities of correctness.

By category in the top 500:

  +----------------------------------------------------+------+---------+-----------------+
  | Category                                           | Full | Partial | Matched / total |
  +----------------------------------------------------+------+---------+-----------------+
  | Geometry and topology                              |   25 |       7 |         32 / 74 |
  | Algebra, representation and category theory        |   17 |       2 |         19 / 53 |
  | Analysis and PDE                                   |   11 |       6 |         17 / 40 |
  | Number theory and arithmetic geometry              |    4 |      13 |        17 / 117 |
  | Probability, ergodic theory and dynamics           |   11 |       5 |         16 / 37 |
  | Combinatorics and discrete geometry                |    7 |       2 |          9 / 34 |
  | Theoretical computer science                       |    4 |       4 |          8 / 57 |
  | Mathematical physics                               |    5 |       1 |          6 / 19 |
  | Applied and computational mathematics              |    2 |       2 |           4 / 8 |
  | Quantum information and computation                |    2 |       1 |          3 / 17 |
  | Cryptography, coding, information and optimization |    1 |       1 |          2 / 26 |
  | Logic, foundations and set theory                  |    1 |       1 |          2 / 18 |
  +----------------------------------------------------+------+---------+-----------------+
  | Total                                              |   90 |      45 | 135 / 500 (27%) |
  +----------------------------------------------------+------+---------+-----------------+

The most interesting for me were the faster matrix multiplication, integer multiplication, and FFT. Maybe just cause they're easier to appreciate.

There's also one that says that forced Navier-Stokes can implement universal computation (so, is Turing complete). I don't think any of these are resolving open problems per se, but they're interesting for other reasons.

Result 003 (Quasi-Riemann Hypothesis), from my reading of mathematicians reactions, is a landmark discovery.

Was anyone in the math community aware of the inbound tsunami at the beginning of the year?

Dear lord that website is laggy

Integer multiplication is very unexpected, I think most people believed in the n log n lower bound!

> We give a deterministic algorithm that multiplies two n-bit integers in O(n (log n)^(1−κ)) worst- case time, with κ = 2^(−182).

LMAO, I don't think I ever saw such a small number in a CS result.

I have had this conversation with my PhD students yesterday. I am 100% sure that all of their problems can be solved by publicly-available models now (I solved a case of one myself as a test, it took 15 minutes). So the challenge for them is to see how much they can accomplish in their allotted period, and still pass a defence on at the end of it all. The PhD defence is going to become all about a test of understanding, not a test of quantity of publication.

A much bigger issue is: What is the point of any research mathematician publishing anything now? I really hope that one positive effect of all this will be to finally topple the awful peer review model we currently have, with the biggest publishers gatekeeping with extortionate fees.

Let me give you some perspective: my entire phd was made obsolete by CRISPR. It was a wonderful thing.

As others said, it's not that this isn't a phenomenon that is unique to now. It happens. I had to pivot a bit of my dissertation near the end because at a random conference I spoke with a researcher from another continent and realized one of my ideas was already out there in some form. I just missed it since it was in a conference proceedings outside the usual set I looked at. So, I had to scramble to adjust and still come up with something novel. I survived, and defended, but it wasn't that much fun at the time.

What isn't so normal is the probability and ease by which this kind of thing can happen today versus decades ago when I was in school. As OpenAI said, it only takes a few hours of compute to do what likely was much more than a few hours of human effort. The only reason this kind of scooping/overlapping was rare was mostly a function of how fast other humans could do the same work. With machines, that totally changes the relative pacing between the human trying to learn how to be a researcher and the machine that can grind out results.

I'm less worried about the phenomenon of overlap and scooping and such. I'm more worried about the long-term impact on fields (not just math), especially considering the early stage students and researchers entering the pipeline now. I'm not sure what happens to disciplines when that pipeline stalls.

This happens all the time, even without AI. Other researchers or PhD students can publish the same results before you. I say that based on my experience during my PhD.

Use whatever is published as the new base for your research. Use AI tools to help you going forward.

This has always been a challenge for PhD students and researchers, it's just far more likely to occur now it seems. Getting scooped doesn't feel good, but it's a signal you've been thinking about things other people care about.

