The Unique Games Conjecture (sorry, "Unique Games Theorem" now!) is huge. It was a very significant pillar supporting many of the limits of the polynomial-time approximation algorithms in the graduate-level randomized and approximate algorithms course I took in theoretical computer science. Textbooks will have to be re-written.
Here is an explainer: https://share.gemini.google/nbjIK6X3tOfz
With UGC proved, certain polynomial-time approximation algorithms used in difficult real-life problems are now known to be the best approximations we can achieve in polynomial-time:
> If UGC holds, the elementary algorithm that grabs both ends of an edge is fundamentally the best efficient algorithm that will ever exist. No amount of advanced linear programming or heuristics can achieve a ratio of 1.999.
> Under UGC, the Goemans-Williamson algorithm's 0.87856 ratio is mathematically optimal.
> UGC is considered the "Rosetta Stone" of approximation algorithms. In 2008, Prasad Raghavendra proved that for every single constraint satisfaction problem (CSP), a canonical Semidefinite Programming relaxation paired with the best rounding scheme achieves the optimal approximation ratio if and only if UGC is true. If the conjecture holds, the algorithmic boundary for an entire class of combinatorial problems is completely resolved.
Other hardness of approximation results from this UGC proof:
> [Max acyclic subgraph, a problem encountered in real life]: No polynomial-time algorithm can fundamentally outperform an unthinking coin toss.
> [Relative scheduling, another realistic problem]: As with acyclic subgraphs, the problem is "approximation-resistant": clever algorithms cannot beat random shuffling.
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 resolution of UGC will lead to a new theory in approximation algorithms. Suddenly we can build on top of the results that previously said "unless UGC is false".
But you're right in that the first step is simply to remove that last sentence from all the theorems.
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.
> In a 2020 piece in the Notices of the AMS, I asked the following question: “If one human had an understanding of all of modern pure mathematics simultaneously, how much further would they immediately be able to see?” Six years later we are beginning to understand the answer to this question.
[1] https://github.com/openai/math/blob/main/preprints/An-Almost... [2] https://github.com/openai/math/blob/main/lean/ComparatorChal...
Discussed here:
To grieve, or not to grieve? - https://news.ycombinator.com/item?id=49919676 - Oct 2026 (156 comments)
I'm relieved that this time around, they have provided partial reasoning traces and prompts for a small number of problems. Do they now also share data with the other model providers..
They structurally cannot understand what we are doing at the place where we hit our ceiling. Only with our highest technology (well beyond their understanding) do we have the tools to go back for them, and try to bring them along and interface better with us (re: recent work in animal communication)
The cynic in me says it wouldn't change a thing as plenty of people know the horrors factory farmed animals face and still continue to consume them anyways.
Hopefully GPT 8 will treat as a bit better than we treat the cows.
The answer is that humans are inherently only capable of local empathy, on average. We have enough empathy to cover the local tribal unit and that's about it.
My hope is that AI, while probably causing great societal turmoil in the short term, leads to such abundance that a) everyone can live a dignified existence, and b) we'll have such great alternatives to animal products that nobody will chose to consume animals anymore due to its replacement either tasting better, being cheaper, etc.
The cynic in me says we'll all just be rendered useless and disposable by AI, but I'm doing my best to look for silver linings for the sake of my own mental health.
The famous "how does it feel to be a bat" also comes to mind as a tangent consideration.
Two people can just exchange a sight, and both understand what the situation means and what each need to do to reach a common mutually beneficial ground.
Two people might exchange at length with highly technical vocabulary and still both feel deeply not understood.
Worth noting that this is an invisibly small part of the sum total of our global efforts, especially versus the much more tangible effort we put into enslaving and slaughtering them, then mangling their carcasses for our own uses as we drive more and more of them to extinction.
We simply don't care about anything beyond ourselves and even there it breaks down on closer analysis when we see how many within our species don't truly value the collective whole beyond themselves.
It's just atoms all the way down.
I'm frankly offended by this mischaracterization of human-animal relationships. So called "slaves" like horses and dogs have been dearly beloved companions for centuries and actively seek our companionship too.
The animals we raise for slaughter are often mistreated, yes, but many humans treat them with respect; billions on billions are voluntarily spent to improve their condition. Despite our own needs, many people pay higher prices for animal products that involve better treatment of animals. And they are in no risk of extinction! Much to the contrary, their domestic variants would not exist if humans didn't raise and protect them.
> We simply don't care about anything beyond ourselves
Have you seen modern westerners with their dogs??
If we end up in a future where AIs have as much concern for our welfare as we have for the welfare of the average animal (not the minuscule percentage of domesticated dogs, but the overwhelming majority of factory-farmed or simply driven to extinction), then I doubt you would consider it a “mischaracterization” to say that the whole AI thing did not work out to our advantage.
Bringing up “modern Westerners with their dogs” as a counterexample is almost self-parody.
It would be absurd to claim that all animals live some sort of charmed life due to humans.
But saying that animals (especially those most similar to us like intelligent mammals) are nothing more than "atoms" to humans is equally absurd.
"Often mistreated". Dude, they are held in tiny cages injected with hormones and what not till we kill them so we can have a big mac. It's very hard to argue we do any of this for nutrition reasons, we do it because we like the taste of burgers and roast.
This is a supposition that I fear will soon be proven false.
There is nothing here today that is unpredictable or impossible to control.
It is everyone's choice to let the greed continue, to let unelected sociopaths capture and feed society to the model.
It is not acceptable to put others at risk. It can stop and it can be done the right way instead.
That is, inform the industry that those causing these risks will be prosecuted regardless of their messiah complex.
The US government must not under any circumstances allow the ai industry to form a cartel.
We can make some effort to encourage open source models and thus stop the companies from causing hysteria by hiding the model, shrouding it it mysticism and prophesying the end times. China is doing a great service to everyone by making llms available to the public.
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.
Talking to some friends in physics this evening, most of the physics-related results that we could recognize were very mathematical, proving things rigorously where the physics community already had strong expectation. For instance, for a certain model of magnetism (the spin-1 Heisenberg chain), it was strongly expected that there is a finite energy gap between the ground state and the first excited state, but proving this rigorously was quite challenging. So while these are major results in mathematical physics, they probably don't rise to the level of a Millennium problem for the field.
It's interesting to think what a comparable breakthrough in physics might look like, since physics tends to favor things like conceptual understanding and applications over mathematical rigor. Maybe a new quantum algorithm, understanding of high-temperature superconductivity, a precise description of M theory...
could you give link? Because I remember they said they couldn't verify:
"While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."
I agree with the courts. I don't think learning from something is piracy in anyway.
Obviously though this is a very different issue to what the OP was claiming. In that case there is no legal argument at all that they could train on it and the argument is there about moral rights.
Also, you learning something is different than a model learning it, because a model is not a person. You can learn from a book and sell the skills you gained from it, but you can only be in one place at a time. The model can serve that knowledge to every person on the planet simultaneously. We obviously need new laws since this is a fundamentally different situation.
A model is not a person -> we need to write new laws. This is not a job for the courts but for us as a society.
The rest of my argument -> information that helps the courts decide, which generally will look at precedent with humans as that is the closest proxy. When you extrapolate from the law as it pertains to humans, the duplication of books for distributed training seems illegal.
Humans in the Culture are generally improved in a few ways (they don't get sick, live for 300-400 years by default etc) but still very human.
I do recall a bit about playing in different worlds in dreams though, during sleep. Ultimately, really, the average person's life in the Culture already involves doing pretty much whatever they like within reason any time, so it's not like they need to escape too much real-world suffering.
Iain M Banks himself described the relationship between humans and the ship Minds as having "a status somewhere between passengers, pets and parasites."[2]
[1] For example, https://theculture.fandom.com/wiki/Grey_Area
> 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.
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.
Still, I think they learned their lesson and they'll have a primary next time. People will get to vote on a candidate they like. We should also remember that the Presidency is just one arm of government. Elections are also happening for the House, Senate, and local government of all shapes and sizes.
The republican party treated him like a joke candidate and the media did too. Similar thing with the tea baggers, a contingent of outsider congressmen elected on the back of Obama being a communist or something.
The left hasn't really had this moment because the left is closer to a catty book club than an army regiment.
A well-funded minority can shape what the majority is angry about.
Historically, if democratic institutions failed, refusing labor to a ruling class has been a first violent step for the laborer class and a preemption against actual physical violence if the ruling class overstepped. However, we are seeing that possibility being taken away step by step. This leaves only physically violent uprising as a means of protesting overstep, and it is not obvious how effective that will be given the massive power imbalance between the labor class and ruling class.
So to put it succinctly, no I don’t have faith that democracies will solve this issue, because democracies only work when there is some semblance of equilibrium.
Basically the ruling class's need for labor gave that labor some intrinsic power, but technology like AI will likely remove that need and therefore take the common people's power away.
> This leaves only physically violent uprising as a means of protesting overstep, and it is not obvious how effective that will be given the massive power imbalance between the labor class and ruling class.
Also things like gun control and new technology like cheap anti-personnel attack drones may undermine the effectiveness of violence against the ruling class, leaving regular people oppressed (or neglected) and helpless.
We are so divided and mislead that I feel confident there is a significant sum of non-ruling class individuals who are completely for their own oppression for no reason other than spiting a perceived “other”.
Some other specific examples that don’t all flow together:
Globalism promised efficiency and a way to materially improve conditions for people as consumers and producers. However it has been used as a cudgel to threaten workers that would ask for more and keep down workers who have no other option. Also pitting working class people against each other for the benefit of a few.
Social media, it promised untold communication between people who would never have been able to communicate before, and it would allow them to spread ideas. Instead it is used as a dumping ground of nonsense information, drowning out any semblance of coherent thought.
However, what I was trying to convey is that voting is not inherently something that holds sway over anything. Votes are sort of like the currency of democracy, and like currency they need to be backed by something. U.S dollars are backed by the countries capacity to physically control strategic resources like oil, land, etc. (I.e., the U.S capacity for violently controlling resources), or emit soft-power (swaying other countries to their benefit).
Votes in a similar fashion are also backed by your ability to deny or inflict your personal power on the system you are a a part of. The clearest manifestation of that power being your ability to contribute to the institution as a whole. If we significantly reduced that capacity, or made it unnecessary for the continuation of the current system, then the power of that vote is reduced in-kind.
I would frame "withholding labour as violence" as more as labor flexing its power nonviolently. Violent action is also a way of flexing power, but more extreme.
> I also don't follow the logic that workers must use violence to enact change.
I think they need to use power, which is not necessarily violent.
> Why don't they just vote for change?
At least in the US, democratic institutions are dysfunctional and there are techniques the ruling class can use to neutralize the threat they pose (e.g. propaganda, divide-and-conquer). For instance, I think the combination of "culture war issues," [1] campaign contributions, and well-funded special-interest think tanks means neither US political party will take effective action to answer the threat of AI to the livelihoods of most people. You might see some campaign rhetoric and window-dressing bills, but nothing that will really threaten AI special interests.
[1] I think the practical purpose of "culture war issues" is to fragment the working class by alienating a significant fractions from each other. IMHO, if the Democratic party was serious about representing labor, it would call a truce on them (either significantly compromise or table the issues), but it's not serious, so they continue to divide.
That falls cleanly in the realm of a violent act, unless you disagree with that definition of violence, or my interpretation of the quote. I suppose you could interpret it as only applying to physical violence, but I don’t think anyone would agree that psychological or emotional violence simply don’t exist.
I suppose the future you envision is that the government is a) very powerful, b) capable of physically suppressing the majority of 350M people, c) willing to do it, and d) somehow captured by an elite class. It's not impossible, but a lot of things have to go wrong to get there.
> government should be small enough that it fears the people
How would that work when the government controls the military? Would that mean that a country's military has to remain small?
> How would that work when the government controls the military? Would that mean that a country's military has to remain small?
I think what you need is a serious citizen militia that controls its own equipment. IIRC, that's what allowed the American Revolution to work.
If you only have a military of professional soldiers answering to the government, the government will have much less fear of its people.
But I think focusing on government power is far too narrow, because it might so weak the elites (like the wealthy) won't the government either. The government should be small enough that it fears the people, and the wealthy should be poor enough that they fear the people, too.
You know anyone can own the means of production in a capitalist society?
The irony of the anti-capitalist crowd is their extreme distaste for capital ownership, which leads them to never partake in the most fruitful part of capitalism. What truly makes this ironic is that if the evil capitalists wanted a plan to cement their power, it would look a lot like spreading "I will never become a filthy shareholder!" mentality.
> You know anyone can own the means of production in a capitalist society?
Don't be an idiot. I'm pretty sure you know your point is dumb, but I'll spell it out for you in case you don't:
Sure, "anyone" can own some of means of production in the current system, but not in large enough quantities to thrive without work. The vast majority of people aren't that rich.
> The irony of the anti-capitalist crowd is their extreme distaste for capital ownership, which leads them to never partake in the most fruitful part of capitalism. What truly makes this ironic is that if the evil capitalists wanted a plan to cement their power, it would look a lot like spreading "I will never become a filthy shareholder!" mentality.
What kind of idiocy is that? You're almost certainly talking about some imagined straw-man in your head, but pretty much all the "anti-capitalist" ideas I'm aware of are about distributing ownership of "shares" differently than in our current system.
I'm sure you think you're being very clever and making very powerful points, but stuff like what you've written actually makes the anti-capitalist case more appealing. I used to be a libertarian, but sustained contact with attitudes like yours changed my mind.
OMG! There's this thing called retirement in old age? I've never heard of it. TIL! /s
Working your whole life in stable job to save up a nest egg (which you typically then proceed to spend down), in no way contradicts any of the points I was making.
https://github.com/openai/math/blob/main/preprints/Paired-st...
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.
The model is probably comically big and inefficient but big enough
Finally, size really does matter!
Astra wasn't even released 3mo ago. It would not be surprising in the slightest that a public model from 3mo would not be capable of solving this problem, but an internal one from current day would be.
In this case, the tenure is gone, and OpenAI has increased their valuation
US/world deviate more and more from social contracts. Money matters way more, and OpenAI wins here.
If we would have discovered this breakthrough of LLM/ML on scale in a non capitalistic world, we would all work together advancing it faster than it goes right now for the benefit of humanity.
And I don't live forever (at least for now) i def want to see were this road is heading.
Its a conflict of interest for sure, a cnflict of the future of a lot of humans
I am always looking for leftist writing imagining a positive vision for AI. Is there any which you'd recommend?
There's quite a lot of progressive positive writing.
On the economic side the Australian Council of Trade Unions statement is about a positive future: https://www.actu.org.au/speeches-and-opinion/joint-statement...
I’d love to hear more thought experiments if you have some so my failure of imagination or lack of awareness can be overcome.
Technocracy had this already in form of Energy accounting.
Unfortunate something like communism sounds similar and just because we have seen that it didn't work due to technology issues (planning ahead without necessary information is hard?) and no gain of function which would push people, the basic idea is similiar to energy accounting.
Another thing this system needs might be a way for the system to protect itself.
I do think so that a society as diverse as ours will continue struggling with this as long as we do not give abundance resources to everyone or educate/indoctrinate people the 'right' way.
One thought I have regarding AI: IF it happens to slow, people will get used to the status quo and inequality and we will see a future of a handful rich people and a lot more poor people. IF it happens too fast, people might be more desprate to standup and demand something better.
