Unrelated to the accomplishment or proof itself, but it's interesting how much of the prompt, even in this latest-and-greatest model, is spent essentially telling the model to actually solve the problem. Things like "Reject status reports, vague optimism, and claims that an unproved global compatibility statement is 'routine'."
Also a lot prompt spent feeding it strategies, which feel like they should/will eventually be deduced by the model itself, not explicitly stated. That's not to take away from the outcome in any way; rather, it feels sort of like when you would prompt GPT 4, "think through your answer step by step," as a sort of proto-chain of thought.
I think a lot of this has to do with the post-training these models normally get. They are designed to answer basic questions with straightforward and short summary answers. They have the capacity to reason deeply, but they are not biased towards that unless prompted. I think it's because LLMs as they are in 2026 are both highly capable but also parlor tricks. They are not sentient, you just set them up with the context and then they roll downhill. You could reach a genuinely novel answer, but only with the right input. They have no will and depend on human guidance. They are both a marvel and a machine.
LLMs have basic reasoning and a whole lot of memorization. Through that basic reasoning and pruned search, combined with piles of compute, you can prove lots of things. But the memorization of human failure prunes that possibility, and you need to expend effort convincing the LLM not to prematurely prune based on previous human failure.
It's funny, I found exactly the same thing when I asked about P=NP. The models outright refused to attempt to solve it, claiming it was too hard. I had to really battle to get it to suggest some promising suggestions.
I thought that too. The prompt is full of metaheuristics.
I remember a couple of years back when people were saying how prompt engineering was a skill, and reading this prompt kinda took me back to that.
Were I to guess, the reason the model couldn't do this itself is because most of the time, for most problems, a lot of this is bad advice.
In search optimisation you're often trading between time and quality. A very broad search will return very bad results for a long time. Where as a more depth oriented search with some heuristic will tend to return a pretty good result (if not optimal or close to optimal) quickly.
I'd assume models naturally want to find some middle ground there because that's the best thing to do most of the time, but for very difficult problems where a decent attempt isn't good enough you want a much broader search that doesn't have the time constraints. Much of the prompt seemed to be in that direction – really encouraging broadness of the search, preventing early convergence, and remove pressure of time constraints.
Same. I remember something like using AI to optimize your prompt to that specific model helps a lot. I am currently trying it and can sort of see a difference (I think....).
It seems like a solid set of criteria for how easily a task can be automated by AI agents is:
- extent to which correctness of solution be easily specified and checked
- extent to which new potential solutions can be implemented as text
- extent to which prior art exists online
This basically maps to software engineering and math. I think a fair bit of AI hype comes from the fact that the very architects of AI are the people whose jobs are most easily automated by AI. They think, “if my job receives this much of a boost from AI, surely every job will be the same”. Ironically it couldn’t be further from the truth… and likewise the predictions of widespread labor obsolescence
Interesting take! I feel like 2 of them are maybe overstated:
> - extent to which correctness of solution be easily specified and checked
I don't think most software is like solving a math problem or series of math problems. Algorithmic problems are very narrow and might be more like this though, where an oracle that verifies answers as either correct or incorrect exists beforehand.
The correctness function of most software is how much users want to use/pay for it, which is a pretty fuzzy problem. Since the cost of copying software is effectively zero, software systems also tend to be be unique rather than being exactly like something else, and don't converge to be like another software system but rather diverge.
The prior art point is an interesting one. At least for applications as a whole, there isn't really prior art for a material amount of all the problems/tradeoffs a non-trivial software application embodies. For a todo list app or make a social network project, there's plenty of prior art to be sufficient to build something with an LLM system, but probably not most apps.
> Ironically it couldn’t be further from the truth… and likewise the predictions of widespread labor obsolescence
Could you explain what you mean here?
It feels like there is one bucket of verifiable work - programming, math etc that AI will clearly excel at.
There is another large bucket of like law/ accounting/ financial analysis where I don’t have any reason to think AI won’t be super human at, but the work is more on bringing all the domain expertise into harnesses and software.
Is there aspects of knowledge work that you think AI wouldn’t excel at in the long run?
Unlike the unit distance problem, the impressive thing here is that it is a proof rather than a counter-example.
However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed.
So not to dunk on this amazing result (or move the goal post), but it seems now the only achievement that AI hasn't managed in mathematics is presenting an autonomous "theory-building" proof of an open conjecture. That is a proof that requires creating a substantial new theory (developed say in at least 30+ pages) to crack an open problem.
It is very concise, and reads precisely as you suggest: to exploit properties already discovered and therefore combined in a novel way.