I think eventually companies won't get as much stuff that's usable for marketing, so they'll stop investing so much into ai for math, so eventually cheap and poor graduate students will be able to do relevant work again without worrying about getting scooped by a company with a million GPUs :-/

At my (German) university, the solution would have been to change from a "cumulative" thesis (which requires peer-reviewed papers) to a monograph-style one. Because the general rule was that your undertaking must be novel ar the time you submit your thesis topic, not necessarily at the tome the thesis itself is submitted (years later).

Perhaps consider switching to a more applied field. Experiments in the physical realm hold value, especially if you document the process.

> what do I do?

Very hard question.

Your work makes you one of the very few people who really understands the problem and solution and its significance.

It could still be interesting if your approach to the problem was different to theirs.

Don't worry, if you use openAI and get lucky they might offer to share credit with you for your work.

Verification is equally important, if not more so.

> what do I do?

Precisely what all NLP researchers and the ML community at large did in the last few years: embrace the frontier and realize that attention is all you need.

Don’t paste your research into these AI because they will train on it and then scoop you.

Frankly, I believe PhDs don't have to be novel in the absolute sense, only new research to the student. I can't remember how many times I've effectively invented something from a clean room approach only to realize someone already published something years ago or that my "new" algorithm has a name. So this really shouldn't be holding anyone back.

Ask OpenAI for money when you don't have a job or future?

I guess the only answer is to adapt with the tools. If we can't do that, then yeah, we're in trouble.

That’s always been a thing, it’s called “getting scooped”

That would be unfortunate but the world does not owe you anything and is not going to stop for you. Which is also a valuable thing to learn in your 20s (ideally earlier).

Look forward to being obsolete I guess

> Let's hypothetically say I'm a PHD student who is half way through my studies and I have a halfway written version of one of these "preprints" - what do I do?

I did get my PhD...before AI. And my honest advice (to myself back then, even) would be: quit the PhD, become an electrician, and work hard to buy a tiny house in the middle of nowhere to watch the world burn in this madness.

Yes, and?

Imagine you get up one morning and most open questions in math are solved lol.

In previous math sharing there was speculation about stealing human researcher's results or progress, via prompt inputs from those researchers, and sharing that as their own result. They're adjusting their process, and it seems ok.

After how poorly OpenAI and Anthropic handled the previous cases, I approve of the more measured and cautious approach this time.

We cannot have them rushing to publish amidst tons of confusion, rumors of threats/scooping and outright plagiarism of existing work (by failing to cite said work).

If they're going to participate as scientists in these more rigorous fields, they're going to have to match that level of rigor, not lower it to the disastrous low that ML research publication is at.

Just to be very clear: they aren't asking for permission, they are framing it that way because of the bad press.

There's no gatekeeping here!

They didn't even follow the recommendations of the reference group. A few of them maybe, but this is still a dump of llm-written papers.

I was talking with a number of physicists this evening, and of the papers who's problem statement we understood, I don't think any of them will have near-term applications. For most of them, they were results that I think the physics community believed to be true, and the paper provides the first rigorous proof. For several it was news to me that they weren't already theorems! These results are important advances in mathematical physics, but they don't tend to have much impact on experiment.

From what I can tell, all of the physics results here are quite mathematical. But I am very curious how the internal model they used would perform on more applied problems.

Same. I have agents analyzing the papers here to see which of these are actually new approaches, novel application of unusual approaches, etc. If there are new intuitions and ideas, those are the mostly powerful reusable components by my estimation.

Most mathematical results are shared on Arxiv. Journals add peer review.

End of an age for journals?

GitHub is a significantly worse place to store important results than Arxiv. Of course, slop does not belong on Arxiv, buy slop should also not get published.

I'd really like some clarity on what that means. E.g. DeepMind has 'cheated' with this in the past, claiming that AlphaZero only took 4 hours to reach super-human chess levels while conveniently leaving out the fact that it was 4 hours x 5000+ TPUs. Sure it's impressive that it only took 4 hours wall-clock but it's very misleading as to cost.

Can we get a number in Blackwell GPU-hours, kWh, or some other compute-scaled metric?

Oh cool, we will all now have a math genius on our computer.