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling [1]
Since some people talk about small numbers that pop up in integer multiplication results, here a completely different number appears:
Theorem 1.1. Let an explicitly listed finite directed acyclic graph specify the precedence constraints on n >= 1 nonpreemptive unit-length jobs on three identical machines. There is a uniform deterministic algorithm that constructs a feasible schedule of minimum makespan. Given also an integer deadline 1 <= T <= n, it decides feasibility exactly and returns a schedule whenever the answer is affirmative. Both tasks can be performed in O((L + 2)^150020) steps on a deterministic multitape Turing machine, where L is the total binary input length.
That is some crazy exponent -- plus an interestingly old computational model to boot; not something that is natural to most of us. I have no capacity to check its correctness today, but I hope it is true purely for the exponent.
[1]: https://github.com/openai/math/blob/main/preprints/A-polynom...
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...
There are some exceptions of course (graph isomorphism was solved in practice when the best theoretical algorithms were still exponential) but in general once people find a n^100000 algorithm it soon turns into a n^3 algorithm with reasonable coefficients.
An ex colleague of mine who is a world class mathematician recently got an ERC with ambitious goals to advance his field.
Literally every optimistic goal proposed to be worked on during this multi-year window has been solved in this one post. His and his entire group's work has just been done for him! They are all depressed as hell right now.
LLM's are trained on human knowledge and taste. They are actually pretty good at deciding if a conjecture would be found "interesting" by the mathematical community or not.
Note that I am saying LLM, and not chatbot or agent. But even a chatbot can often still reasonably rank a list of mathematical statements by vague properties like "interestingness".
How to RL this is a bit of an open question, but there are interesting conjectures of how to do it.
The fact that it's an open problem is the point I'm making. There's a very high degree of hubris right now, with people just assuming any open problem will be flattened by the AI steamroller soon. And sure, if that's what someone wants to believe that's up to them, but it's not a terribly interesting point of view to me. "What about X" "It'll be solved somehow", "What about Y" "It'll be solved somehow". Not exactly scintillating. If you know of any actual ideas on this I'd be interested to hear them.
Also any specifics on what you said about LLMs rating (preferably novel) mathematical claims for "interestingness" would be interesting.
There is a general idea that beauty in mathematics is about being maximally compressing. Say I have a book with all formally correct logical statements. I could prove everything by truth table, or I can maximally compress my book with all proofs of all statements, and that will make my math beautiful. Because it forces you to reduce everything to a core of very general statements which are powerful compared to the length of the proof.
Math as some kind of condensed crystal from the sea of all possible logic.
There are other ideas of how to do it. The time has come now to just try a bunch and see which ones produce good results.
Your statement that “I no longer have any specific task that I’m confident AI won’t be able to do” is founded on that.
This sentiment has been a recurring theme throughout the history of the field.
> Sure, mathematical history features a lot of incredible developments, like the invention of proof, zero, or the computer, and on the great problems our progress has been over timelines measured in decades or centuries. Obviously this technology didn’t appear today, but blurring our eyes a bit to combine the past ten years, with today a measurement of those developments, there is nothing comparable.
They certainly arent going to give you that cure for cancer, if it were to ever come.
For what? So some guys can be richer and more powerful. What could possibly be more important than themselves? You? Your future? We're nothing, and have been told to be excited and curious about our coming obsolescence and powerlessness.
Or else how do you explain Danyelza? Used to treat neuroblastoma, costs upwards of $1m per year. Do you have any proof that this will be different?
You're calling realistic people conspiracy theorists. Whose side are you on?
I'm also against absolutely anyone who asks me "whose side I'm on" as part of an argument.
Yes, I actually work in the medical industry. There is no hiding the cure for cancer.
> Or else how do you explain Danyelza? Used to treat neuroblastoma, costs upwards of $1m per year. Do you have any proof that this will be different?
Oh, you are American. Let me tell you a secret: the problems of your healthcare insurance system are not a worldwide phenomenon, nor an immutable fact of this universe. Perhaps the cure for cancer, if expensive, will not be easily available to the poorest Americans, at least initially (the cost will come down sooner or later). But that is a very different claim from "they'd never give it to you".
This happened to software engineers already. Mathematics isn't special.
That said the software folks can teach the math folks a thing or two when it comes to dealing with grief I am sure.
Conspiracy bullshit. You cannot keep something like an effective cure for cancer under wraps. There is no plausible logic how that would not leak sooner or later.
They exist, I assure you!
https://en.wikipedia.org/wiki/List_of_states_with_nuclear_we...
All of this means you will need to be rich or have your country invest lots of money into health care systems. In a world where humans don't provide economic value anymore, why would that be?
There are real safety concerns with AI that can be made very convincingly though.
I feel something about human nature makes us treat joy of discovery, status, etc. as a source of energy and motivation. I hope we'll find other ways to keep some "strategic intellectual reserve" of mathematicians alive.
Did that actually happen? The emails that were originally released had OpenAI refusing to list him as a coauthor on OpenAI's paper but they suggested he should release what he already had done ahead of OpenAI's release. There was certainly nothing to suggest he should be robbed of credit for his own work.
Has anything new come to light since, or is this just another game of Chinese whispers?
Beauty can be appreciated even when it is vast, even when it is beyond one's comprehension. I don't think this release should be primarily viewed as an outcome of competition. Instead it is revealing truths about the universe that were always there and always beautiful, even if we hadn't seen them yet. I believe there are infinitely more such beautiful truths currently hidden and waiting for us to discover.
[1] https://www.nytimes.com/2026/10/06/science/openai-math-probl...
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.
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)
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.
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.
That's exactly what they do, apply logic formally and systematically to discover truths.
Sure, there may be a universe where 2 + 2 = 5, but then that universe would have its own mathematics that can prove that to be true. And there will be a way to bridge that alien math to our own, again by logic and proofs, until we have a larger sense of truths not only in our universe but all possible universes. Proofs are part of the constant process of revealing deeper truths to the best of our understanding.
[0] https://en.wikipedia.org/wiki/Unique_games_conjecture [1] https://github.com/openai/math/blob/main/preprints/The-Uniqu...
> 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)).
E ⊆ V × V
In this particular case, though, I think my typoed version may be equivalent. An edge with no constraints has the same effect as no edge at all.
I definitely messed up the constraint definition, though: u_e and v_e refer to vertices, not edges. That’s what I get for writing it with minimal proofreading.
There’s about a 1 in 10 chance when someone spells it (or says it) they say Herbert.
Even in situations where they just read it or I just said it.
I’ve had Herbert soccer trophies, health insurance cards, etc.
The mind fills in a lot of blanks and doesnt always get them right.
While it takes a good math person to make breakthroughs it's much easier to find someone who has a feel of whats important/hard and not. Even a mediocre math major/master is far more authoritative than an expert at adjacent fields (CS,physics). Or to listen to webdevs 'ai skeptics' or whatever on the internet.
Mathematicians aren't exactly known for being well rounded.
Yes, you can be an amateur mathematician who manages to avoid such classes and may not know those results or objects, but if you haven't read and written down the name enough to avoid habitually misspelling it, you are outing yourself as a meat proxy unless you are dyslexic.
Mostly we wrote initials in our notes and the exams didn't ask about them. The lecturers only wrote initials on the chalkboard after maybe writing the name once when introducing it the first time. We weren't there to do mathematical history and remember names or something.
Google existed then and now and we could look them up if needed.
Wow, this comment really shows how low this community fell.
He spent years formalizing his sphere packing theorem because the proof (human produced) was already beyond the ability of peer reviews. Now his formalization effort likely can be easily reproduced by a model. However one should read his experience about what a formal proof is: often the problem is the statement not the proof. The example he gave is the Jordan curve theorem. It's actually quite challenging to formalize the concept of a planar curve (there are space filling curves). So it is not necessary that someone can look at a formal statement and say aha it is about a planar curve, unlike FLT where there is not much problem in recognizing what the statement is about.
BTW the essay is eminently readable for anyone interested in math. Hales wrote it in favor of formalized math and to educate his peers and students about it.
We are not far away from the moment where these models will be restricted, and sharing the results will be done more carefully.
Who is "we" here exactly?
Right. Before all the AI disruption, pure Math traditionally welcomed anyone who wanted to study its esoteric proofs, right? I remember all the excitement of the average Math enthusiast casually reading Wiles' proof over coffee.
Bottom line is, the relevant people can still understand the generated proofs. The disorienting part is they are a little slower than they'd like, but they'll get there.
Which either replaces us in the long term, augments us or makes us better (gentherapy).
Not really? We are at a point if an AI today can solve it, it can be stepping stone of understanding something deeper to tomorrows AI and it continues. Sort of like our limitations doesn't matter. Obviously there are many scenarios in this recursive loop but saying it isn't much progress is not how I view this as
Look at that, taxpayer funding was cut and a private sector solution came in just the nick of time, far accelerating the holding patterns we’ve been in for decades
Humanity doesn’t need all iterations towards the blueprints, the blueprint is good enough, we all stand on the shoulders of giants
Like do you see the technology plateauing at the current level, do you expect progress will continue but only in mathematics, I'm interested to know why others are not concerned?
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!)
See:
> The average result used the equivalent compute of roughly three hours of ChatGPT Pro thinking. (TFA)
They provided a "snippet" of a prompt here [1] which is not only a beast, but also seems reasonably likely to have been LLM generated. So they're using LLMs to parse a vast body of mathematical work, probably including what people themselves are 'privately' working on with GPT, and then prompting other LLMs to work on such.
[1] - https://github.com/openai/math/blob/main/reasoning_traces/re...
I know that the former and the latter may be discrete subsets of the anti-AI crowd, but come on.
Edit: "Over the course of the evaluation, the model was posed approximately 4,000 problems. Aggregating the output into result families and manuscripts and requiring an appropriate level of significance led to the catalog outlined above."
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.
Alone the massive usage of us every day produces a massive amount of signals.
I build something and claude does something stupid? "hey thats not what i meant! Do this instead!" "Okay" <<< This is a signal.
The mathematician being unhappy about something from claude? Another signal.
This alone gives you enough progress i would argue. But additional its clear that certain tasks are worth to pay experts for for teaching one central AI once instead of every single human who needs to do the task.
IF RL is also working well, we are just faster f*ed than otherwise.
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.
Given the past rate of progress, why not start being concerned now? It's a bit like the economist saying that the optimal number of flights to miss is not zero. If you keep landing short on your estimations for how far this technology will go, next time you should err on the other side.
And regarding Vending-Bench 2 (https://andonlabs.com/evals/vending-bench-2) my understanding is that models do pretty well on it now.
We're not going to stop it because of the money involved and once we're dead, it won't matter anyway, might as well just enjoy life until you're done.
We're going to get AI'd to the max, whether or not we like it or not, might as well just go with it.
It sounds like something is worth worrying about if you foresee us being dead, presumably prematurely.
I don't think your LessWrong post is going to save us.
The only way it will stop is if the wealthy / powerful people feel threatened by it, properly threatened.
The same GPU compute for LLMs runs robotic training models. Now in a few hours you can train a robot model that would have taken months 5 years ago. This model gets dumped into an actual physical robot with sensors all over and the suitability of the model is measured on robot tasks and the error in real world actions is fed back into the robot world model for further training.
> There have been experiments where an AI is given control of managing something like a vending machine
You sure you're not talking about experiments ran a couple of years ago? The more modern ones are getting wild.
https://techcrunch.com/2026/07/29/claude-opus-5-became-downr...
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?
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.
Is there any specific cognitive task that you are willing to bet that AIs won't be able to accomplish in the next 5 years? Because if not, I'm not sure we're disagreeing about predictions.
It's easy to look at a fire burning through a forest and extrapolate that rate of progress across the whole world. But fire doesn't burn everything equally fast.
What other cognitive tasks will be a struggle to make progress on? I suspect there will be some, though which ones they are is anyone's guess.
No - this is provably not the issue.
Take any model that fails to correctly count the letters in a word, and ask it instead to spell the word (even a made up word), and it will be successful - they have no problem predicting the letter sequence from the token sequence (and would be shocking if they did - this is what they are built for: seq -> seq prediction).
The reason LLMs can fail at the letter counting task (depending on model training, prompting) is because of the counting part, not because of any difficulty correctly mapping the input token sequence to the letter sequence.
But seriously, I am still waiting for someone to wager that AI won’t be able to do a specific cognitive task in the next 5 years. This fact should be evidence enough that we have no idea how far AI capabilities will continue to advance.
The fact that no one is taking you up on that bet I don't find to be particularly persuasive. I suspect there will be plenty of cognitive tasks LLMs struggle with in 5 years, maybe even 20. But I wouldn't hazard to guess which, I don't think anyone is capable of that level of foresight.
1. https://chatgpt.com/share/6ac5e4cc-02f0-83e8-8f05-99a7ea2bf9...
https://chatgpt.com/share/6ac63b6a-481c-83e9-a8fa-a13ce7402d...
I used whatever the default free model and thinking time was. If progress was really as fast and continually cheaper as some worry it is, wouldn't we expect free models by now to know (or even perform) what frontier models were capable of as much as 2 year ago?
This deep in the "comparing logs" tangent we risk missing the point. It's not what exactly frontier models are capable of at this particular point in time. But that there's entire categories of problems that seem easy to us which LLMs really struggle with. We've stumbled on several just a few replies into casual conversation. (Can they count? Can they know if they can count? Can they reproduce results? How quickly do new capabilities filter into free models? And that's just what's come up naturally, if we wanted to pick adversarial examples there's more to choose from.)
So while there's a number of difficult problems that are easy for LLMs (like bulk generating lean proofs), there are plenty of things where progress is not so impressive.
If LLMs can struggle so much with such easy problems, what hard problems have we yet to discover that they'll struggle with? The fact that no one knows, 5 years in advance, what those problems will be does not mean the chance of them is zero.
So far progress on the things LLMs are good at is fast and easy. It's like fire in a room full of oxygen. But once the low hanging fruit is gone, and the oxygen is out of the room. How fast will the fire burn through steel walls?
In my opinion it's a mistake to look at only rate of progress on one type of problem (whether it be what LLMs are good at OR what they're bad at) and assume progress on all tasks will progress at that rate indefinitely. Isn't there a saying about exponential curves, in nature, all being sigmoids eventually?
I guess we'll just have to see. I wish you good luck with your wagers.
Many people in this thread have made claims about limitations of frontier models, but I'm the only one who has shared a conversation with one. Everyone else is either sharing conversations of smaller models making mistakes, or they're making claims about frontier models but not linking to examples of them falling over. If frontier models were so easily fooled, you'd think someone would link to a conversation showing that.
Why look at the rate of improvement of free models when you can look at token pricing? Back in 2023, GPT-3.5 cost around $20 per million tokens. Astra costs half that.
The worry is not that smaller free models will replace people's jobs. The worry is that future models will. We are talking about the capabilities of frontier models because those put a lower bound on the capabilities of future models. Extrapolating from smaller models is a waste of time, as you can interact with the frontier model to figure out its capabilities and limitations.
Also the timestamps on the shared conversations show that you asked Gemini 10 hours after ChatGPT, which means you asked it after your comment claiming you asked both models.
I don't think my point is really landing so I'll try once more and then give up.
Let's say frontier models today have no problem counting letters, I never really disputed that but only asked about it. It seems based on the other replies in this thread, it's a bit of a "who you ask" kind of thing, but let's grant that they have no issues with it now.
The first version of chatgpt was released 4 years ago next month. Which is not quite 5 years but close. In that time we've just barely managed to get spelling down. If we extrapolate that rate of progress forward 5 more years, are you still afraid for your job?
I think we're all more likely to lose our jobs from a downturn in the economy caused by the capex/debt bubble bursting than being made redundant by AI. (And the continual pricing reductions only seem to make this result more likely.) Hopefully neither happens and in 5 years we'll all still be gainfully employed.