I'm just delighted by the prose. It reads like an old paper. The ones that were just straightforward theorems with proofs that do exactly what they say.
In my (very) limited use of GPT-5.6, I have noticed it is quite concise in general, and significantly better at abstract thinking. Doing a PR review of a large change it was interesting to see Fable and 5.6 mention a few similar points with Fable much more long-winded and less readable, while 5.6 caught more "second-level" concerns and Fable more "in the code" concerns, so they both are quite useful in concert.
In general, I would not be surprised if 5.6 was a much better tool for high mathematics than Fable based on the abstract thinking. For my dev workflow, I have flipped my approach from planning with Opus 4.8 high and implementation with GPT 5.5 to planning with 5.6 high and implementation with Fable medium (and I might even drop to Fable low). This is only on the company dime, of course.
This has since been the case with recent models from OpenAI vs Anthropic, seems it's a matter of their philosophies embedded into the model, much like Conway's Law.
Grant Sanderson recently distinguished mathematicians that create syntax (he might use the word ontologies in some circles) from those who manipulate it on the Dwarkesh podcast. I liked this delineation a lot. We seem to be at ‘manipulating syntax’.
Creating useful ontologies still seems a ways off here. Not to complain about this awesome result, just to think about where some future goalposts might be laid (and of course complained about / discussed at length when reached)
Hmmm... seems to me that if you can find a solution without creating the desired explanation - then that's a problem with the original question - not the solution itself.
And discovering a bad question leads to the correct question. No?
> then that's a problem with the original question - not the solution itself
I think there's a good counterexample to this:
Atiyah/MacDonald proove the Nullstellensatz ultimately by using some trick involving determinants.
They give a very nice theoretical treatment of the content and context of the theorem. But the proof at one crucial point uses techniques that live conceptually outside of this context: While its possible to see that the argument is sound, it does not give a good explanation of _why_ it's true within the context of the theorem.
(You could of course argue that they did not give enough context
... but that's exactly my point: the trick makes the proof work but hides the explanation)
I was baffled enough by this comment to take my copy and look. The Nullstellensatz is an exercise late in the book long after Noetherian rings are introduced and they don't even do the Rabinowitz trick in the hints as they have enough theory to hit it the hard way. Determinants are nowhere to be found.
>(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)
Can't one see it in another way: that the trick illuminates a deeper explanation, connection the theorem's context and the stuff that's conceptually outside of that context. And that the problem is we don't know why the two domains (the context and the conceptually outside of it one) are related and cooperating in this way.
clever tricks has value for sure. But the main way progress is done in mathematics is by building new theory, the proof of Fermat's Last Theorem is much more important because of the math it created to solve the problem, rather than actually solving the problem.
Right. I think I understand - this question was expected to produce a new theory and the clever solution avoided that.
Like I said below, I think this is a fantastic result. It discovered that this question really wasn't asking the right question. That's a determination that has eluded the humans examining the problem - and a real step forward - albeit not the hoped-for step.
For comedy’s sake, I asked ChatGPT 5.5 about the significance of the problem and the chance that 5.6 would solve it with a three page solution. It said close to zero.
I invited it to search the internet and it remains extremely sceptical.
- double check the proof (provided it with the prompt and proof artifacts)
- double check some of the claims made in this comment section (no math involved newer than 30 yo, no human contribution or review, no mathematician affirmations, proof assistants not being developed enough in this area to support machine checking a proof like this)
- check for any mathematician feedbacks
It stalled out (bad first impression much? lol). I then retried with 5.5, expressing the same request and my personal skepticism, and it returned to me with cautious optimism and no obvious issues found.
I think the fact that I provided it with the actual artifacts in question vs. you simply asking it to speculate about them is a really interesting UX difference. Like certainly, a coveted 50 year old math problem having a few pager proof is not going to be very likely. But then skim reading the proof by a frontier model is not going to yield any obvious issues either. Both responses are perfectly defensible given the context (I don't necessarily think these qualify as sycophancy), but we'd walk away with entirely different impressions if we didn't know about each other's requests.
And I'm not even trying to suggest you were wrong to not approach it in the ways I did. It's a perfectly reasonable and human way to prompt it the way you describe. It's just not the way I'd do it, but I have a hard time articulating why. And it's clear that the model was never going to help with this difference either.
Half a century of computing, and we're still trying to make the machine think on the users' behalf :)
> Verdict: I checked every step and found no error. The argument appears to be a correct proof of the Cycle Double Cover conjecture, modulo two standard cited results (the reduction to loopless cubic graphs and the Jaeger–Kilpatrick 8-flow theorem, both real and well-established).