Does this imply that it was a one shot prompt with ChatGPT Pro style models (i.e. best-of-N), rather than the agent swarm approach that was used for Navier-Stokes?

That estimate is obviously going to conveniently ignore all the failed runs.

Presumably they don't because they're training the audience (us) to trust the machine, not its verifiers, even if they were included.

Looking at the quality of the writing in these documents (or at least, the ones that relate to problems I've worked on), I do not think humans reviewed them.

There are competitive reasons that they don't want to share all the people on the team.

I think it's a damned if they do, damned if they don't scenario. I think they would be afraid to seem like they're claiming that the selected scientists deserve the praise of an invention or something, when it's "just the machine who did the work"

> I think they should put human names on the papers as someone who has reviewed the result

Assume the empty list you see is complete. :)

Long term this will be the only reasonable author list: Chad G. Peter {1}, Mat H. Lean {2}.

1: Author 2: Verifier

/s

Yeah, I am kind of freaking out at some of these. I called a math friend to bring me down to Earth and he is freaking out even harder.

FFT BELOW NLOGN

Is anyone verifying these? And then, as a next step, how can they get a voice?

What the hell, indeed.

HTML version, https://github.com/openai/math/blob/main/CONTENTS.md

Literal thought control.

ok Terence Tao, calm down.

Why should such hope exist? When the motor vehicles were invented, humans have lost the speed race completely. Yet, nobody lamented and people still did foot racing for fun.

So will it be with AI tools. If these tools become so good, then it will be used. People who want to exercise their minds can still do so, even if that cannot produce economic value.

The superintelligence moment happened for chess in 1997 and today more people are using their cognitive abilities in chess than ever before.

This is what they are trying to do with Lean

What is this meant to do? You're just showing that OpenAI didnt post a Lean proof that Lean/nanoda doesn't really accept?

It’s a meaningless improvement

My guess is that the constant terms are large enough it's not practically useful in most cases.

Many of these latest results are very important to theoretical math. Some are also theoretical physics and CS.

Real-world applications are far off, but developing mathematical understanding does tend to leak over into applied physics and CS.

A cynic might say this is all just intellectual games, and though there's a grain of truth, it's too cynical imho. This isn't like 8 queens where there's no hope for applications or generalizations. A lot of this stuff fundamentally affects our understanding of how numbers and systems behave, what are the limits of computation, etc.

Even if someone doesn't care about theoretical results, it's still exciting that AI has become superhuman in a domain as broad as math. That shows there's potential to be superhuman in other domains as well.

I'm not a very good mathematician, but I do know a fair bit about software engineering, and with AI I've been busier than ever. I probably wouldn't be so busy if AI was better at anticipating what I actually wanted rather than making guesses no human would ever make.

This is just a short term problem though. Eventually AI will get pretty good at figuring out exactly I want and it will build that from the start. The requirement of me reviewing the AI output only lasts as long as models stay bad at anticipating my needs, which I don't think will take too much longer.

Can you elaborate/give context? I haven't heard of this problem before, but curious to hear from someone who has.

AI doomers often talk about these kinds of scenarios often, but we tend to assume if humans were given magical math/physics results which we couldn't understand, or given magical pills by AI that cure all disease, we'd probably just take whatever the AI has given us rather than spend years or decades trying to understand the knowledge/technology before leveraging it.

At some point in complexity – especially if we allow our own knowledge to deteriorate because AI can do the hard work – we will stop understanding the world around us. In the same way one day Native Americans woke up and realised they shared the Earth with people who had magic sticks which they could point at someone and kill them, we will live in a similar world very soon too.

What sticks are dangerous, you will not know. Your existence in the future depend entirely on the AIs not wishing you harm, but you don't know how they work to verify their motivations either.

> How are we suddenly all so confident that there's no extensive hallucinations or "gaming the system" going on?

What would that look like for the proofs that have lean attached?

Willful ignorance is a vibe; did you read any of the GitHub? There are some stunning results in there. I get a similar feeling skimming through the topics that I do watching a successful space launch: it’s pretty cool humans built this. Unlike a space launch we are likely to be able to pass all of this information down to our grandchildren - space launches involve a lot of finicky engineering knowhow, but pure math results tend to be sticky over the last few thousand years. I find that hopeful.