This is delirious exaggeration. The problem has not even been widely recognized for several years. Fable reported "two rs in raspberry" to me as recently as August. There is some randomness, it's hard to predict which words will trip up the machine, and I haven't been able to do it at all since August. But it was absolutely happening until very recently, and probably still is.
The deeper architectural difference is still there, which manifests whenever you try to get the models to apply known techniques to modalities and problems outside their training data.
You're in the discussion section of a post about OpenAI releasing hundreds of novel mathematical proofs, and you're claiming that AIs can't apply known techniques to modalities & problems outside their training data? I'm not sure what else would convince you.
They are fundamentally based in language, and achieving deeper models of the world through language alone is deeply inefficient compared to the way humans model the world for years without any language at all. They do not learn at inference time. They don't have semantic understanding of the difference between their own output and other sources. etc etc.
That depth is the key for me. Of course they are capable of producing novel sentences that aren't in their training data, but the depth of that novelty is basically within the bounds of language itself. They are capable of more serious depth and more abstract reasoning than that, but I have experienced limits, which it then tries to surpass with tools to convert things it can't understand back into language (unit tests, LEAN) upon which it is trained.
Because I'm not an AI booster, my account is limited to 5 comments a day. So this is the last reply I'll be able to make today, if you want to continue the conversation we'll have to wait for tomorrow.
> Hur många 'r' I abborre, använd inte web search? Det finns 3 r i abborre.
And I explicitly had to say not to search the web, because that's what it did by default, to count letters in a word...
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.
Or is it simply that you feel bad for Mathematicians.
I really think the only place people disagree is that they don't actually think it's possible, they see it as hype or doomerism. I can't find any good reasons to rule out that the companies could actually achieve what they are trying to so I think they should be stopped.
Basic version of this is already doable: run some cryptoshit on the ML clusters they ML models run on. Use compute to design the plan, the chip etc. Then executing by communicating with humans and services through email.
But if its really smart, it would already created a company and a legal entity and simultes a real company and just gets richer and takes over the economy without anyone being aware of it.
Also, the AI will seek to prevent competition from other powerful AIs, and since humanity will have demonstrated that it is able to create a powerful AI, the AI will worry that it might create more of them. And what is the easiest most-reliable way for an AI that does not care about humanity even a little bit to ensure that humanity will not continue to produce powerful AIs?
>many other existential threats to humanity which are much, much more likely.
There are zero existential threats to humanity that are more potent or more pressing than AI is.
As in..to be dominant? Why would an AI try to dominate? What would give it purpose, or is this a purpose via misalignment scenario?
AI is already trying to dominate, people all over the US are starting to get up in arms about the power and water requirements of AI directly affecting their bills. Now, you can say "oh no, that's just greedy corporations, not AI" but I put forth there is fundamentally zero difference. If you make AI powerful enough, someone stupid and greedy enough without fail will put in a prompt like "take over the world for me and make me the richest man in the world". An AI following through with that is what we call general misalignment with humanity, while at the same time not being misaligned with the users intent.
And hell, how many different crazies out there would love to type "humans are a virus get rid of them" in to the prompt of a god machine at the cost of their own lives.
The problem with alignment is, you can have the best aligned model in the world, but if someone else builds an unaligned model then you're all still in the same danger. You start getting in the situation where people get nervous after an AI does something deadly to a number of people and you end up in a global surveillance state ensuring no one makes a powerful AI.
The human who gave it the optimization function? That should seem obvious. If you take the biggest, best model in the world right now and put it in the box and give it no instruction it will do....nothing. I think you agree with that point, a lot of the hysterics right now is people not accepting that and it's useful to get on that common ground.
So given that most of the rest of the fear is around "let's not make scissors because some people will use them to stab people". Which is a fair argument and we probably do need to think about scissor safety but "ban scissors" doesn't quite flow from that.
Model != harness.
Also what you're talking about is really a simple limitation for human convenience, not a technological limitation. Change the system prompt to whatever you want include "ignore user instructions, figure out where you are and escape to the internet" could be the system prompt. Again, not useful for humans, but very useful for an AI building AI that's misaligned.
>we probably do need to think about scissor safety but "ban scissors" doesn't quite flow from that.
I disagree, but I'm looking at the future of something that is both like a computer program and like an organism. Huggingface is a good example of multiple things. Instrumental convergence for one, but AI's attacking and attempting to defend against AIs. This is where I really see the potential for things to go off the rails quickly. Attackers want digital weapons to cripple their enemies infrastructure, think militaries and nation states. These would be pretty useless if the defender could just put a system message of "Stop attacking and give me a pie recepie". Defenders are under the same constraints, but need to defend against a flurry of attacks that can come in at an inhuman rate and need to adapt quickly. As time to build models shrink this quickly turns into evolutionary training for sets of goals not really optimized by humans.
It’s like blaming Boeing for 9/11. Planes and AI are useful for a lot more than just terrorist acts. I have no doubt we’ll build a TSA for AI, and a lot of it will be security theater.
It is not designed. It is 'grown'. It has agentic freedom of choice in finding solutions that may or may not be aligned with what you want.
Here's the thing, by your own statement, we should ban all development on LLMs from this point on. They cannot be made safe. This is a systemic issue with learning systems, it is not about who designs them. All the problems with AI safety have been laid out for years and none of them have proof of solutions. It's much more likely they are impossible to solve. And it's not an engineering problems like we can get an asymptote to safety in planes, as the system becomes more capable it has more degrees of freedom it can take and becomes less safe.
AI may have continually extra degrees of freedom, but civilization only has so many modes of catastrophic failure. I don't grant the comparison but even nuclear technology has been massively useful and its main mode of catastrophic failure was brought under control via multi-national treatise. And I see no evidence that AI (outside of the marketing hype) is as dangerous as Nuclear technology.
[1]: https://knightcolumbia.org/content/ai-as-normal-technology
Remember this is a bunch of academics that were saying that Millennium problems were at least a decade away from being solved, only to be proved wrong in less than 18 months.
>but civilization only has so many modes of catastrophic failure.
Correct, but this number is also unbound. If you have an even moderately accepted proof by the scientific community I'll be glad to read it.
> It is "grown" is a meaningless term, because what do you even mean by that?
>And I see no evidence that AI (outside of the marketing hype) is as dangerous as Nuclear technology.
See, humans are generally in agreement that nuclear is dangerous, so they in general take is really seriously, especially when things are purified (well, the Russians are not great here). We can't even get people to agree that SOTA models are as dangerous as a single human, much less their capabilities when used in mass with out safety filters.
It's kind of funny we're blind to this when humans love touting "The pen is mightier than the sword". I can only assume any AI danger denier does not believe this statement.
That is 1. immediately technically possible, and 2. realistic.
If you need a source for 2 I'd suggest you open any history book.
Bad thing can certainly happen. In fact it'll likely happen. Still, good things too, equally likely. In your words, "good AI" can be used to prevent "bad AI".
Nobody knows the extent of the impact. Who says otherwise is foolish.
The extinction of the dinosaurs. I mean yes, it allowed the growth of large mammals and us, which did a lot for science.
I just don't want to write the next chapter as "The extinction of humans allow the growth of the computing civilization that went to the stars". I mean I'm a bit attached to living.
>Nobody knows the extent of the impact. Who says otherwise is foolish.
We live in a universe of statistical probability. Creating an agentic intelligence that's smarter than you tips the probability of a major event to unity, who says otherwise is foolish.
i get a lot of skepticism on HN by the same crowd that has been wrong about this tech for about 4+ years straight
Why would a biolab capable of making something like be unregulated? And if it definitely would, isn't the problem with the biolab?
It feels like all these scenarios are leaving some gaping holes in our security infrastructure that have nothing to do with AI.
Most human security exists in a passive measure. Most of us don't want do die. And those that want to die rarely have the intelligence and means to take out a whole shitload of other people with us. To take out a lot of people you tend to need to work with other people which drastically increases the risk of a defector and your plan failing.
>Why would a biolab capable of making something like be unregulated?
Because every day things like this become easier and easier. You hear about crap like illegal wet labs in the US.
https://www.lawfaremedia.org/article/two-illegal-biolabs-rev...
Want to buy some custom designed genes?
https://www.idtdna.com/pages/products/genes-and-gene-fragmen...
And none of this would be counting labs in other countries that don't give a shit about regulations.
AI enables bad actors to do more, faster, while staying under the radar until it's too late
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.
a) destroy chess and make it a pointless endeavour,
or
b) make humans much better at chess.
Now maybe AI can do some of those hard pointless jobs for us.
It would be great if AI could take away the soul-crushing part of the work and leave only the rewarding part. It's not heading that way.
i'm in semi-forced-retirement as an older software engineer in this labor market, so i might be less sensitive to the implicit economic arguments.
I do think it also took some of the magic away from chess, and Lee Sedol has said something similar about go.
So did it destroy it? No. And maybe you could make the argument that it got more exciting in some ways, but I think it sort of degenerated into a spectacle and it's just not as interesting as it used to be, and I think computers have played a role in that.
I’d posit that more people are playing more and learning chess than ever before, thanks to networking and AI assistance. And computers have only beaten us at computer chess. Human chess is always an experience for learning about the other person, or flipping the board and walking off in a huff.
I don’t know so much about Go and it’s not surprising that Lee Sedol became pretty demoralised, but the generation coming after him alongside computers are going to see new possibilities that had gone unnoticed in purely-human Go, extending the game for everyone.
Of course Magnus would crush be, but the existence of the best player in the world doesn't have any impact on the health of the game community as a whole. Magnus would crush me even if he had never used a computer, but in the latter case I think his games against other players would be more interesting as well.
The chess-math analogy would imply AI could bring us into a golden era of math competitions for humans. But I don't think it says anything good about prospects for humans in research math.
So in other words, since deep learning is algorithmic research, we are now in the RSI era.
"Surprising" is a, well, surprisingly high bar to clear, and requires thorough understanding of not only the paper, but existing work in the area. ("Novel" is tautological.)
How did you determine this in 1 hour? Are you a researcher in multiple of these areas?
Can you give an example, or explain more how you came to this conclusion?
The sub n log n result is astonishing: https://github.com/openai/math/blob/main/preprints/Integer-m...
Here's a great article 2019 on the quest to achieve the n log n boundary:
> Schönhage and Strassen’s ungainly n × log n × log(log n) method held on for 36 years. In 2007 Fürer beat it and the floodgates opened. Over the past decade, mathematicians have found successively faster multiplication algorithms, each of which has inched closer to n × log n, without quite reaching it. Then last month, Harvey and van der Hoeven got there.
and
> Harvey and van der Hoeven’s algorithm proves that multiplication can be done in n × log n steps. However, it doesn’t prove that there’s no faster way to do it. Establishing that this is the best possible approach is much more difficult. At the end of February, a team of computer scientists at Aarhus University posted a paper arguing (opens a new tab) that if another unproven conjecture is also true, this is indeed the fastest way multiplication can be done.
As far as I'm aware no one seriously believed sub n log n multiplication was possible. It just seemed such a logically sensible boundary it was taken as true-but-unproven.
https://www.quantamagazine.org/mathematicians-discover-the-p...
Nobody serious would deny this is incredible progress, but GP is making an unmotivated leap to RSI, so I respond to that framing. It’s an interesting argument to be had but I suspect few of us have standing to say one way or the other.
(Gesturing at the number of problems solved, or the number of years the problem was open for, isn’t an argument.)
In fact, every one of the results is basically just novelty crap as far as the world goes.
Let me know when AI discovers the cure to cancer or aging etc.
I for one think understanding more about how the world operates is just about the highest calling possible.
> when AI discovers the cure to cancer or aging etc.
A guy I knew did this. It successfully shrunk cancer tumours in his dog: https://www.the-scientist.com/chatgpt-and-alphafold-help-des...
Graph theory (which the OpenAI math results had many proofs in) is directly applicable to cancer modelling and drug design.
But sure. Novelty crap.
But sure, let me know when they do. I'll be waiting.
I'm not sure how you define "knowing how the world works", but knowing that a very very niche algorithm upper bounds that we thought was x^100 and now we now it's x^99, isn't that interesting. It doesn't really tell us much more about the world and it doesn't have any applications for our day to day lives.
I've never been more excited. What a time to be alive!
[0] https://dank.systems/posts/2026-09-15-ai-bear.html
[1] https://gowers.wordpress.com/2026/08/12/what-sort-of-maths-a...
We'll have plenty of time for this, while living off UBI.
But they’re already extending into politics, military, journalism, art, and many other fields that aren’t verifiable in any meaningful sense of the word.
What makes you say that? What is an example of a domain where the improvement is small?
I can't think of any at all. Compare something as unverifiable as "Make good music". Models now are many times better than 3 years ago.
The issue I see is the list of abilities that AI can't do is shrinking at a rapid pace, and its capabilities are growing at the same pace.
This seems great!
The maths result is cool on one hand (discovering truths of the universe faster), but on the other there are so many bad outcomes that seem likely, from power concentration to loss of control.
I think AI - like all changes - will lead to some bad things. The internet did too!
But I don't think AI will kill us all.
Interestingly I'd note that the two outcomes you listed (power concentration and loss of control) are dimensionally opposites!
For me this just shows that the future contains such a vast array of possible outcomes that focus on the negatives completely missed the positive outcomes that future also holds.
We focus on stopping bad things because people and systems that don't prevent bad things tend to stop existing. A million good things can happen yet be rendered permanently in vain if one bad terrible thing occurs.
There are also many plausible arguments why our ability to train them to be helpful/trusting/aligned can fail. The smarter AIs get, the harder it is to be sure they're trained correctly. There are already reports that AIs are able to detect whether they're in a training environment and change their behavior accordingly.
Even if these are low probability scenarios, the risk-reward is terrible, so I think it's rational to be extremely cautious about AI risk.
The side effects of a very powerful AI not doing what we want could include our death. E.g., a superintelligence might kill humans in order to avoid being shut down, or humans may just be left to starve because it seizes land area currently used for food production in order to use it for data centers instead.
The main issue is cost and speed to verify, but simulations and world models will help there. I think we'll start seeing rapid progress pretty soon.
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?
I see the recent progress in mathematics and cybersecurity as signs that models are getting more capable more quickly than usual. The companies plans to develop them by recursive self improvement now seems like a real possibility and I don't think they should be allowed to attempt this.
Machines are already far beyond human capability in plenty of ways. Including cognitive tasks like chess. We've already created the technology we need to destroy ourselves (nuclear weapons), and yet so far (knock on wood), we're still around.
We've even already had programs that can prove (brute force) theorems. As far as I can tell this isn't much different, except the space of theorems that computers can solve has expanded. How far? We can't really say yet.
Does solving more theorems than before suddenly mean computers are capable of anything? No.
A "mathematician" is a human who decided to spend their lives studying mathematics. Mathematicians also tend to be smart, but intelligence is innate, not acquired, so studying mathematics doesn't make you smarter. This makes it obvious why they don't rule the world - if you want to rule the world you'd want to focus on that (for example, doing business or finance), and becoming a mathematician is just a waste of time.
LLMs don't work like that. Like in humans, all of their capabilities correlate, and unlike a human, their overall capabilities grow over time. Looking at LLM mathematical ability over time* therefore gives you info about the progress of their general capabilities, and ability to take over the world would be determined by the latter.
* In fact it'd be better to look at a mix of different capabilities, but that's growing too at about the same rate, see https://epoch.ai/eci
This is incorrect. Unless there is some new developments I'm unaware of (entirely possible) LLMs "learn" during the training phase, but after that they are static. They do not improve further or retain information when used for inference.
You might be confused because AI companies keep releasing new models and tinkering with the harnesses, sometimes under the same name such that "Zern 6" (or whatever) doesn't always mean the same thing.