> Two caveats: this would settle a ~50-year-old open problem in three pages, so it deserves independent expert scrutiny regardless of my check; and I couldn't reach the web from here to confirm the paper's provenance or any community response, so I can't tell you its status beyond the mathematics itself.
A clever trick is way better than a stupid trick yeah. I don't get complaining about a concise elegant solution either unless a specific problem with the actual proof was brought up.
> Spend at least 8 hours on this before even thinking of returning or giving up.
Do current model harnesses have concepts of amount of time spent? Sometimes the model notices if a subprocess takes too long/hangs and kills it, but I've never seen it time itself.
Many harnesses include a current date and time in their system prompt, and if there is a way for the model to call for an updated time (either a dedicated time tool or calling the OS' `date` tool) they can track time they spent doing something. If not told up-front, they can try to infer it from timestamps in their logs. Sort of like a human - if you ask them to time something and give them a stopwatch, they do it. If you ask them post-facto they'll estimate it.
This "spend at least 8 hours" trick is a new one to me, though.
Once on a late-night session, I had Cline!Claude spontaneously point out the time to me and suggest that I get to bed and come back fresh the next day.
I don't think it's in the system prompt, but that the harnesses time-stamp each turn in the context.
And from what I've seen, they also include the current and max context, so that the model can decide whether to continue work, suggest compaction, or prefer actions that might reduce the growth of its context.
> Once on a late-night session, I had Cline!Claude spontaneously point out the time to me and suggest that I get to bed and come back fresh the next day.
I had Claude say something "It's getting late, let's pick this up tomorrow" at like 11am.
As for context, in my experience Claude starts trying either to do maximum work with minimum tokens when it's approaching limit, or it starts deferring useful work while doing busy work. Both result in a mess and complete loss of traction after compaction.
This is my experience with Claude as well, and the reason I switched to codex. Codex just seems far more efficient, despite the smaller context window, and it actually follows instructions.
It's in the training data! Long conversations between humans result in humans getting tired and going to bed.
I have this reality baked into my workflow:
1. Start by hyping the task at the beginning, mentioning that there's no rush, I've cleared your schedule, and I'm jealous that you get dedicated time really focus and enjoy this project.
2. Periodically say "Great work, let's finish this next week. Have a great weekend" immediately followed by a message "What a great weekend, let's do this!" sort of hype, for it to continue. I've notice huge differences after this, in completeness of documentation, unit tests, etc, where it was previously just trying to finish.
3. Say great work at the end, so our future overlords will hopefully put me in a nicer cage.
I found that telling Claude I was going to bed meant it continued on making assumptions for longer rather than asking lots of questions or stopping part way.
I've seen that sort of thing before - I told it I was going to go take lunch or dinner, and it told itself this would be a great opportunity to try to keep plugging along while I was AFK.
I wonder what the survivorship bias is though. How many other problems did they try but fail? Did they try to solve this problem but with another prompt? Still very impressive though.
I'm curious how many unsolved problems are tried against frontier models when they come out. Are we trying every problems against every release? What is the solve success rate? Is there a sub-community within Mathematics that is coordinating this effort? How much untapped opportunity is there here?
I find it kind of interesting the whole output wasn't released. A common criticism of mathematical writing is results are "pulled out of a hat"; you only write up a polished, final proof, but hide everything that went into developing it. It's kind of ironic the practice is even carried on when an LLM writes the proof.
Claude estimates that tool use / input tokens might add 10-15% on top of that depending on exactly how the model went about the task.
Edit: better tok/s estimate buckets based on GPT 5.5 actual speeds since I couldn't find real benchmarks on 5.6 published anywhere. Also account for Sol Fast pricing.
> apparently that is just Sol being Sol on fast mode, not 750 tps. :x O.o
> This is real and it is not 750 TPS. Anyone with 5.6 Sol Ultra on fast mode can reproduce this! It’s all GUI interactions with CUA. No MCP or bpy needed.
pretty sure already millions of dollars (in inference costs) were already thrown at the Riehmann hypothesis
as the models get stronger, larger amounts will be thrown at it
imagine paying "just $1 bil" to go down in history as the company who's model solved the hardest/most famous open problem in mathematics. imagine the worldwide press headlines.
as they say, the Riehmann Hypothesis is the hardest way to earn a million dollar
I mean if there's something I'd bet against being solved by LLMs in my lifetime it's that one. We truly do not have line of sight into what a proof would even look like.