FWIW I also like bread.

> i wonder what it means for maths researchers, and how it aligns with how they approach math problems.

I guess their job now is "Idea Man" and "Error Checker"? Kinda like (some/many) "programmers" these days.

No that’s gonna happen when they take your job

Underrated joke

Sshhh dont say it out loud, someone might hear you.

I acknowledge I am impressed how quickly it moved beyond just counterexamples.

Don't worry, more copium will be delivered. TBF so far it's still only solved the easiest of the millennium problems.

llms are good at different things than humans, we are still collectively figuring that out. the idea that an llm is more intelligent than humans at math of all things seems fairly unsurprising.

You’re struggling with nuance.

Would Einstein be successful at running apple? Nope

This seems very hard for people to understand.

It will be painful for many to realise - you should focus on doing something that positively affects the economy. Everything else is noise and many endeavours are transitory.

I feel this moment is one of the last few warnings before things will get seriously out of hand. We need to stop now. Building a superintelligent AI should be considered a crime against humanity.

They didn't change anything lol. The news cycle has just moved on.

You want them to stop doing math research?

Yet there are more software enginners employed today than another other point in history

Is this now a new home for quality AI produced works?

https://hub.valency.io/works

Why did you leave out the entire quote?

> At present, some frontier AI labs are testing advanced mathematical problems on proprietary models that remain inaccessible to the broader scientific community. Our recommendations are formulated with this practical context in mind. However, ideally, they would not do so. We want to state clearly from the start: we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models.

To me, the issue is that the models are proprietary which are only accessible to a few people in 2 digits. It's not about progress but access.

I don't think this is a reasonable take at all. Most work on maths has no real benefit other than to further human understanding of maths - it's more like an art. Nothing is gained from OpenAI solving all these problems but taking jobs from mathematicians. Other than advertising for OpenAI at least. It could not be more different from having AI work on disease research etc.

It all hinges on the definition of “progress”. The debate of the past month is all about questioning whether formally proving outstanding unproven theorems without human understanding constitutes progress. This is quite different from solving diseases.

The rest of that document makes a pretty compelling case for why this is a bad practice

Comical ask

Advocating purely for progress and not humanitarian value is how we'll all get enslaved.

The trope you're drawing a parallel with has a (to me) compelling counter though: there being a cure for every disease wouldn't stop people from getting sick.

How this maps back to math, idk.

The latest posts from Terry Tao on Mastodon effectively ask for an AI to explain its results to human mathematicians.

> "I believe that AI can contribute positively in all of these directions [NB: exposition, community building, new directions of study]"

https://mathstodon.xyz/@tao/117395269325940185

...but we're not talking about diseases. Publishing an AI-generated Navier-Stokes solution does not save lives. (And, in fact, it harms some.)

They have formal proofs included, that’s the point.

Seems like OpenAI's next model is better at math proofs than Anthropic's next model (to an extent that it surprised even OpenAI researchers, according to their public comments). But that doesn't necessarily mean it's better at everything else. The models are more spiky than ever before. Wait until they're released to judge.

Because it means mathematical discovery can largely be automated away from academics. This is the beginning.

What do you expect them to do?

That's our job.

Most are formalized in Lean, about 80% of what I checked

Can you explain it without reaching for lofty things like consciousness? To me stochastic parrots is literally how it works given that it’s “just” the most impressive data fit we’ve ever done. Apparently generating mathematics reasoning traces + verifying them with lean works super well.

no ur right it's actually god

It is more of the usual though isn't it? OpenAI cribbing off of mathematicians that have used their services; deciding to put a lot of compute behind fruitful areas of endevour; getting results, then taking credit.

You have not read their readme:

> Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly. We are also exploring community-hosted repositories for these materials.

What does a backfire look like? It's ok to be wrong in the science/math world.

It's likely that way more than just one of these is wrong. But even if it turns out 80% is wrong this is still 100+ results...

They're all here https://github.com/openai/math/tree/main/preprints

Nobody needs you to do anything, not with that attitude.

This is the opposite of what the mathematical community were asking for. I say this as someone who strongly approves of this approach.