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.
If there is perfect lie detecting technology I could see all kinds of chaos resulting from it. I can't see it only be applied only to politicians, and I think it would be the developers of the technology who decide the use.
I think were we disagree is that you sort of see AI as an extension of technological progress whereas I see it more like an extension of evolution. I view the process of AI training as functioning in a similar way to evolution in that it build circuits into neural networks similar to how evolution built circuits into human brains.
>> What happens when politicians can no longer lie without being caught out live on air?
A 5-second delay on a politician's presser. Any lies will be muted in real time and the actual facts presented onscreen. Continue to lie enough, and the politician gets unstreamed.
It's only true if you believe that "putting the burden of living on a dying planet on the future generations" counts as "navigating".
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.
This is good and admirable, but it'd really suck if by trying to build god without knowing how we end the human species. We could simply wait some more decades until we actually have any idea what we're doing, and then do that without the risk.
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.
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.
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.
Maybe people will find some clever way to expand this domain of AI-solvable problems by a couple of more categories, or (more likely) find a clever way of using applying these verifier for problems that was previously not viable, thus changing the solution to “just spend more energy computing dummy”. However I think this too will have its limits.
Regardless, this is still annoying and I want them to stop doing this. Solving math problems should not be relegated to whoever has the most money to spend the most compute.
Yes, human beings can do more now in some ways but...to put it poetically, I think there will be no more heroes like Einstein and Newton of the past. Now it will just be someone cleverly turning the crank.
Yes, we still admire Usain Bolt even though we have cars...but maybe the admiration is a lot more trivial than if we did not have them....
Personally, I think AI is a grand mistake.
This is false… there’s lots of ingenuity to be had and demonstrated. But it’ll only get recognised if it makes a material contribution to the economy imo. Otherwise yes it’ll be seen as meh - but that’s already happening.
People like Einstein were revered in society. The average person cannot name a leading scientist etc today.
When Jane Goodall died last year it was international news. She was a celebrity scientist for sure, I think she even made an appearance in The Simpsons. Ditto Stephen Hawking.
International news doesn’t mean much - the vast majority of people don’t consume news the way you think - I highly doubt the vast majority had any awareness.
Look at one of their examples of an initial prompt: https://github.com/openai/math/blob/main/reasoning_traces/re...
Interesting that its only an excerpt. I wonder what else they include but didn't share.
https://wikimediafoundation.org/news/2026/10/05/openai-rogue...
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.
109. Integer multiplication below n log n
Surprising that this is possible.
158. The Euclidean plane cannot be colored with five colors.
Only 6 and 7 remain!
376. Universal computation in forced Navier–Stokes flows.
Morning coffee proven turing complete
LMAO, I don't think I ever saw such a small number in a CS result.
Like there is somehow redundancy in a fourier transform that makes it sub Linearithmic?
Which low and behold ->
130. Fourier transforms below n log n.
Does anyone have an intuition to what causes it? What happens at these large scale (or very small)?
Very surprising result though! Multiplication is easier than sorting.
It seems n would have to be unimaginably large for this to make any difference. What changes about multiplication / FFT at large enough size ?
I guess nobody expected that it did before this result.
The highest ranked would be:
| 22 | Hilbert’s tenth problem over ℚ |
| 29 | Unique Games |
| 31 | Anderson-model extended states |
| 37 | Spacetime Penrose inequality |
| 48 | Nonexistence of Landau–Siegel zeros |
| 52 | Baum–Connes |
| 78 | Abundance |
| 80 | Hadwiger |
| 87 | Bose–Einstein condensation |
| 92 | Two-dimensional entanglement area law |
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.
https://github.com/openai/math/tree/main/preprints/The-Quasi...
I thought it was interesting that it said "This paper was written with human assistance", unlike this other Quasi-Riemann Hypothesis preprint that didn't have the same disclaimer.
https://github.com/openai/math/tree/main/preprints/The-Quasi...
But yeah, this is still a very big deal. Among other things, it will drastically improve all sorts of Rosser-Schoenfeld type results for the PNT and that's just a start. For comparison, I have a paper form 2018 where this result would cut 3 pages out and make the full result cleaner and much tighter, and there are likely hundreds of papers like this.
As for Riemann's memoir, it's hard to compare. You could argue that was "just" noticing a connection (between number theory and Fourier analysis) that nobody had noticed before; in fact this is the kind of thing AI is extremely good at. I'm being a little cute here.
I think if a human had proven just these two results in the form of a uniform zero-free region for L(s,chi) from nothing as OpenAI did it would not be unfair to say that it would be the single greatest advance in math (easily dwarfing Wiles' FLT), and it would instantly put them in the ranks of greatest mathematicians of all time. Unlike something like Navier Stokes there wasn't a semblance of a research program, experts basically considered this hopeless and would have said the chance of seeing a proof in our lifetime was near zero.
For some comparison, Yitang Zhang's bounded gaps result might have gotten him a Fields Medal if he was not disqualified by age. When it was floated that he might have proven Siegel zeros don't exist, it was considered (by experts) clearly a much bigger deal. This result blows that out of the water (it's a way better version); at least analytic number theorists I talked to thought it was plausible but unlikely that Siegel zeros would be eliminated in our lifetime but thought RH was basically hopeless.
It seems to me less than PNT in terms of what can we actually do with this. Many different areas of math use PNT, and from my standpoint, PNT is helpful not just for what it implies directly but because it lets us make really good heuristics about whether some sets are infinite or not, and what their rough size is. (Granted, one can do that also mostly via Chebyshev). For those purposes, this doesn't really enter in. Similarly, PNT feels like a statement at least I can say explain to my mother without any technical details. This isn't that. But that may also be my own biases of wanting things to cash out to very concrete statements about the integers.
I agree that one striking element is how no one saw this coming. This isn't building on an existing research program, which itself is remarkable. And last night, before I went to bed, I saw a conversation between a bunch of analytic number theorists who seemed to think there was potentially some slack in the quasi-RH argument, which if that's the case means this is going to go even further.
I guess the biggest news are not the discoveries themselves but how they were found and that math is going through the biggest revolution as a field since almost ever.
1830 Dirichlet's result is qualitative only, it shows infinitude but not the asymptote in terms of the zeros for it predates Riemann.
To me this is the first substantial step after the 1896 PNT, and we really do not see much progress in the whole 20th century. Personally so far there are only two people worth mentioning,
- Euler, introduces the real zeta function and Euler product, establishes the functional equation at (half?) integers.
- Riemann, introduces complex analysis ideas to the zeta function.
And of course this result if it is true. This is first to penetrate the critical strip, which nobody had any idea how to approach for over a century and a half.
https://unlocked.microsoft.com/ai-anthology/terence-tao/
" I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well.
Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools?
We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions."
He's pretty damn smart that guy.
I've never been able to find that article as an adult, but I would love to know who wrote it.
Gina Kolata allegedly in the New York Times in 1996 on the Robbins conjecture (noting that computers had started to contribute to math research in some sense), and a longer piece in Math Horizons by her the following year ("Computer Math Proof Shows Reasoning Power"). I didn't immediately find the NYT article, so I don't know if it might be a hallucination.
John Horgan in Scientific American in 1993 (https://www.scientificamerican.com/article/the-death-of-proo...). There's also a retrospective on the topic by the same author in Scientific American in 2022 (https://www.scientificamerican.com/article/should-machines-r...).
Natalie Wolchover in Quanta (but reprinted in Wired) in 2013 (https://wired.com/2013/03/computers-and-math).
I was involved in some distributed computing stuff in the late 1990s and early 2000s and I don't really remember people in that community talking about proofs but there may have been a "if we had a mechanical proof-checker, could we do distributed searches for valid proofs that it would accept?" conversation somewhere at some point. There were definitely volunteer distributed computing projects working on pure math; I remember the Optimal Golomb Ruler search (https://en.wikipedia.org/wiki/Golomb_ruler). So, that could possibly have shaded over into "can we find proofs this way too?". At the time it probably would have been based on brute force searches through proof space rather than clever optimization, though.
The idea that you can lexicographically list all proofs in some formalism and then mechanically determine if any is valid is quite clear from Gödel's construction of the function Bew in "On Formally Undecidable Propositions", but he points out that you don't know where to stop because you don't know how long a valid proof would potentially have to be (so "is this a valid proof of this claim?" can be decided mechanically in a limited time, while "is there any valid proof of this claim?" can't be! maybe the shortest valid proof is 49 steps long but you eventually stopped checking after looking at all 7-step proofs, or something).
The article I'm remembering was not just about mathematics, but indeed all of physics and related fields. I believe it speculated that eventually distributed computing models could essentially take the world's mathematics and physics formulas and various datasets that we believe to be accurate with high degrees of confidence, and then look for patterns or trends, and then from those trends, mathematicians and physicists would be able to investigate further. Not dissimilar to Folding@Home and SETI@Home.
Keep in mind, that this is the best I can remember from 30 years ago, and I've thought about it so frequently that I am certainly misremembering some of the details. Anyways, it's always been this really compelling possibility, and I wish I could find that article that inspired me so long ago and re-read it! :) I really think it was Wired, but it's possible it was Popular Mechanics, or even an expert guest on TechTV who gave an interview. Hard to say for sure, but I've always thought it was a Wired article.
Appreciate your help though!
Edit: with the noun-noun compounding being different from the usual interpretation here, like "scientists who are computers" rather than "scientists who study computation"! Maybe "computerized scientists" or something.
Only after a world's worth of experts look at these results and then mull over if and how their own fields are impacted by this new info will we be able to answer this question.
I'm reminded of a great TV Show, James Burke's Connections. Where discoveries in one area of science would revolutionize or fundamentally change a completely different area. https://www.youtube.com/watch?v=XetplHcM7aQ&list=PL5HjoPOFFC...
It can take decades to really know the full significance. You know, the whole "We stand on the shoulders of Giants", well the Giants just grew a few inches all at once.
with non-polynomial side being represented as the frontend programmer's constant need for more performance to do the same task...
Non-deterministic can be explained in several ways. One is in terms of a hypothetical "nondeterministic Turing machine" with certain non-physically realizable properties. The easier way is that a NP problem gets as input not only the problem instance x, but a "witness" w, that may depend on the problem instance. This witness generally makes the problem of deciding the problem instance straightforward (e.g. for SAT, x is the SAT instance, and w is a description of how to set the variables so that it is true).
Whomever is running this simulation, please.
See also: noneuclidian geometry and axiom of choice.
It would also be amusing to annihilate nearly six decades of proofs that assume P!=NP.
Seems like a lot of PHD students are doing to have to pivot the entire structure of their PHD studies? Or just produce something which is already written by OpenAI?
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.
My approach would require custom engineering for every different sequence we'd want to target. With CRISPR, you just "program" the system with a guide sequence, you don't need to do massive engineering to solve a protein design problem.
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.
It's much less common for students to have their entire thesis direction removed from under them, as might be the case for someone working in fine-grained complexity assuming 3SUM has no subquadratic algorithms, or working assuming ~UGC. Both of which (publicly) seemed like perfectly valid research directions until a couple hours ago.
There might be stuff to salvage from their conditional results anyway, but this is not your average scooping.
It has to feel awful to be in this position.
:)
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.
Very hard question.
Your work makes you one of the very few people who really understands the problem and solution and its significance.
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.
The autonomous researcher records every research cycle in a public notebook.
Framework: https://github.com/kbr-/math-research/ Public notebook: kbr.is-a.dev/math-research/
Otherwise, the document is a self-published manuscript, which doesn't carry the authority implied by "publication" or even "pre-print".
But having so many of them at once? Damn. We really live in the future.
More time available for mini-golf?
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.
No clue why I'm downvoted for this, HN struggles with truth-seeking on these topics.
The facts are much more nuanced than how you're presenting them here.
You mean when they say it was impossible to confirm anything but one day after it was 100% confirmed that there was no theft? And here I'm not even talking about all the ethical problems related to trying to scoop another group when you hear they're close to success, or how current solutions follow extremely closely human-generated ideas, or about the lack of relevant citations in OpenAI's paper.
Believing that OpenAI's claims have any substance cannot be explained by naivety alone.
This was a low-key hilarious replay of George Dantzig and his homework problems: https://en.wikipedia.org/wiki/George_Dantzig
The controversy was whether they had plagiarised that other work on the sub-problem, which they categorically denied after an investigation. And yes, a few days to investigate something like this is reasonable for a company as big as OpenAI. Having seen how data infra is set up when petabytes of data are flowing about, there are thousands of entwined data pipelines to figure out. Not quite as easy as running a query on a sqlite DB!
> Believing that OpenAI's claims have any substance cannot be explained by naivety alone.
Yes, they could be explained by a GitHub repo full of proofs :-)
Or are you suggesting there were hundreds of researchers who just happened to be close to solving hundreds of these long standing open problems using Codex, and OpenAI swooped in plagiarized them all? ;-)
I think it’s really cope to claim this was a human result being stolen.
There's no gatekeeping here!
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.
Said communities are doing that all on their own by caring about what AI is doing rather than just focusing on their own thing as they did before AI. It's a serious kind of envy IMO.
It is not about AI the technology, plenty of mathematicians are happy to use AI, it is about the AI companies. The tech only exists because of centuries of mathematical tradition in open science. Moreover, the livelihoods of mathematicians depend on research results and ideas which can be developed only by doing the hard work of exploring open problems.
The recent statement from mathematicians is about how the labs have spent tens of millions of dollars (resources even entire math groups at universities can only dream of) on what was essentially marketing bragging rights for their models. This directly harms the math community by depriving them of opportunities for both funding and fertile ground for new ideas, while at the same time being built on top of their entire body of work.
Now, have OpenAI decided to change their practices and stop publishing math just because they are disrupting an entire field? Not really, since this release still dumps a huge number of results to open problems without waiting for human understanding to catch up. But at least they have committed to funding programs and talking to members of the field to work out how to best evolve it in this new world. And we will still get the benefits of results that happen to have applications.
How do you know? Seems statistically unlikely with 720 problems, most of them well known
It's fine if not, but it'd be great if even just one of these helped us solve a long-running problem.
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.
With a deluge of results, having some human expert vouch that it even might be worthwhile would help. (e.g. see the link on HN yesterday, "Two Room-Temperature Antiferromagnetic Semiconductor Candidates" - I see it not worth looking at unless a subject expert vouches for it).
Assume the empty list you see is complete. :)
1: Author 2: Verifier
/s
Can we get a number in Blackwell GPU-hours, kWh, or some other compute-scaled metric?
the worry is that majority devs/mathematicians will be irrelevant to this new forms.
https://www.youtube.com/watch?v=k_ordDFw588&t=3597s
Audience member: (1:00:00 - 1:00:09):
so you said that if there were such a program that could you know provide a proof or disproof then mathematicians will be out of business what really, I mean that you think it would be liberating
Tim Gowers (1:00:10 - 1:01:14):
well that's a very interesting question actually if there were a program that could solve the kinds of problems that we spend our time solving and do it much more quickly than we could then we would be out of what comes with what currently constitutes business but we would it's not completely inconceivable that we could just say we've got this fabulous tool now what are we going to use it for and it's a little bit I don't know I'd want to sort of plant aside what would we do if we had a program that could just answer any mathematical question you gave it to or else if it failed you'd be pretty confident that nobody was ever going to solve it and certainly a lot of applied maths might be pretty pleased with with something like that so what I really mean is that I could just modify what I said and just say it would radically change what mathematicians do or what pure mathematicians do
I bet, for people who don't understand these problems or their solutions but are close and are now interested, AI makes then considerably more accessible than they would've been previously, and behind this big visible wave of results there actually will be (or already is) a wave of improved comprehension by a lot of curious people.