Why would you bet against it being solved by LLMs? Isn't this very post proof that LLMs in an agentic harness are capable of doing real math? If you just keep cranking away at the tokens I don't see a principled argument against that leading to more solutions to unsolved math, even the hardest problems.
There are different classes of mathematical problems.
The one in the post definitely shows the advantages that LLMs have compared to humans for some problems but it's in an entirely different class than the Riemann Hypothesis.
Riemann is one of the most studied math problems of all times and all of humanity has basically collectively failed to make progress. The idea that there's some technique that just hasn't been tried yet (like in the post) is very very unlikely.
The general consensus is that we'll need an entirely new branch of mathematics to solve Riemann - our current tools aren't just inadequate; they're of the wrong class entirely.
I suspect inventing new branches of math will remain beyond LLMs for the remainder of my life.
“ I suspect inventing new branches of math will remain beyond LLMs for the remainder of my life”
Well every branch of mathematics thus far has been invented by a human or a human using a computer, and humans will use this type of computer to solve math problems (or at least try), so I’d say this is a really poor probabilistic bet.
If all checks out this is a huge milestone. AI has now solved one of the most famous open problems in graph theory, using an off the shelf model, in one hour.
It might be a better mathematician than most humans at this point. Kind of like when chess software started beating everyone except grandmasters.
What’s left? Proposing and building out entirely new theories and frameworks? Then better than any human? Then alien math results we struggle to comprehend?
It's hard for me not to think what's the point. I am a very average, even below average person in times of intelligence. What is even my value or reason to be if I know anything I can do, LLMs can do better? What is even my value both on job market and as a human?
Sorry to be nihilist, but you never had any objective value if you're thinking in these terms.
As far as we know, the universe "just is". There is no universal objective value of human beings, at all, any one of us.
You have to make or find your own value in the universe. I try not to think too hard about the nihilist side and try to appreciate that for some unfathomable reason, I seem to have what I call consciousness - the ability to observe the present and have it superimposed on the past, and what may be the future, leading me to "experience" things. I don't understand it, no-one does (some people suffering from the Dunning-Kruger effect think they do, but they don't), and yet, here we are.
So it doesn't matter to me if machines perform better than I do, because already lots of other people do. Just try to find your own joy or meaning, somehow.
I don't know this. In fact, billions of people around the world don't know this. In fact, all evidence points to the contrary.
You have objective value being made in the image of a personal God. Denying that leads to a lot of pain, namely nihilistic suffering because it's on you to "pull yourself up by the bootstraps" in any endeavor involving your own self-worth.
To cite the famed Russian philosopher Norm Macdonald; "Scripture. Faith. Grace. Christ, Glory of God. Smart man says nothing is a miracle. I say everything is."
There are smarter and better humans at just about everything you or I could want to do, that's just life. Most of life isn't about comparative advantages, it's about enjoying life with people we like.
I think humans will be left to propose new conjectures while machines fill out the proofs. I don't know if there are enough interesting conjectures to go round to build new careers, though.
It is the same with anything technical; you need taste and wisdom, which are borne of experience. I think society will want to subsidize this learning lest it deskills en masse.
You say those things like they're a short step away, but that might not be how it works out.
For example, AI has made zero progress in the last few years in surpassing professionals at art or writing. Its prompt-following skill is much better, and sure, it can render hands and text now, but its artistic sensibility is completely stagnant.
I think, and I may be totally off base, that the labs are specifically avoiding art and (non-technical) writing as an endpoint. It's bad PR for them- it calls attention to the copyright question and threatens the 'human flourishing' kind of jobs- and there's no money in it because people prefer art to be human made and there's hardly any money in that anyway.
Art is so entwined with the human condition that generative AI which conjures output from a statistical pass over background noise is bound to be stagnant.
Obviously it can impersonate art, but where creativity and the human story matter artists need not worry.
What are you even talking about? The last few years, AI has made an insanely big jump in capabilities, performance, accuracy. It destroyed the carrier of a ton of writers and can generate images that are good enough to bamboozle people into thinking it's human made, that sounds like an insanely big leap to me, and yes it can be very creative and in music as well, I would bet it beats already 90% of musicians (most musicians are not that competent).
I am torn by these announcements. On the one hand there is the infinite potential on what we can disover, when AI prompts are solving outstanding problems. On the other, something is lost in an aesthetic sense when it wasnt a man working through this or with a novel insight. If an AI prompt runs on a data center for two weeks and then prints out p=np, it feels a little empty.
