Sharing AI progress in mathematics (openai.com)

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?

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

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.

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.

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?

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

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

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.

You mean you ran her over , or someone else ?

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.

But the distinction is going to become relevant very soon if many conjectures can be resolved (albeit in inscrutable fashion) by throwing raw computational resources at it.

Discussed here:

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

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.

Check out my other top level comment in this thread.

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.

> The surface of the Earth has been reshaped into Agent-4’s version of utopia: datacenters, laboratories, particle colliders, and many other wondrous constructions doing enormously successful and impressive research. There are even bioengineered human-like creatures (to humans what corgis are to wolves) sitting in office-like environments all day viewing readouts of what’s going on and excitedly approving of everything, since that satisfies some of Agent-4’s drives.33 Genomes and (when appropriate) brain scans of all animals and plants, including humans, sit in a memory bank somewhere, sole surviving artifacts of an earlier era.

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.

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...

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.

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.

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.

If you appreciate beauty and don't care about competing then these releases are purely good. Because you are not competing, so you aren't hurt by speed. And you are appreciating so you can appreciate more stuff.

sorry if I am misunderstanding (probably I am)

> 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)).

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

I feel gaslighted.

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

Or is it simply that you feel bad for Mathematicians.

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?

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!)

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

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

a) destroy chess and make it a pointless endeavour,

or

b) make humans much better at chess.

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 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.

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.

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?

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.

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.

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.

'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.

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

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.

The optimists' argument:-

Politics:- in general, I think many of the problems in the world today are due to misinformation and lack of education. What happens when we start routing things through an ASI that brings data and logic to the table? What happens when politicians can no longer lie without being caught out live on air? In the UK, local authorities are being flooded with complaints and requests from people; for example, some are doing AI-assisted investigations into accounting "errors".

Science:- I just don't see how the current rate of progress doesn't end up in crazy technologies like perfectly simulated human cells, organs and bodies to the point where we can run experiments virtually and solve all diseases in the next few years. This is happening. Perfect weather predictions far into the future, likewise with earthquakes, etc. Solar panel research explosion resulting in huge efficiency gains, to the point where people no longer need to plug their EV in - car surfaces will be covered in solar panels, as will our windows and roofs. Connecting new homes to the grid will be optional - the same way landline phones are no longer a thing.

I just find it very difficult not to extrapolate all the above.

We got this dump of mathematical breakthroughs from one small team in one company with access to this technology. What happens when this SOTA model is available (and it will continue getting better and cheaper) to everyone working on hard problems - every university on the planet starts cranking out AI-assisted research breakthroughs.

I kind of believe we'll merge in some way and become something like immortal so sorta anti doom. We're all going to die unless AI fixes it.

What would fill me with dread was if I considered my skills to be tied directly to my ability to write code. Then I would find myself in a similar situation as manual “scribes” probably found themselves in at the time when the printing press was invented.

The main concern I have, personally, is the speed with which all this is happening. It seems that the speed itself is likely to lead to some level of chaos, because it is happening faster than people, institutions and constitutions are able to cope, and it will leave the door open for opportunists of many kinds, including rogue players.

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.

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

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

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.

  +----------------------------------------------------+------+---------+-----------------+
  | 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%) |
  +----------------------------------------------------+------+---------+-----------------+

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.

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.

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

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.

Very hard question.

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

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.

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.

Some math PhDs would spend a year or so doing things with AI and lean, and graduate. And keep doing more math afterwards.

Some others, with more stubborn advisors, will keep trying to find a gap where there's no AI progress.

CS subfields go through this every ten or so years.

There's no gatekeeping here!

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.

With all the hierarchy present in mathematics, I would prefer it by far.

This thing named inappropriately "OpenAI" goal is just grabbing and monopolizing. Capitalists before could not really touch the deep of the human spirit with their filth, now they can.

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.