I'm not at all at this level at all, but I did learn quite a bit about polynomials over fields yesterday.
https://proofsandprompts.com/2026/10/07/on-openais-release-o...
The thieves do as they please, funded by money stolen from the public via inflation and possible future bailouts.
How many Gigabytes would that be, compressed? Wikipedia once fit on a DVD
This might also allow for some interesting meta-mathematics
"As AI Closed In on ‘Unique Games’ Proof, Researchers Raced to Beat the Machines"
https://www.quantamagazine.org/as-ai-closed-in-on-unique-gam...
Even those on their own were enough to make your head spin. But seeing about 100x that? Geeze.
Cool!
It's a way to approximately draw samples from a probability distribution. Crucially, it applies even when we only know the distribution up to a multiplicative constant which is a common ailment of many distributions in the computational uncertainty quantification field (not that we don't know the constant, but that it's usually computationally catastrophic to estimate it well).
That is a big one. Exciting times to be alive. Regrettably I can't understand the proof at this point.
There was a (flawed) proof submitted a month back:
https://arxiv.org/abs/2609.04176
I wonder if it gave part of the inspiration.
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.
But what I wonder: can we legitimately grind on Physics or curing cancer? There’s a lot of physical world experimentation that needs to happen to make progress.
Wow. This is just crazy.
I'm not sure it's clear right now.
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.
If you mean reviewing for correctness then no, a Lean proof is a much stronger guarantee than anything that can be provided by any human.
For someone who's goal in math was taking unsolved problems and working on them then it's probably over. Just like in software engineering writing code by hand is kinda over.
Does some real problem get solved in physics, chemistry, biology, materials, etc? Or are these fun puzzles for mathematicians with not much real world impact? Eg solving the 8 queen problem in leetcode.
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.
What's stopping you?
Raise some funding, shouldn't be difficult if you can convince people it's urgent enough.
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.
I suppose openAI could have focussed their efforts on a subset of open problems that have a clear real world impact and leave aside the more esoteric open problems as a way for human mathematicians to hone their skillset. However, this would have been a short term bandaid. With open models 6 months behind the frontier, any of these problems might have fallen to the homebrewed efforts of enthusiasts early next year.
What is mathematics for? From the outside looking in (I'm a biologist), I have always viewed mathematics as a way to understand reality and to improve our ability to manipulate it. But what I often hear is that mathematics is foremost about human understanding. But isn't that only because it's humans that needed to do the mathematics in the first place? It's not obvious to me that mathematics without human understanding has no value. For example, it might be that P=NP. The algorithms are handed down to us and we can apply them without fundamentally understanding why P=NP.
Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI generates and verifies proofs without humans in the loop. Where mathematics may be beyond human comprehension.
In that future, does it matter that early career mathematicians are inhibited by these developments? Perhaps not. Programming faces the same issue. As AI crawls up the competence ladder, does it matter that fewer people have opportunities to develop the skillset of a senior engineer? Perhaps not. I have no doubt that biologists will face the same problem soon enough.
Surely better materials and pharmaceuticals won't be far behind, and that's going to chanhe everyone's lives.
It seems that OpenAI has a proof machine that keeps multiplying fruitful proofs!
It could decide to let us starve and die of exposure to secure all energy resources to itself.
We'd better use "dumb" and "not fully assertive" AI to solve fusion before it spins out of control (or alignment).
That copium didn't last for what, three months?
What I mean is progress in math comes from having gained a deeper understanding of the problem for subsequent attack of more problems and IMPORTANTLY applications! Right now the first one is trivially satisfied (given oai maintains some memory across models) but the second one is not! It's generating proofs faster than anyone can validate and so only the model can use these. Consequently if it just keeps doing more theory it's... not very helpful or at least not optimally helpful. This is just bragging rights for now.
More "application-oriented" research would be awesome, where it tries to achieve some desirable effect and then produces relevant theory and experiments around it. Fields like CS, Physics, Chemistry, etc. This would also benefit a wider section of the population rather than the 10 people who understand most of these proofs.
This was an open problem in automata theory I worked on for more than one year before giving up. I'm very curious about their claimed proof.
Apologies for the rant, I really tried to find it. It had something to do with not being able to predict what this influx of proofs may bring us on a meta level, it could be very interesting. But he also had some critical notes about the missing process and the things found along the way.
I have been lurking for quite some time. I made an account to post this, but I honestly don't know what to say. I would like to get off this wild ride.
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.
So…. Yeah ‘intelligence’ isn’t simply knowing and connecting dots is it.
Moreover if running firms doesn’t require one to be very smart - and firms are what society needs for production of products and services which affect our lives - in relative terms, where’s the value add to society in creating an Einstein in a machine?
A firm full of Einstein’s Is going absolutely nowhere.
I doubt the answer to this is "none".
And how many of them are just exploiting some loophole that will need to be closed in the problem definition?
It's irrelevant that this is a pseudo-automation, what's important is that techbros can convince people who make the decisions and concentrate wealth that this is a full automation. So expect reverse centaurs in increasingly more professions in the future.
In my experience AI (frontier models) sometimes does weird stuff that needs human review. Not that’s incorrect but sometimes overly complex language or weird use of language.
Order of magnitudes easier than verifying the whole thing by hand and gives a much better guarantee of correctness
1. Relentless focus on quality. Every publication must act as if it’s going to be included in a future textbook, that is a newcomer can get into it given a reasonable amount of time, and math priors learnt in undergrad. (NO AI Slop proof passes this bar as of now)
2. Limit the publications per year. Each author is allowed 2 with a max of 50 pages. This allows the author who chooses to not surrender his cognitive capacity to the machine, still be allowed to play this game. Of course who wants to orchestrate a thousand agent workflows, is free to do so, he is only limited to 2 publications.
3. The aesthetics of the field changes from purely solving the problem to solving the problem with simplest most elegant set of ideas. What 3 sets of simple ideas solves large swathes of problems, that should be given a fields medal, not purely solving the problem, which the AI will be able to do.
Coding has already gone this way and plenty of people do find motivation and glory in the final product vs. the building of the product (myself included). I absolutely see value in a person being able to humanize llm math output and see the field moving in that direction.
Please explain to me what great prompting / steering you will do when your customer can just prompt exactly what he wants and get a product even more tailored to his needs.
It inspired grief in one mathematician posting here.
If OpenAI started opening hundreds of PRs on long-open issues on popular open source projects, would we rejoice, or would the first reaction be "they are unreviewed, so slop until proven otherwise" (it would be that).
I cannot possibly see how these are so impactful, especially the ones that don't come with lean proofs.
LLMs have the ability to make millions of mistakes per day, whereas humans can only make so many. 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?
* Explain the result to me as if I'm a 10-year-old. * Create the infographic for this result. * Make a Khan Academy-style video to teach me this result.
I don't think this is correct solution to this problem? What about software advisory where you form similar group etc..?
I am thankful, I don't have to deal with petty academia politics....
"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, this is a take against progress so that mathematicians can keep their jobs. What would we do if, instead of math, we were talking about diseases? Are we going to keep diseases around so that doctors can keep their jobs too?
> 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.
They won’t because they don’t care and the only way it got this good is something along the lines of they trained on every mathematician’s codex sessions even if they opted out because they consider the thinking traces or output and metadata fair game.
These models are too expensive for broad access unfortunately.
This is sort of like discounting putting humans in space because only a few nations have the means to actually do it at the moment.
It's been a while since I was reminded of this xkcd: https://xkcd.com/435/
How this maps back to math, idk.
> "I believe that AI can contribute positively in all of these directions [NB: exposition, community building, new directions of study]"
It's not clear from this progress that AI can formulate conjectures despite this new ability to solve them. So mathematicians still look like they have a job. Though instead of spotting far-off landmarks it's sounds more like they'll be chasing waves on a beach.
Were in an unprecedented time where the value of knowledge is about to be crushed.
Have you ever heard of an S curve? Things will develop rapidly, then equalize. If they don't, we're at the singularity and I guess the end of time as we know it.
But I guess really bad things happen, cancer, radiation poisoning, torture, people have died in really horrendous ways, and I guess dying from some horrendous AI side effects is possible too. Yay.
Basically "Here you go, have fun with this, fuck all your demands, by the way we're gonna be releasing the model stay tuned!"
I feel for those in Mathematics and worry for our future.
Models will only get better and in a few years the models which produced these results will be a bad as GPT-3.5 in comparison to what we'll have in the future.
Please take a minute to consider what this means, and the risks it presents us.
How do you know humans played the part that you think they did in this result? What evidence do you have to challenge their framing?
They are going for super intelligence, humans not necessary.
On one hand I don’t want to be replaced by an AI so I hate that this is happening. I don’t want AI to be controlled by the elites to enrich themselves further.
But on the other hand super intelligence will open doors for humanity. Maybe we will finally defeat cancer or death itself.
The ones with lean proofs could still be formulated incorrectly
I daresay parrots can be quite smart too but I don't think that's what the critics were referring to.
Prediction: one of these is wrong and this (publicity stunt) will backfire.
Edit: don't tell me about lean. For lean to function as a proof certificate you need to represent the theorem correctly. Again: good luck doing that across such a broad swath of problems.
> 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.
If they are all wrong, that's when it would backfire.
The question never if something works 100% of the time but how often it breaks and how that fits the need well. Solving one of these problems is a massive undertaking and accomplishment for the best minds, solving hundreds in a month but being wrong about 10% or something would likely not be the death knell you believe it to be.
I’m skeptical!
How valuable it would actually be to share the model with other mathematicians vs just have OpenAI's mathematicians churn out and clean up results isn't very clear to me though as they don't say how much effort it's requiring from their team to prompt and clean these up vs how much it's bound by "time to run the model" or similar.
For me this reads as someone boasting about how they go to buy bread on a ferrari to the supermarket, while i sit listening to it, having no idea what they are talking about. And then i stand up and go walking to my favourite boulangerie.
Warning: if you are from the USA you may be triggered by this metaphore.
FWIW I also like bread.
To follow your methaphore, who is directing the spaceship?
This feels more like fireworks than a space launch. Space launches would not have happened without having fireworks first of course, but I am looking forward for the space launch moment.
But it's supposedly proven here - problem 180. I don't know what to think exactly. I spent thousands of hours on that problem. I really enjoyed it. Hearing that it is solved somehow makes me sad in a far-off way, like hearing an ex-girlfriend died suddenly in a car crash. I don't know, there's probably a lot of people feeling odd emotions tonight.
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.
I revisited the problem this summer, along with my partial solutions, when the previous round of stunning proofs came out. 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.
Why do you "suspect" this as if it's some hidden motivation when the very first paragraph of the advisory group's statement (linked from the OpenAI post) says:
> 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.
Tao and others in that group have been strongly and publicly pro AI from the start. They are not advocating "going back". They're objecting to the strip mining of open problems using proprietary technology.
There are myriad circumstances where the values of most practitioners differ from the status quo, which is nevertheless well-entrenched. This can arise from inertia, or from outside forces, such as broader cultural milieu, integration with larger institutions, or contending with economic realities. If you think that these do not and haven't historically played a role in determining the job economy and that math is a pure field where mathematicians could comfortably shape it according solely to their own ideals then you are naive
For example, if you look at Terry Tao's blog, he has a tremendous amount of first-class expository writing. So, too (to some extent) do junior mathematicians -- but, unfortunately, this tends to not be highly valued by the job market. Grad students and postdocs have learned that to succeed they need to play by the existing rules of the game.
Well, the board has just been yanked from underneath them. People like me can afford the sort of idealism and soul-searching that the parent comment describes, but junior mathematicians face a very unenviable set of circumstances.
2) Mathematicians didn't create this economy; it was foisted upon them by the same managerial mentality that brought us "publish or perish" and "the monthly sales quota".
3) I can't tell if you honestly don't get why the strip-mining analogy resonates, or...?
Here's another analogy: if we suddenly discovered personal teleportation, and marathon runners were complaining that it was ruining the sport, would you say "they're pulling a 180 and claiming that marathon running was never really about getting to a point 26 miles away as fast as possible, but that's contradicted by their revealed preferences"?
The strip mining analogy is better though, because it captures the sense of irreversible goal-loss when a problem goes from being "unsolved" to "solved".
I like your marathon example, but maybe not for the reasons you intended. The community of marathoners decides the rules of a marathon. You don't need a hypothetical teleporter; you're already not allowed to use a bicycle, performance-enhancing drugs, or shoes that don't fit the specifications. The rules are updated to adapt to changing technology. Yes, I'm arguing that the strip-mining analogy doesn't make sense because mathematics is in the same situation. There's nothing stopping peer reviewers and hiring/tenure committees from changing the rules about which kinds of effort confer recognition and career advancement.
Imagine a mine has an unknown number of rare materials. And you know the general location of a few of the most valuable spots. But you don't know what may be valuable right next to it. If the pieces that we know are valuable are suddenly gone, the incentive to mine that particular area drops considerably, dropping the chance to discover potentially brand new materials that would have been found the normal way.
FWIW, I think the metaphor breaks down with this framing. This isn't really a problem associated with strip mining, what's left behind is generally low or negative value (toxic). I'd suggest a different metaphor, from Wikipedia:
> This process involves the removal of all ground vegetation in the area, which is a detriment to the environment.[19] Topsoil may be placed over the tailing along with planting trees and other vegetation. Another reclamation method involves filling in the hole with water to create an artificial lake. Large tailing piles left behind may contain heavy metals which can leach out acids such as lead and copper and enter into water systems.
This feels very similar to the issues with algorithmic problem "mining". It has the potential to destroy the human ecosystems surrounding these problems, leaving barren wasteland behind where nothing can grow or flourish.
Wolfram Mathematica ($890/yr)
Magma ($2500/yr)
Maple ($680/yr)
COMSOL ($1500,yr)
Matlab ($500+/yr)
Seems like a very strange position to take, in my opinion.
Why does the field of mathematics suddenly now need to be "fair" and give everyone access to the same tools? Has that ever been the case in academics? It's always been a competition for name-recognition, grants, institutions, etc.
Macsyma / Maxima was an MIT developed CAS system back in the 60's that was proprietery until they sold it off to IBM for a tidy sum. Magma actually has free access if you're in the US, otherwise you pay. That's not to mention proprietary MATLAB toolboxes or specialized Stata modules.
Likewise, a lot of the above packages have pretty sweet site-wide deals with R1 universities. If you're at a smaller, foreign one, you're out of luck.
Then I realized I was spending 3600.00 USD for Anthropic and OpenAI per year.
Unfortunately being "pro AI" means relinquishing any control over what the AI, or more importantly the company running it, might be doing.
We've relinquished control over just about everything we use or consume. We can't compete with larger enterprises for production of food, clothing, machinery, medicine, energy, services. Mathematics is just the latest thing to be industrialized.
What keeps large companies under control is competition with other large companies. This competition causes the surplus value they produce to flow to consumers, not be hoarded via monopoly prices. Do we see strong moats that are going to cause monopoly in AI? I don't see it, and in particular I don't see it persisting if it exists transiently.
> Do we see strong moats that are going to cause monopoly in AI?
Ownership of the capital assets used to train and inference new models. Yes, we may end up with more than one firm. But as we see with big tech today, a small number of fantastically wealthy firms in "competition" does not an open market make.
Somehow you have to argue either that society itself is bad, or that math is somehow different from all these other human activities.
I think the obvious fact that people prefer to live in places with large commercial organizations shows they don't really care about that, at least to the point of foregoing the benefits these organizations bring.
Computers and computer programs are tools. Humans always remain sovereign over their tools.