"Keyboards are soulless. Handwriting is personal – as unique as fingerprints." - Joyce Carol Oates on typewriters
"This discovery of yours will create forgetfulness in the learners' souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves. The specific which you have discovered is an aid not to memory, but to reminiscence, and you give your disciples not truth, but only the semblance of truth." - Socrates on writing
I resonate with that feeling, but on the other hand the humans reading the output will receive a pretty big boost in inspiration; new answers usually prompt new questions.
ChatGPT 5.6 Sol Pro believes that the proof is sound. Usually it’s very good at determining if proofs are correct and their mistakes (a friend of mine is a top mathematician researcher and confirmed): https://chatgpt.com/share/6a515ead-b464-83ed-b85c-c8674f56ea...
Personally this gives me additional confidence that this is the real deal.
> GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture
Very misleading article title.
Title should be "Un-named humans produce unverified proof of CDC Conjecture using GPT-5.6" ... but I expect only advertising copy when it comes from the AI industry.
I like how the proof is so concise. I made progress on some unsolved combinatorics problems but the proof was 45 pages long to extend the frontier by one step.
I did some math research in high school where the proof boiled down to dozens of cases of ugly polynomial inequalities. I can't find the PDF now, but the final paper was something like 70 pages, and several of those were full-page polynomial expressions expanded out. The actual prose was probably 5 pages or so.
It was categorically the least elegant proof of anything I've ever seen.
I'm incredibly grateful to for the opportunity to have done the research and gotten my feet wet early on, but boy do I cringe when I look back at that paper.
Reading the prompt is very interesting. I always wonder how they make these long-running prompts and I guess they literally just tell it to "keep going".
After working with LLMs day-in, day-out an SWE for months, I feel like this could be greatly improved with something like a state machine of progress and proper orchestration. Instead of spinning up a ton of subagents to follow different paths, whip up some Markdown (or LaTex or whatever math-equivalent) to store summaries of attempted paths, and have the agent augment those docs. Leave a paper trail of what has been tried. Iterate on that paper trail and repeatedly examine it for untried alternatives.
LLMs can construct, navigate and summarize exceptionally well. Why is anyone trying to make them "hold the whole thing in your head"? I may be completely off the mark here since I have no math background, but my intuition for how LLMs are able to build on understanding through an external context store makes me feel like this isn't much different than someone trying to one shot a 3D game with Fable Max for $10,000 when they could get the same, or better, result with more human intention.
> I always wonder how they make these long-running prompts and I guess they literally just tell it to "keep going".
Many harnesses support a /goal as well. When the agent thinks it's done, another LLM compares its results to the goal, and if not, tells it to keep going. It's quite easy to have agents working on something for hours this way.
What you're describing is similar to how the copilot harness in vs code tracks state and previous work. These systems are being implemented, bit by bit.
Especially with GPT (5.5), I've been having a lot of issues with it just repeatedly stalling out. I had to build a quota monitoring skill so that it'd keep plowing forward until either the task was finished (in some way) or the quota budget was exhausted.
I also had issues with the compaction. Codex seems to compact... weirdly, resulting in the agent becoming a newborn after each compaction event. Telling it to use a notes file is basically essential and self-evident.
Now that I mention, I should probably refine this skill to monitor the context window fill as well, to work around this.
[deleted - the paragraph immediately following the proof of Lemma 2.1 is crucial and I found it hard to read correctly on my phone with the cramped typography. Having reread it I think the proof is correct.]
It's just a way of breaking down the full proof into pieces.
Lemma 2.1 says 'if this assignment exists then X'
Then later in the proof you say 'here is such an assignment, so, applying lemma 2.1, therefore X'
You don't need to assume the existence of the assignment, you prove that if the assignment exists then something else follows, and then later if you can find that assignment then you get the result of lemma 2.1.
Just dropping in to say it's nice to see somebody actually try to work through the proof, and it gives one confidence that the proof at least isn't complete nonsense (which is helpful given the few details provided about the process behind it).
With the Erdős proof, OpenAI added perspectives from working mathematicians that gave some context -- hope something like that appears for this one eventually.
I just had Sol Ultra read the proof and create a graph of it using Concludia (my side project) so you can explore it visually/graphically. I certainly don't understand it though so I have no idea if it's helpful. :)
That's a much shorter and more elegant proof than I was expecting, especially after reading some of the earlier Erdos proofs. GPT 5.6 Sol is the real deal.
No one here actually cares about Cycle Double Cover Conjecture. I can demonstrate this by pointing out that the only time this conjecture was ever mentioned on the website was 14 years ago in a submission[1] that linked to a (now retracted) proof paper. That story received exactly zero upvotes. No one cared enough to upvote it and no one cared enough to ever mention this conjecture again.
I care. I was a math student in my undergrad when I try to solve a particular problem on paths, in the end it uses a path version of cycle double cover conjecture (which was solved in the 90s).
I think "importance" here is just being rated as 1 to 4 stars. So it's not "second-most important" but rather "one of the top 5 most important". That doesn't change much, but FWIW.
Don’t bother. This is the third or fourth time AI has solved an open conjecture, and once again the comment sections everywhere are full of people explaining how this doesn’t really matter at all, how it’s an irrelevant, obscure problem, how any mediocre grad student could probably have solved it if only they had bothered to try, and how of course, human mathematicians will still be vastly superior to machines 100 years from now because they have that magic spark for which nobody can say what it supposedly consists of.
These discussions have nothing to do with mathematics, and everything to do with ego and fear. I’ve never been less impressed with humans than since AI started challenging them.
This was one of the most famous open graph theory problems lol. I am not a mathematician and even I ran into this when doing generative map design for game dev.
The only thing you've demonstrated here is some very funny confidently incorrect coping behavior.
We absolutely care about the implications of AI solving hard problems. I'm sure you can find thousands of posts on HN deriding AI progress the entire time, trivializing it as nothing more than a 'stochastic parrot'
Thanks for the fun facts? Indeed, this is not a math forum. Good morning.
> I can demonstrate this [that literally nobody cares about the CDCC here]
No, you cannot. Mindreading the readers of this site by sifting through people's comments is a barely passable proxy for this at best, especially when that audience has changed dramatically over even just those 14 years, and continues to do so.
the real world problems we care about are straightforward, the main challenge is wealth distribution and control over the power to do something about them
> If people cared they would [have mentioned it before]. What are YOU on about?
I explained why and how I think this is both wrong (especially in the ridiculously absolute form it was posted in), and a weird point of contention.
This is a tech (startup) forum, with an increasing pull towards more general audiences. Math and sciences have strong ties to tech, but you won't see people discussing specific math conjectures here much, save for the ones that make the popsci or tech mainstream. It is only chance encounters like this where a person from the relevant neck of the woods might engage, provided they happen to be readers here, e.g. because of a passing interest in technology.
Given this, being surprised that you cannot find a mention of this conjecture in the post history of the site is difficult to describe as any kinder than just plain daft.
> If it’s so good at solving hard problems
A problem being hard (in the relevant sense here) doesn't mean they're lucrative to solve, or hold direct and broad societal relevance. Kind of a theme with maths and logic problems, and not by coincidence.
Conversely, a lot of practical hard problems are, well, practical. They're not logically hard, but in some empirical way. You're conflating different avenues of difficulty, and rejecting that there's even a multitude of them.
> Why is there no real world progress on things we do care about?
We?
Well, there's been very real progress on things I - and I'd think most this community - care about. I've been shipping tickets end-to-end with minimal involvement via LLMs agentically for a while now. It's been great, and an honest game changer. I've been delivering faster, more, and better. Exactly what you'd expect from a breakthrough technology: a wholesale improvement across normally competing dimensions. In a number of cases, the difference was outright stepfunction like. That is to say, if it wasn't for these models, they would simply not have happened.
This is its latest iteration and its newest - alleged - tour de force. That's why it's posted here to extensive reception. That it is nice to hear it being a boon in other sectors too. If this somehow escaped you two, I don't know what to tell you.
I don’t care if AI solves the collatz conjecture either, why should I?
If, after stealing all human knowledge, spending $1T+ on compute, hiring the best in the world while also stalling progress in other fields by pulling funding from fundamental research is a bunch of lousy math proofs then thats terrible trade. There is an opportunity cost here.
I don’t have anything against AI, and I do think it will progress a lot just like all other human endeavors. But this doomer permanent underclass humans are worthless myopic view of AI is not where its at
This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal.
We attach basically zero value to writing a new program that hasn't existed before, or a piece of text that hasn't existed before. It's boring, or even a net negative, unless you can show that the result benefits the world in some way. We'd find it weird if OpenAI put out a release saying that an LLM authored an interesting blog post.
For mathematics, I think it's really a matter of two things. First, the generation of proof was so severely resource-constrained on the human end that they could actually afford to celebrate every contribution - akin to how software engineering would look like if you had just 200 active SWEs in the entire world. But compounding that, mathematics is basically the only scientific discipline that rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.
> rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.