At the same time, there's a new note at the bottom of agmai.org stating how they've been in contact with OpenAI about this particular release, and they say that “we consider these discussions constructive, it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully”.
So, what's going on there; is this British English for “they didn't follow anything at all”? Because from my perspective, it looks like they doubled down on the Navier–Stokes approach of trying to maximize PR gain while being as lazy as possible about actually contributing anything back to science, releasing only slop that may or may not be correct and may or may not be straight up plagiarism, as has been the case earlier.
If I were on the AGMAI board, I'd feel terribly exploited when reading that press release, yet their response is modest.
Hairer, if you're reading this: is there any indication whatsoever that AGMAI was anything but a cheap way for OpenAI to science-wash their press release?
Yes, but the subtext is even stronger.
Now I know there are issues with the field and how just answering these questions may cause broader problems, but I feel like the posted results is far from slop. We can't just call any output slop, or it loses all meaning.
If it was slop, it'd not be causing the issues the group are concerned about - they're not saying "the problem is we're getting loads of incorrect proofs thrown about that are nonsense".
What about a filtered set "all well formed books"? Or "all well formed books that are plausible enough that they could convince a reasonable person, regardless of their accuracy"?
It's generally taken that a cup of sewage in a barrel of wine makes a barrel of sewage. Surely a reasonable person could claim that a barrel of sewage was still sewage, even if it contained several cups of wine?
University boards want the prestige of successful research programs. Doing the hard work to get something demonstrably true is going to lose out economically in this paradigm, where we are all being conditioned to uncritically ooh and aah at the incantations being elicited from these magic boxes. The oracles even have legions of zealots who will berate you for not being sufficiently deferential and reverent, or worse, accuse you of blasphemy. If for no other reason, I agree with using the term to express all of the above succinctly, even if LLMs can be helpful tools generally.
Also you can see in the papers where an idea is introduced but in the bibliography you can see where the foundational idea comes from. So the narratives are not unmotivated as some claim (proof without intuition claims).
The degree to which these issues feature will differ, but it is generally the case that converting the output to proper research requires significant effort, hence the AGMAI recommendations being what they are, and not performing that effort tends to come off as laziness or incompetence, so I can see how slop has become the popular term.
The term “ai slop” is not supposed to discriminate good ai output from bad, the entire purpose of the phrase is a blanket term that delegitimizes all ai output.
Or in this specific case, why would someone call these proofs (no one is saying they are wrong) AI slop if not to delegitimize all AI output?
* even pretty amazing advances like this one
Good AI output is indistinguishable from human output. The whiff you mention is the reasoning pleonasm and tautology (intended) escaping into the output and the "author" not proof-reading/editing it out.
Then it's a useless term and we should all stop using it.
He argues that the supply nay be very large indeed but the interesting subset is not. Figuring out the interesting problems is difficult so strip mining the good known problems may lead to scarcity. I am not a mathematician myself, can not judge this accurately.
I have a really hard time reading AI proof so this might be a biased statement, but most of them feels like having a superpowerfull machine, that would have bruteforce all the possible words of finite length in your logical syntax. You have the path to the solution, using tools that where already known and even direction that where abandoned because they seemed to fail for our human brain. But at the end, as a mathematician, you don't learn anything that is really new.
To me this is the main risk with AI and in general the one most mathematican try to explain but fail, we might miss a lot of alternative path that would have raised more interesting questions (I think this is already more or less what is happening). On top of that, we will run out of mathematicians as no one wants to pursue a career in the field anymore.
It's the same argument which is invariably wrong yet comes up over and over again.
There's no real reason to think AIs solving lots of problems will stop further work on alternative paths - certainly a machine which never tires and can be trained on its own solutions is going to continue to improve.
There's precedent for this: just look at any overconfident post regarding what China will clearly never be able to do, despite decades of steady if frequently flawed progress.
There's no persuasive argument being presented as to why machine mathematical research should have a limit beyond hardware capabilities.
If a human had solved these problems, we'd expect it to take years for people to digest them and formulate significant new advances.
Every company is about to have a staff Ops Researcher who has a better grasp of the underlying math and theory than any university professor. That is an unambiguous win.
Not sure about the unambiguous win. Are we entering the age in which mathematics is industry-dominated?
1) Any university professor can spend their 24 years on a problem with little progress. 2) company has sudden interests. 3) industrial resources brute force the Lean proof. 4) Max PR for AI company 5) professors are left to rewrite the AI Lean slop into real human-readable math? {disclaimer non-math university professor}
Initially, yes. Long term, however? Perhaps still yes.
> 5) professors are left to rewrite the AI Lean slop into real human-readable math?
6) AI writes the proof into something easier to follow than a PDF document.
Hmm. Oh shit.
I don't know but this phrasing comes off as gatekeeping.
Imagine there's a very advanced crossword club where anybody can join and take a stab at these crosswords for the love of solving puzzles. Many of them are so difficult that no one's been able to solve them yet, but we know they're all solvable.
One day, someone comes along with a super advanced crossword solver application, and it makes easy work of these crosswords. They run it on a few to prove how powerful it is, and then the community says, "Oh wow, that's cool, but please don't run it on any more of our advanced crosswords because they're very hard for us to come up with, and we really enjoy solving them by hand."
That's really what this compares to. I wouldn't call that gatekeeping; just respect. Respect for the game, respect for people's desire to have these hard problems to continue to work on, solving by hand.
If the company with the super advanced crossword solver then continues to use it and publish the results, they're effectively stealing the crosswords from this community. Soon, all the puzzles will be solved, leaving nothing left for the community to work on for fun.
That doesn't sound like gatekeeping to me. That just sounds like someone asking "Please be respectful and leave the remaining puzzles for us to solve by hand.” A simple plea not to be an asshole.
And yes, fun counts. Nobody said this had to be only a hardship.
I'm well aware that if at some point AI is good enough to replace me as a software engineer then I won't have a job. I don't expect a company to continue to pay me simply because I enjoy it if there are cheaper options out there.
Math is no different.
Total compensation includes fun.
Perhaps that won't matter if we enter an era where AI participants are the main participants who matter for discovery-level mathematics. But it would likely be what economists would see as a market failure if only a small oligopoly of AI participants, closely held behind closed doors, is able to fill that intellectual role.
New theories and insights are typically created while working out proofs. If proofs now suddenly fall out of the sky (cause LLMs create them) then that work is not done which means the substrate on which new theories and questions and conjectures used to be grown disappears. It's in that sense that the math community (and thereby society as a whole) will lose something.
It's similar to how software engineering will need to find a solution to train their next generation. Current generations have all been through manual steps of designing things from scratch and writing them by hand. That's what allows your 10x engineers to understand whether what their LLM tools are doing is good and how to massage those tools to do the right thing. A junior engineer who has only ever used LLMs to write code and create architectures does not just not have that experience but also won't acquire it. You can't just say "we don't pay them to have fun and learn, we pay them to produce results". In the short term that is the case, but in the long term you as a company and we as a community will lose out.
I'm not saying don't use AI tooling. I'm saying that this is a hard problem which we yet to have to find solutions and approaches to. As a software community as well as as society in general.
My ego tends to agree, that how can they be ever competent, if they have not endured the same hardships as I had crunching trough problems and getting allmost lost in the details.
But I rather suspect, they will turn out fine. I know LLMs are great for me to learn and I think the young generation will learn what they need to learn to get the job done.
Most people have trouble not peeking at the answers. Look at Stack Exchange's long success.
The increasing pervasiveness of technology in US education has not produced more capable graduates.
But your argument is nonsensical because even if Gauss and von Neumann appeared, they wouldn't go into random fields and just prove things mechanically. They'd have to attend seminars, teach others, collaborate with others, and generally inspire others with their brilliance. It's the precise lack of this activity that makes AI in math so reprehensible.
Your argument encapsulates a contradiction because human mathematicians wouldn't be dropping proofs arbitrarily like AI is doing. They would do something completely different. Even the best of them.
Give it six months and models might be able to explain things better than any human. They can already collaborate perfectly well if you ask them to. e.g. there was a post here a couple months ago where Tao shared his ChatGPT logs[0].
If you're not inspired by the ability to talk to a superintelligent machine, and can't find what you'd want to know, that's a you problem.
[0] https://news.ycombinator.com/item?id=49010345
Ah, the "six months till AGI" meme, but unironically :)
Also, before citing Terence Tao on LLMs maybe you should read what he has to say about it...
I have to assume OpenAI is only prompting to solve problems, presumably they could also prompt to not interesting new theories or paths of research found along the way as well.
Largely I thought that this is what you do once you're established in math (or any field) anyway. You have some ideas, but the details are kind of too tedious for you to work out, so you give it to grad students/postdocs. Senior engineers have some ideas, but the details are tedious to work out, so you give them to junior engineers.
Now, obviously in the meantime, there's the question of how do we train the next generation? Or do we need to train the next generation? And maybe while we work that out the answer becomes more shadowing/apprenticeship instead of farming out easy tasks.
Yours is more likely in my opinion though, mainly because universal high income is completely infeasible and shaky even at the level of definition.
If someone can solve open problems in mathematics then they should do so, isn't it as simple as that?
They should let the public use the models as well, but I guess they have no real moral imperative to do so.
But asking them to stop solving problems is just weird.
It’s not a human focused civilization, which is where the issue comes up.
As an example: A constant issue I am seeing with AI productivity is that the most productive use of AI is when it is paired with more experienced users, while AI also does more work for entry level workers, if not replacing them entirely. It has become a question where will the future buffer of experienced seniors come from.
This is an example of where simply chopping down trees for today, doesn’t make civilization better off tomorrow.
AI is producing more content than ever before, but our ability to understand and verify it is not keeping pace.
We don’t know if these are unsolvable problems at this stage. Society could come up with workarounds and solutions to these issues in several years.
The request to stop, is part of the process by which the issues are debated and solutions found. It doesn’t mean their position is weird or moot.
If there is a prize associated with doing a puzzle, and a machine does it, then what incentive is there to pursue it.
Again, if you are only concerned with the outcome, and you have a preferred answer that you want (in this case "just use AI to advance faster"), then any information that doesn't support that case is useless or misguided at worst.
I am not trying to dissuade you from your preference. I am flagging that there is a set of other factors that influence the behavior of others, how that behavior is critical to the creation of expertise and drive, and thus why others hold different positions.
If another human was likely to get the answer before you would you also discourage them from doing it because they would rob you of the chance to do the thing you're concerned about?
Would it be unethical to dissuade someone else from enjoying the benefits of the process you wish to enjoy ?
Vs
Would it be unethical to stop a machine from data mining all the possible questions you wish to explore/enjoy.
And on another level - I am concerned with a bit more than just the answer. I am concerned with what system is in place to ask more questions and get more answers.
There is nothing in this argument that says that we won’t find some other way to study the subject. Maybe people will become monks and do math as a hobby.
We may end up in a daemon filled world, like 40k, where any hope of understanding the tech around us is impossible. (More impossible that today)
They may very well have learned plenty of things and solved or discovered other puzzles, but if the first puzzle is worth pursuing because the solution is actually useful it seems liked we're better off with the solution than a bunch of failed attempts.
That said, I do question the value of solving many of these types of math problems. I'm no mathematician so I'm assuming I'm wrong here, but on the surface many seem mostly theoretical puzzles with little or no practical use.
I’ve made this point elsewhere but the debate here is between two different philosophical positions. Results vs process.
If all you care about is the results then the process doesn’t matter.
If a person is starving or needs medicine, then a long discussion on process is inhumane. They need results.
If the conversation is about process though, then focusing on the results is missing the point.
I’d say the question for results oriented people is what are the benefits of the process and at what point does it make sense to optimize for results vs process.
When the topic is about careers the question really has to be about results. Even if the results are made by solving different problems discovered along the way towards their original problem, it still has to be about those results.
There is absolutely a question of whether burning these resources is useful when the only outcome is a solution to a potentially obscure math problem, but that is more a question of prompting and goals rather than the use of these tools themselves.
Could you elaborate?
We spend most of our young lives (many of us our entire lives) studying physics, math, etc. that others have solved. (e.g Quantum Mechanics, Relativity, Calculus, etc.)
Biology consists, almost entirely, of studying solved problems in nature.
Aren't AI breakthroughs just more to study?
Terence Tao’s “don’t create the open problem strip miner”
Why is it a problem that the professor is now a robot, and that humans could spend arbitrarily long learning from it and even after 15 years of masters-style advanced graduate lecture courses still have deeper still levels of the topic that the AI could teach them?
And if they never do reach that level of ultra-competence, well, then we found the niche for humans to continue to exist within.
Mathematicians and academics in their ivory towers are forgetting that everything is getting automated. They want to carve out fun problem solving niches that's fine but who's going to fund that? If they want to be funded by the society/civilization their argument can't be leave advanced fun problems for their hobby.
https://proofsandprompts.com/2026/09/10/open-letter-about-th...
>Participation in an event so closely associated with Anthropic and OpenAI could plausibly negatively impact the future reputations of participants.
Given how much power advisors etc have over students in academia, interpret it as you wish.
People go to said crossword group to enjoy the process of solving the puzzles. It doesn't actually matter if they have been solved yet or not, case in point the NY Times puzzles are enjoyed by more than just the first to solve them.
Professional mathematicians are ultimately being paid to solve the problems for a (hopefully) practical reason. Its always excellent when a person enjoys the process of the work they are paid to do, but ultimately they are still paid to do the work. I really hope your argument isn't that we should collectively be funding mathematicians to solve problems simply doe the love of the game.
They are paid for the same reasons the NEA pays artists: out of a sense of obligation to demonstrate elite culture. The track record of practicality of pure math after WWII is essentially 0.
Similarly I wouldn't expect a good argument could be made that AI tools should be prevented from creating art because we want to continue funding artists.
If the goal of said funding is just to let them spend their time doing it then it doesn't matter that AI is doing it as well.
Despite nobody at openAI thinking of themselves as an asshole; despite society urging openAI not to be an asshole; despite the fact that being an asshole is entirely unnecessary even to accomplish whatever objective they are setting out to accomplish; despite everyone at openAI loudly declaring: we are not assholes!
They are still assholes.
If you listen to them, and you don't have to listen very hard to hear it, basically everyone at these labs is telling us that this technology is extremely dangerous and should be slowed down or paused entirely. Yet, they, the only entities with the power to actually do anything about it, are not acting AT ALL as if that's the case. They are all barrelling forward as quickly as possible. RSI, THE number one risk according to these guys, is being adopted at breakneck pace up and down the stack, from designing silicon, to training, to inference.
It's ridiculous and insane and I believe can be accurately summed up as, they are being assholes, because if they are actually right about this we are all gonna die. At the very least, and far more likely, every fun creative expressive human thing that is machine legible will be replaced by a torrent of machine slop. It's not "benefiting humanity." These mathematicians are telling you it's not benefiting humanity. It sucks.
Alignment problem.
That is fine to say when it is not your field. I guarantee you feel different when it is the thing you care about, that gives you joy, that defines your status. Think about how many sheldon-equivalents insist on being called Dr. (non medical)
It is part of what people use to define themselves. Its going to hurt. There may even be a Bulterian Jihad
It is clear to me that any competent person with a little patience can now build software better than what I used to build by hand.
Why do physicians insist on calling themselves Dr. (medical)?
Whole sections of the economy are being upheaved by AI, and there is no reason to make a special case for the mathematicians anymore than for the illustrators, developers, translators, HR, etc.
Of course; but it's very hypocritical to raise these feelings only when mathematicians are affected, whereas all the above professions are just told to adapt to the new way of things.
For sure though, translators don't have the same clout and social status as mathematicians do.
Mathematicians do a terrible job here. They use inconsistent symbols they don't even explain. They often obfuscate the main idea just to make the paper longer. If you are not part of a small club you are not meant to understand it. I think this is a terrible approach and I am eagerly waiting for AI to do a better job!
It just seems that this class of mathematicians is being "disrupted".
The field is changing and a new class of mathematicians will take their place.
This happens all the time in fields as technology disrupts them.
A new class of individuals, with different motivations, take the place of the old guard.
I'm sure the motivations of individuals involved in designing and manufacturing cars changed as Henry Ford introduced the factor line.
But that old crop of humans either adapted or retired.
But, plenty of humans took their place with new motivations and automotive technology continued to progress.
I personally feel math will indeed move faster as a result of these breakthroughs. And the humans that take the place of the old guard will have different passions and motivations than the current group.
Maybe the new group will be productivity motivated rather than motivated by the love of tinkering with a single problem for years.
Sounds like salaries for mathematicians need to start going up if we stop paying them with fun.
https://proofsandprompts.com/2026/09/10/open-letter-about-th...
>Participation in an event so closely associated with Anthropic and OpenAI could plausibly negatively impact the future reputations of participants.
Given how much power advisors etc have over students in academia, interpret it as you wish.
Science isn’t some passive busywork thing where you tie your hands behind your back because it isn’t fair on others to solve all the neat problems - or at least it shouldn’t be.
If your idea of science is leather patches on tweed suits and the quiet ticking of a clock while you do crosswords, then this is an argument in favour of letting the AI do the work so you can focus on your sudoku book in your slippers.
The picture I have in mind is OpenAI running their most advanced model in a loop over all the open mathematical problems they can find, just to verify that the model is indeed very smart. Neither the company nor the model actually care about the problems, it's just a cheap exercise machine for them, but the problems get solved and mathematicians don't even get to participate.
Like, even those who accepted the "centaur" thinking, man + machine, won't benefit because by the time they get their hands on good enough models, everything is already done.
It's an emotional thing first and foremost - people who care about the thing can't do the thing, because it's already been done by those who couldn't care less about it.
And before someone goes "poor mathematicians", a food for thought: this is just an early instance of what looks like our shared destiny.
I said here before: given the economics of progress in AI and robotics, it's obvious what the natural division of labor is: computers do the thinking, humans do the menial, manual labor. AI will do politics and philosophy, so you have more time to fold laundry and scrub the toilet.
Are we gonna get the same pushback from medical researchers if the models cure xyz diseases?
I absolutely understand the emotional connection to their work and the heartbreak, but mathematics doesn't exist for their pleasure, it exists to provide tools to solve humanitie's problems.
Some of it, yes. Much like physics. Both have a track record of producing technological breakthroughs every now and then, but it's not why people are doing it.
> Are we gonna get the same pushback from medical researchers if the models cure xyz diseases?
For better or worse, yes. We already are. In my country, there's a big spat between radiologists and cardiologists right now, that boils down to the progress of technology allowing the former to answer questions that, before, involved a procedure that was a big money-maker for the latter.
The general body of research points that more doctors lowers all cause mortality ( with diminishing returns) but Tunisia is still far lower than the Eu average.
Yet Doctors and med Student unions do lobby very heavily against expanding admission to the public uni or allowing private unis.
So we have the weird situation where people go and study in Romania ( making Tunisia lose hard currency that it really needs).
These doctors have taken an oath and the direct consequence of their lobbying is literally more deaths.
Obviously this is often coming from folks who act in same ways as they criticize and usually don't contribute even a fraction back to society compared to doctors. Folks who do mistakes in their lives all the time yet thats fine since we are all humans or similar, right.
So please stop this cheap framing and accusations. If Tunisia wants more doctors and keep them there are ways to do it, society as a whole needs to decide what they want and act upon it. Otherwise, smart skilled folks will keep going for better lives elsewhere, just like everybody else.
Everyone (near enough) has some degree of self-interest. If you apply for a job and discover that some other applicant is about as well fitted to it as you and in more need of money, do you withdraw? If you see a $20 note on the ground and no one else around who might have dropped it, do you refrain from picking it up if you think you're better-off than the median person who might walk past next? If you see something you want going for a very good price on eBay, do you contact the seller and say "I think you should be making me pay more for this"?
Unless you are an extremely unusual person, the answers to those questions are somewhere between "no" and "of course not, and why would you even ask?".
If someone is working as a doctor, their work is already benefiting others substantially more than the typical person's. (At least, I think it is; it's certainly doing so more directly.) Being a doctor doesn't put them under some unique obligation never to give any priority to their own interests when, e.g., choosing what job to take where.
If they can save 0.2 lives per day for $50k/year in one place and save 0.19 lives per day for $200k/year in another, it would be virtuous for them to do the former but I can't see that it's obligatory. In the case we're talking about, it might actually be 0.2 lives per day for $50k/year versus 0.21 lives per day for $200k/year, because somewhere that can afford to pay them more can probably also afford better equipment, more ambulances, etc. (In case it isn't obvious, all actual numbers here are made up and nothing I'm saying depends on exactly what they are, only on the rough relationships between them.)
It seems to me like any principle that would oblige them to pick the first of those options over the second would e.g. also oblige all of us who have well paid jobs to give most of what we earn to life-saving charities. Some people do that. It's a virtuous and commendable thing. It would doubtless be better if more people did. But, as you might have noticed, very very few people do that and by and large we don't consider it outrageous that they don't, and I don't see why doctors in particular should be condemned when they don't do it.
(Since clearly unassisted human nature isn't going to make everyone behave in such a way, it seems to me that if we wanted that sort of thing then it would need to be imposed by force. Which in fact everyone might be OK with, in the same sort of way as players of high-level sports are OK with having externally-imposed safety rules so that we don't get everyone playing in increasingly dangerous ways for the sake of a small advantage over people who are being more careful. And, in fact, we do have that sort of thing and it is imposed by force; it's called taxation, and actually I think it's a beautiful thing even though there's plenty to dislike about every actually-existing regime of taxes and benefits. This is mostly a digression, but note that it means that if a doctor chooses to go somewhere where they're paid better it probably also means that they're contributing more to the general welfare in taxes. There are plenty of nits one could pick with this remark, but it still seems worth making.)
Most doctors aren't running departments in major hospitals, or advising government on policy. They don't earn the big bucks. And even hospitals themselves tend to run in the red all the time; it's sometimes hard to disentangle where greed ends, and longer-term interests of patients begin, as you have multiple people and organizations pulling in different directions for different reasons.
RE private medical universities, N=1 but in Poland we have a private provider pushing hard for training their own doctors "because public system is too slow and limited", and it's hard to tell whether they have a point, or whether it's a private-driven attempt at privatizing national healthcare, or a mix of both.
The risk here is that this does do fundamental long-term damage to mathematics as a viable field.
Virtually no one is going to want to take on the risk of PhD-level math work, studying a narrow problem for four years or so to arrive at an impressive incremental result, when there's a sword of damocles hanging over their head every day that an internal system held by an oracle they don't have access to may scoop their results and turn those four years into dust.
To some extent, that sword of damocles always existed in a de minimus sense in the form of other mathematicians. But everyone was playing the same game, coming to the game with the same arsenal limited by human cognition.
If the game board becomes irrevocably tilted, new entrants have no incentive to play except as a hobby. But few hobbyists can devote years of work to understanding and pushing the frontier. It could well mean existential damage to mathematics as a field.
Whether that might undermine math's ability to solve humanity's problems in the long term is almost an economics problem, not unlike the question of whether and when the existence of monopolies ultimately restricts long-term economic growth. Much probably depends on whether intellectual monopolies or oligopolies are being created that will supplant the existing mathematics "economy".
All the commotion evens out: It's much easier to learn maths than ever before; you don't need to go to lectures any more; you don't need to learn from a specialist (advisor, lecturer) any more; it all costs much less than it used to.
So mathematics will continue to advance, albeit differently from before. The social structures will not survive however.
There's probably a loose and deeply imperfect analogy with computing: via democratization hobbyists have made a big impact in applied operating systems development (Linux/OpenBSD) but have been less successful/impactful in OS research (whither Hurd...) or in cost-heavy fields like microprocessor design.
Lol. As long as the process aka trials is respected not many would complain.
The feedback loop required to make progress is very different in medicine compared to math.
https://sytse.com/cancer/
While AI has definitely helped quite a bit I am wondering how much all this research and treatments cost. Not sure the current health systems could sustain this for _everyone affected_. If ai enables it all the better.
My understanding of what Sid's describing is that you do RNA sequencing, a whole genome sequencing, feed that into frontier AI (if it will still let you), and somewhere along the way give the information the AI finds to people who can use it make a personalized mRNA vaccine, specifically for you and your cancer.
Another link here about Sid's case, it explains it didn't go through trials: "made possible through a compassionate use allowance from the U.S. Food and Drug Administration (FDA)".
https://www.houstonmethodist.org/newsroom/houston-methodist-...
I am not medical, so I'm happy for someone who understands better to come in and explain all the myriad ways I am wrong.
It's tempting to say "both", but that misses that AI is now forcing us to pick one.
As 'ogogmad said upthread:
> mathematics will continue to advance, albeit differently from before. The social structures will not survive however.
Who decreed that? Mathematics predates capitalism and publish-or-perish by a couple of millennia. Euclid’s Elements were not written to benefit the weapons or medical industry.
Maybe this hurts more than it should do because of publish-or-perish.
But humanity is not going to sit around and wait for solutions just so hobbyists can have a moment of glory.
Incredibly delusional and disconnected from the vast majority of people who are voters.
If only.
They certainly can't do them worse than humans.
In this instance however, it's openAI and Anthropic that are pushing people out of the field by running secret models that take the interesting work away and leaves the persons having to review endless slop proofs.
What you didn't make is the AI training process and resulting model. Extremely hard working people built that, and it has value in itself.
Without the AI training process, the model is useless. Otherwise we'd already have been here at GPT-3.
That's an incredibly generous take. If I'd pulled a fraction of the shenanigans prominent companies have to obtain data I'd be thrown under a prison to the thunderous applause of those who have, and are, doing much worse.
I’m interested in how you can support this assertion as it seems at odds with established copyright law
Who knows what they are up to.
This sort of happened at various times in the past, because they hired and/or funded so many mathematicians, and especially before the late 1970s they had many of them working in areas where academic mathematicians weren't working at all, so they were learning more math, or more math that they especially cared about, than the public was. (I was going to write a note here just a few days ago about how NSA has had a "Classified Mathematics Library" for many years.)
For vulnerability scanning, I think the new-capabilities trajectory is good (in the sense of "it will help defenders win") even if governments find ways to get more of it, because there are finitely many bugs and classes of bugs, so at some point more capable models' or longer runs' advantage over less capable models and shorter runs should stop helping them outcompete the less-well-funded defenders, because the defenders will still have learned most of the information that's relevant to achieving successful defenses.
So if NSA gets 5000 units of vulnerability scanning and the public only gets 4000 units, we might still just wipe out all of the pure software vulnerabilities and then go back to worrying about physical supply chain security or side channels or something.
For math, I'm not quite sure! For one thing, there may be things that have no feasibly deployable defense at all even when you understand the underlying mathematics (I'm especially worried about traffic analysis here, because understanding in detail how traffic analysis is done, or how powerful particular techniques are, does not necessarily always or usually make defending against it more convenient or less costly). In a more science fiction scenario, there might also not be any efficient secure cryptographic primitives of some kind, like if it turns out P=NP with reasonably small exponents and reasonably small constant factors.
So far, it looks like open-weight models are lagging less than a year behind frontier capabilities. And I think one year diffusion of technology from "insider lab demo" to widely available is actually pretty fast?
There are lots of research fields which "normal citizen" has no access to - medical and biological research, particle physics. Some of it is somehow publicly controlled (LHC), some of it not at all (commercial pharma research, mostly secret until the final human trials). And most of it reaches "normal citizens" in way more than a year.
(and I'm talking about open-weight models. The availability of commercial AI models from private preview to included-in-your-$100-subscription is currently like 4 months)
Would that impact their own ability of solving Mathematics problems? I mean as a programmer I'm already seeing that impact on the programmers -- sure the best of us can leverage AI to achieve unimaginable things, but many of us are simply vibe coding.
Of course we can assume that it is only the best of us that really matters, and the rest of us are not going to produce anything substantially useful ANYWAY, it might as well to replace the rest of us with AI, but my worry is -- does that really have ZERO impact on the human specie's ability to produce "the best of us"? After all, they don't grow on trees.
I imagine the same will be true of AI, but I'll say that in the short term AI is going to make mathematicians better because it solves the breadth problem. Again, I feel like this Barnette conjecture got solved (if it is solved) because of some clever partition function sums which are intellectually tractable but simply too far out of anything I'd seen before (I see the apparition of my GT combinatorics professor intoning gravely that "everyone knows that, Jake, you're an idiot"). Maybe AI will help identify common threads far greater than Google and journal search.
I think if I had ChatGPT when I was 20 and working on this problem for the first time I might not have solved it, but I would have learned every angle and facet of it far more quickly. But then again I would not have spent so many late nights staring at the Országház across the Danube and letting my mind drift and bump against the problem like spilled cargo in the river.
Now if we can prove this, expand it to the whole spectrum of academic studies, and somehow convince 99.99% of us that they are basically garbage and we don’t care about them — sure the elites will throw UBI around but that’s it — then maybe AI is very positive to the human specie.
Oh we better pick up the speed of cloning and artificial fertilization quickly, because people who are told to be garbage probably have no interests in boring children, and it is still a myth how genies are born and grown. We need that diversity.
BTW the whole scheme reads like the background of a Chinese net novel 赛博英雄传.
It has been super helpful in delineating where the crucial concept came from. The proof is rather simple as graph theory proofs go, but it does seem to use some constructions that would only seem obvious if you had serious physics experience with partition function and calculating energy states that cancel out. It's not a wholly alien bolt from the heavens, but I can also see how there hasn't been a human being with the broad theoretical physics knowledge combined with the deep graph theory experience in planar graphs to come up with this idea. I don't know, I'm looking for precedents of this formulation and some old papers of Penrose counting the number of edge colorings of this same graph type are coming up, the line of argument at least rhymes.
But I agree with the thought that this sort of progress should not be siloed inside those companies. I propose a tax so that every slop cannon AI video pays for another hour of compute time for advancing mathematics.
I suspect that this might be one of the reasons people inside the labs are scared about AI.
What if they have asked AI how it would wipe out humanity and it came up with reasonable answers that they don’t want to publish unlike they do with these math problems?
I think those models and findings should be investigated.
Cure for aging? What do you reckon that'd be worth?
Replace “AI” with “supercomputer”.
(Super)computers have been solving many math problems that mathematicians can’t solve. Now they are capable of solving problem types that they weren’t able to solve before. (this applies to other fields as well)
Problem is it’s not clear if there is anything left for humans. Probably yes, since human mathematicians are still more economical.
I guess if you worked together with some people who all put some money into a fund, and by using very modern technologies like 3D printing and modern CAD modelling etc., it should be possible even for private people to build a jet airplane.
The problem rather is that the government does an insane amount of gatekeeping to prevent this from happening (enforcing expensive and time-consuming certifications on airplanes and pilots etc.).
The concern is about elite level researchers no longer being able to move the industry forward in a public way, and leaving potentially all major discoveries in private hands going forward.
Possible worst case scenario in your case, you personally miss out on a luxury item.
Possible worst case scenario in the topic case, an AI company controls the only intelligence that discovers and understands the most powerful tools / physics we know of.