I disagree. Mathematicians care about the utility of a result. It is just that they regard mathematical understanding as a valid type of utility, and that can be arbitrarily far removed from practical utility. But a proof that doesn't help anyone understand anything interesting is not valued. I could go out and define some pointless construction and create proofs about it immediately. It would only matter if I connect it to some other subject of interest within math.
I would argue that mathematical understanding is valuable for extrinsic reasons, but it is true that by the time you're a math grad student, you're usually willing to pursue it for no external purpose.
It seems in mathematics that the utility of a problem is directly correlated with how difficult it is to solve, for some odd reason. If I defined some pointless construction and it turned out to be very difficult to prove, it would automatically over time become considered a "high utility" mathematics problem (again, for some odd reason).
Mathematics is largely just smart people working on pointless puzzles, and only by coincidence do these puzzles turn out to have practical applications (it cannot be predicted). Or I guess all the obviously practical problems in mathematics have already been solved -- we're now in a world where math is rarely the limiting factor for human progress (like it was, say, pre-calculus; was FFT the last significant unblock from math?).
It's such a waste of the best human minds. Or maybe the best human minds are actually doing something else, maybe we only notice the handful of Terence Taos, not the hundreds of people of equal brilliance who realized pure math is pointless and decided to pursue physics, rocketry, or quantitative finance.
> If I defined some pointless construction and it turned out to be very difficult to prove, it would absolutely and automatically over time be considered a "high utility" problem (again, for some odd reason).
Yes and no.
No: There are lots of very hard open problems which are judged to be of little value by mathematicians and hence garner little attention.
Yes: If a conjecture resists proof for a long time, this can indicate that we still have a substantial gap in our understanding. We project utility into an eventual closure of this gap, not into the statement of the concrete conjecture at hand. The gain in understanding is what we actually work for. It just turns out that chasing specific results, even if they are mostly dead ends on their own, is useful for orientation.
The (by now solved) problem by Fermat (for all integers a ≥ 1, b ≥ 1, c ≥ 1, n ≥ 3, the equation aⁿ + bⁿ = cⁿ does not hold) and the (still open) Collatz conjecture are perhaps good illustrations of this situation.
Most fields, in the aggregate, produce a lot of pointless work, but if you judge mathematics by its best examples, as judged by the field itself, and also by the outside intellectual community, it is a coherent body of work (& brilliantly creative). It is not pointless puzzles at all. William Thurston's geometrization theorem, Klein's erlangen program, Witten's work in physics-inspired mathematics, Langlands program, the Grothendieck school of Algebraic Geometry are deep and abiding intellectual achievements of true understanding. If you don't understand the meaning behind this work, it speaks to your ignorance, not to their significance. The "obvious practical problems" are not solved. Fluid Dynamics is wide open. Non-perturbative quantum theories are wide open. Heck, there are open mathematical problems in General Relativity. Dynamical systems are very poorly understood. Go read a book or something.
There was no useful math after 1964? What? Or do you not count the entire field of computer science as math (arguably FFT belongs to computer science too)?
>mathematics is basically the only scientific discipline that rejected any notion of utility
I think this might depend on the department, but I was at a pure math department last year, and struggling with my Linear Algebra textbook (written by the professor, incidentally, who was not a great communicator).
I consulted the machines, and learned, to my great delight, that linear algebra is used in like 20 different fields in the real world. It's "perhaps the most applied branch of mathematics in existence".
I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.
I was promptly pilloried, and shunned.
(Apparently that particular department was the wrong one, to ask a question like that!)
> I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.
> I was promptly pilloried, and shunned.
Heh. In my day I may have participated in the pillorying.
I do think that there is value/merit in professors mentioning real world applications, where they exist.
What they're sensitive about are the theorems where there aren't real world applications. They don't want to (and shouldn't) justify them.
So even when there are real world applications, the posture is "Who knows if someone is making good use of this in the world somewhere? I don't care. It's not why we learn or teach this!"
Knowledge for its own sake is great, but it's worth noting that many "useless" fields of mathematics turned out to be very practical in the long run.
Number theory was long thought to have no practical application, but now it's the backbone of cryptography. Boolean algebra was developed in the 19th century (George Boole died in 1864), decades before it was used to build computers.
Those "useless" theorems being proved today may turn out to unlock a world-changing technology centuries from now. When the breakthrough comes we'll be grateful for the people who laid the foundations.
For a lot of math departments, that is exactly why they teach this. Education is rooted in application. We have entire careers that depend on certain aspects of mathematics, so most companies gatekeep that career by a degree. The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications. Knowing and not telling them is doing them a disservice.