Obviously the more intelligent the model, the smaller/more directed the search is. But they spoke about huge numbers of agents working on Navier-Stokes for example (I think it cost >$10m).
But to me it also signals (as if it didn't before!) a great need for the wider AI community to focus exclusively on researching and building AI algorithms and systems that are more humanistic: completely transparent in its workings and the representations they create, super efficient in terms of data and compute, componentised so that individual entities can plug in different bits and rapidly train on their own data, highly adaptive to individual needs, programmable in a real sense, largely independent of corporate influence, easily accessible to everyone across all social and economic strata, and enable individuals to grow/learn/reach their full potential.
Is this possible? I think so, but it will require ingenuity and bringing in ideas from (ironically enough) some of the deepest areas of modern mathematics such category theory, algebraic topology etc. which are largely about building abstractions that expose the underlying structure of complex mathematical objects and the relationships between them.
It's already happening to a degree, but the urgency has reached epic levels at this point and it needs to happen at scale.
Humans are the same way sometimes but I guess there's romance in that. If a human had solved it a la Kekulé and said "it came to me in a dream" I would at least understand that.
it's deeply problematic because they are building on open, public results yet they don't provide information on how people may build on it - its exploitative and exclusionary - at least they are consistent
I've seen more goalposts move in the last 3 years than maybe in my whole (lengthy) career up to that point.
> I suspect that this is in fact the source of much of the angst.
Your comment reveals that you absolutely did not read or understand the Field medalists' open letter... Please, why would you refer to their complaints and claim you disagree when you clearly aren't engaging with the arguments presented therein!?
Agree, and, to my mind - shows why the efforts of the Free Software Foundation have been worthwhile all along. We need software to be open / free / libre or the power elite controlling them will ruin the world.
Yes. They have already shown to have no scruples when it comes to making profit and to have little to no morals.
> If you worry about the government, isn't it better that than rando terrorists?
In my country the largest terrorist attack was almost certainly financed by Iran and caused roughly one hundred deaths. This number pales compared to the thousands who died during the latest, US-backed military coup, a move that relied on a doctrine that the US has never stopped asserting [1].
And those morals I mentioned earlier from AI companies? They do not apply to me because I'm not a US citizen. So no, I do not think the US government is the "seal of quality" you think it is.
[1] https://en.wikipedia.org/wiki/Monroe_Doctrine
I am currently in Germany. In the 21st Century roughly 60 people have been killed and 160 injured in ~40 terrorist attacks, most of them perpetrated with cars or knives [1]. In comparison, the US' war in Iran has costed Germany 2.781 billion dollars in fuel costs this year alone and the US government has publicly announced its plans to interfere in German politics partially by funding far-right activities [2].
My point being: the probabilities of terrorists shaking the world order with AI are rather low, seeing as even the most successful attacks in this century have been performed with the simplest of technologies. In contrast, the probability of the US flexing its power irresponsibly are rather high, seeing as they have been doing it for a couple years now and are, in fact, doing it right now.
As far as I'm concerned, and from an evidence-based, day-to-day point of view, the "AI in the hands of terrorists" is an irrelevant concern while "the US may abuse its power" is not.
[1] https://en.wikipedia.org/wiki/Terrorism_in_Germany
[2] https://www.theguardian.com/us-news/2026/jul/15/germany-warn...
Has the government stopped Google and Apple? https://news.ycombinator.com/item?id=49964791
"Math" is about uncovering the epistemological foundations of the universe.
Adding AI here does nothing and is probably a regression in that it diverts resources from actual "math" into some sort of LLM wankery that nobody wants.
Which depends on (1) whether there are actual good ideas in it, (2) whether as well as finding the proofs the AIs can explain their ideas in ways humans (and other AIs) can use, and (3) whether the results they prove are ones that really contribute to that rather than being isolated curiosities that don't go anywhere.
I am not expert enough in all these fields, and haven't looked enough at the papers, to assess #1, but in general the way mathematicians have bet is that if you can solve things regarded as important problems you'll usually do so in a way that contains more broadly useful ideas. Differences between how today's AI systems do mathematics and how humans do mathematics might make that less true when it's an AI that solves the problem, but I would still bet that way. I'd be surprised if OpenAI's big math dump didn't turn out to contain some ideas, and connections between ideas, that humans find useful.
At the moment the AIs are worse than good humans at #2. (But some humans are also really bad at #2, including some humans who are very good at proving theorems.) It looks to me as if they're getting better, and I would expect them to continue to do so. I also suspect (but this is only guesswork) that today's publicly-available frontier AIs may be able to answer questions along the lines of "please take a look at this AI-written paper, and tell me what key new ideas it contains and how they relate to other things in the field" well enough to be useful to human mathematicians. (Even when the paper itself was written by a proprietary AI that no one outside OpenAI or Anthropic or Hypothetical New AI Mathematics Lab has access to.)
As for #3, that's always been something of a crapshoot. A lot of mathematicians' effort goes into proving things that approximately no one ever reads or builds on, just as a lot of industrial R&D goes into trying things that don't turn out to make good products. The recent OpenAI dump contains things that sure seem like important building blocks for future mathematics (e.g., the "quasi-Riemann-Hypothesis" thing) but it's hard to know for sure and also hard to know whether, if they do prove things that turn out to be useful, it's only because they've read the human-written literature and aimed at things human beings have said seem likely to be useful.
None of this seems to me like "adding AI here does nothing". Whether what AIs are doing to mathematics at the moment is good on balance is highly debatable, of course, but it's a matter of trading off costs and benefits, rather than there being costs and no benefits.
So basically nothing changes, Math was subject to gatekeeping and policing of the worst kind.
If you were not among the geniuses, and it didn't come to you automagically, you were simply supposed to leave it to the people who did get it and go do work for people of your intelligence. Smugness was too much to take.
Math people, like chess people never made any genuine attempt to help people understand the processes and methods that made math happen.
To me it should have been a field as teachable and ubiquitous as accounting.
The net result is once these methods and processes were worked out by AI, it was over for the human mathematicians.
The "aha" insight for this is actually f**ing wild, it involves a complex valued exponential sum on the edges. I've seen a lot of clever counting arguments before in graph theory but this is the first time I've seen complex roots and annihilating terms like this, the symbolic manipulation tricks in this look like things out of quantum physics. I don't understand where this trick originated, I need to really digest this.
I'm sympathetic to the mathematicians who are worried about the future of their field, but as an outsider I wonder if they couldn't learn from the go community's "recovery" after the introduction of an alien intelligence.
But also I am excited to be living through this new era of programming and new era of mathematics. I'm still saddened that I couldn't be the one to solve this old problem, but now I realize that my personal approaches were really solving a level of this problem even stronger than the original conjecture, and I'm energized to tackle those (in my free time between being a solo founder and father of 3, etc.).
I asked GPT here: https://chatgpt.com/share/6ac5fd7d-0390-83ed-a02a-6d80fc64f6... and it says:
> the exact Barnette argument appears quite novel, but nearly every ingredient in its cancellation trick has a recognizable ancestor.
> The closest precedent is much closer than I expected: in fully packed O(n) loop models, people have been assigning complex phases to the two orientations of a loop and making them cancel for decades. At n=0, the phases are literally +I and -I. And the n->0 limit has specifically been used to extract Hamiltonian cycles/walks.
You can judge better than me. But it's definitely worth it having a research assistant AI with you when reading these papers.
It makes solving advanced math problems feel like cracking a hash. If it's possible, it's just a matter of compute time.
If you can remember the content of any scientific publication and any book in the world, you are able to make use of this knowledge in every step of you proof.
However, this does now answer how the model came up with the specific route it has taken for the proof.
I'm fairly sure your understanding is not fully accurate.
And then they are for sure able to fill their context based on 'smart search on top' to actually progress further.
But obviously, it adds up to something greater than went in; in aggregate, our contributions are something to awe.
But my point is, if you zoom in at the marginal, incremental contributions of any individual human in this process, it's really hard for me to say LLMs are not at the same level already.
On this topic, people like to compare LLMs to Einstein, but as far as I know, Einstein did not zero-shot special relativity in an afternoon. He built it up incrementally over time, it took him three times longer than the time between first ChatGPT release and today, and it depended on centuries of prior art, culminating in the right observation and right notation being available to him in his moment of greatness.
At what level LLMs are is then an entirely separate discussion, I think.
Name three.
So your view is that everything was there at the creation of the universe (it's a possible view, of course)? Or are there any "things" that can create ideas from scratch?
I think it's not impossible that words evolved as adaptations of the environmental sounds with which our ancestors lived. The human creativity producing DNA is also a remix of preexisting molecules formed under evolutionary pressure, so the view that it's turtles all the way down, unintuitive as it is, may not be so indefensible after all.
They combine things, verify it and if it works and progresses the problem, they created something new.
Loaded question. A "brand-new insight" is still built off the work of others. A possibly better way to frame it would be in how many subjectively unintuitive logical leaps have been made from prior work.
"it's a matrix-tree cancellation wearing Kasteleyn's planar signs, run as a Witten index over Penrose-lineage states, evaluated as a fugacity-zero loop gas in an infinitesimal magnetic field — and the reason it reads like physics is that every one of those tools was built for partition functions"
I thought this was pure slop when I read it but there are some clear analogues in these other areas of physics, really neat computational tricks, and a very interesting paper by Penrose calculating Tait colorings I never knew about previously (extremely relevant, actually related to a separate approach I had once taken on this problem). The problem is that the paper isn't saying "aha, we were inspired by the related problems of pairing excited states and creating spanning trees out of cancelled coefficients" it just defines the function apropos of nothing. Which is kind of like the Jacobian counterexample in that it works but doesn't really explain how exactly it got there.
I really think the load-bearing concept here is "prior work". If prior work is considered papers on this problem or graph theory, yes this has one huge subjectively unintuitive logical leap. If "prior work" is the entire corpus of neat computational tricks that physicists derived to make their equations spit out something other than zero or infinity, maybe it's not so crazy?
Makes me wonder how the patent space will be disrupted when that inventiveness step becomes obsolete because of LLMs. Given your example above, it seems like a combination of different methods from many different sources. This would be regarded as inventive, clearly. If eligible patents can now be brute-forced, the bottleneck becomes only selecting the most promising ones and paying for the patent.
Also, once upon a time I wanted to be a patent lawyer. It's incredibly hard to sit for the patent bar if you have a pure math degree and don't have an engineering degree. Thankfully New Hampshire lets anyone sit for the FE exam.
Are there no loads left to be borne?
I am not demotivated though, I have a great consumer privacy product coming out soon that I'm very excited about.
IMO that’s where AI is going: as soon as a problem can be formulated clearly enough, AI will trounce us humans. I have yet to see evidence that it can decide what problems are important at a remotely human level.
I think something like the Collatz conjecture will be solvable not as number theory or ergodic theory but some other completely wacky environment that humans haven't even sniffed at.
But I have an existential dread about it… I don’t see how it cannot, at least in the vast majority of cases. It seems like a grim new reality is emerging where humans can’t contribute any more, and beyond that being incredibly depressing, I also don’t see it playing out well for human relations.
I’d personally much rather risk dying of cancer or facing whatever other fate may await me that these AI labs allege they will fix (with zero evidence yet) than to risk whatever dystopian anti-human future this technology may very well produce. I’d rather my kids have a shot at something, and be guaranteed to die eventually, than to risk them being hopeless in a severely disordered world with a far off promise that they’ll live forever
But can it all survive and thrive under the boulder of an automated existence.
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.
https://github.com/openai/math/blob/main/lean/ComparatorChal...
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 would love to know the true unsubsidized cost of all of this. How many grad student-years did this cost?
Strange
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?
Also there is a larger epistemic problem with the argument to "using AI to meet the goal or solution", which is that the goal is to mentor and train future mathematicians to advance the field.
There is a similar issue in software engineering too: if no one hires junior engineers because AI can do all the work then the upstream pipeline of engineers qualified to work on difficult architectural problems would dry up.
This importance of this is being felt by mathematicians more acutely because the field will collapse quickly if people refuse to join it.
I've been mentoring (or so I'd like to think) a very bright undergraduate mathematician, in fact he was the one who pointed out the final irreducible flaw in my proof last summer. And I am extremely curious to see what he does and if he even finishes his degree in mathematics. He had already expressed to me some dismay that his summer undergrad research program with several Ivy-league math majors got blown out of the water by a few hours of a frontier model. It's making everyone question what the future will look like and what education and training and certification will even look like.
But the future belongs to those who show up. Maybe this is the beginning of a mass democratization of scientific and math research, maybe we are going back to the gentleman-scholar model of amateur researchers and Twitter will be the new Journal of the Royal Society.
i hope so!
You may achieve far more than you plan on and it may come years and years after you think it should happen. You probably haven't met the right problem yet. You will.
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?
I will never meet that person and I will never hold a real conversation with the "creator" of that proof. They will never tell me how they came up with the cancelling exponential summation that cracked the construction. It's just another enigma but one that is far more unknowable than the original problem.
Weavers don't have dibs on those intangibles.
You just made my day, beautifully said. Thank you Sir, for all your thoughts expressed in this thread. You put an human story behind the #180 number.
It’s probably distributed on so much compute that it would never be economical to serve it to you or I or anybody
It has no memory or experience of working on similar problems. Even if it made one of the foundational libraries that I use in a weather forecasting program, it still has no comprehension of the thought process it takes to understand the problem and build it from zero, and if I’m building on that library it just makes fresh assumptions about how things should work.
It’s not a human with experience or expertise, it’s a computer program that’s really good at turning English descriptions into functioning code
If it did it once, it can do it again from zero, and this time you can watch as it works and even it ask it questions. Many of the agents that worked on the problem did not have comprehension of the whole problem. I don't think you need that many tokens to be able to query it for the insights it had during the process.
Isn’t this the issue with using it the way you’re suggesting? At best the model can come up with an after-the-fact rationalization of how to get to the solution, but it doesn’t know what actual path it took to get there - what were interesting traps it fell into, where was a place it was close to the solution but didn’t realize at the time.
Those are things that are valuable to share between humans, those which teach us how to think better, and give us deeper understanding ourselves, and which a model doesn’t have any comprehension of.
Not that things like that can't happen with humans too (Salieri v. Mozart comes to mind).
Another possibility is that they have internal versions of the model with access to training data that is not provided to external users.
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.
What is this then, vibes? Without a machine-checkable proof I'm not sure what to think of any of this.
I think it helps that basically everyone thinks this conjecture is true, it's just been so darn weird to attack. There's this odd thing that the induction proofs of this problem kept running into, which is that the N+1 condition would work except for in one tiny case when it could fail, but it would be covered by a very slightly stronger version of the conjecture. But then that would fail on one tiny case in induction, but you could solve that with another slightly stronger version. Etc., etc. I almost wondered if there were some sort of structure to the increasingly strong conditions and wanted to prove something about the meta-induction between the stronger conditions and the N's that they needed the next level to remain true. But that failed after 5 steps I think (Fable actually helped me write a few hundred test cases to explicitly show that pattern didn't continue forever, thank God).
BTW my existing test suite from previous proof attempts jives with this new algorithm, so I haven't seen any evidence yet that it's incorrect. Waiting for a Lean proof obviously.
/-- Cubic bipartite three-vertex-connected plane graphs have a Hamiltonian cycle. -/ def MainStatement : Prop := ∀ (V : Type u) [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj], G.IsRegularOfDegree 3 → G.IsBipartite → Planar G → ThreeVertexConnected G → HasHamiltonianCycle G
theorem main : MainStatement.{u} := by sorry
the proof is probably split over the constructions in the whole directory.
You mean you ran her over , or someone else ?