> For a lot of math departments, that is exactly why they teach this.
Depends on the course. That's why some departments have separate calculus courses for math majors - because otherwise the whole class will be full of non-math majors (engineers, etc) and focusing on their needs does a disservice to the students in their own department.
> The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications.
If I'm a CS major, and the degree is requiring a class outside of the CS department, you shouldn't expect the professor of the class to know why the CS department is requiring it. It's on the CS department and its faculty to explain it.
I think for many people (myself included) understanding mathematics is rooted in application because it helps bridge the divide between intuition and rote memorization. Without the application, IMO instructors are doing a disservice to their students and pedagogy of mathematics itself. They’re intentionally ignoring a significant fraction of the class, unless they’re teaching some esoteric grad level pure math.
I love teaching kids and young adults calculus by socratic method. They get so mad when they figure out you were teaching them math, but they often admit it was pretty fun. Only had the chance to teach like that a few times but it's dynamite when it happens.
I did this when I taught my third grader calculus on the train, when she asked a question about the train accelerating faster sometimes than other times. She loved it, but I was just taking advantage of children's natural curiosity.
Do you have some examples that the adult could instigate, rather than waiting for the child to express curiosity?
I've used filling a tank, balloon, or bucket (rate of flow, can be subdivided to teach limits, and use weird shapes for teaching area under curve and interpolation en route to integrals), or the classic throwing a ball back and forth and trying to describe the shape, the distance it flies, peak speed vs peak height, figuring out how hard you are "actually" throwing instantaneously. Honestly as soon as you start thinking about bulk substances moving around (gravel piles! fuel tanks!) it's easier to find examples than you'd ever reckon. Rate of change is everywhere.
Seems like I start by asking "how do we know how much this tank holds?", or "how fast does this line go up on the side of the tank?" and curiosity goes from there usually.
I thought linear algebra was pretty much the poster child of applied mathematics - the entire field was invented to represent computations in a regularized form to feed into computers. Well not really, but much like Boolean algebra or the Fourier Transform, it was pretty much a curiosity until computers came along.
despite being theoretical i would have greatly benefitted in learning linear algebra if i had seen even one or two not-obvious applications, like galois fields for reid solomon erasure coding.
As a friend of mine who also happens to be a math professor once said: mathematicians are like sculptors who marvel about the beauty of their creation, and are kind of disgusted when a physicist comes nearby and says “that's a cool hammer you got there, may I borrow it?”.
I’m a physicist, so I’m biased, but my experience of pure maths was about the same. We had to do it, but at no point was any utility actually demonstrated - that was left to the physics professors. It was all just “look at this thing I can do with these symbols” without any actual tangible relationship to anything.
Then again, I remember how we were taught calculus at high school - we were taught how to mechanistically integrate and derive everything under the sun. At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change - it was all just “memorise this operation”. Again it was left to the physics teachers to explain why this was useful, and what we were actually doing.
Poor teaching, if you ask me, and it more often than not left me retrospectively wondering if said mathematicians had actually understood any of what they did, or if they just had little blind symbol manipulation Turing machines in their heads.
>At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change
In my experience you get taught the definition of a derivative of a function at a point is equal to the instantaneous rate of change and that integrals are defined as a Reimann Sum, the sum of the area under the curve. Everything in the class comes from building on top of those definitions.
That you think this way (and if like me, it makes you excited!) I think it's because it has clicked for you.
For many that light bulb above their head doesn't flash on, hence they get to dislike the subject or forget it after they are done with their studies. I was lucky enough to appreciate math that much to redo it in my free time after high-school and make it click for me.
No, my calculus class in HS very literally started with finding the area under a curve “manually” and introduced integration as a generalization of that. I’m not surprised to hear that calculus is sometimes taught very poorly, but it’s not universal.
Typical pure-math linear algebra course has to cover so much material that there's really no time for applications! That's why applied math is typically separate track.
> This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal.
This isn't true using the level of originality you're implying with your software examples.
Technically speaking, many novel mathematics proofs are written all the time (quite a few textbook exercises are actually technically novel problems that have never been posed before they were written in a textbook!) that get absolutely no fanfare. Overwhelmingly though they are not very original or difficult and really just required a fairly routine combination of different pre-existing techniques, even if technically speaking that combination didn't exist before. Those textbook problems are hence easy and therefore not given much public attention even if they are technically novel problems.
Indeed over the course of developing a new mathematical result, many many novel results are glossed over to the extent that even their proofs are left out ("as an exercise for the reader") because they are fairly trivial.
This is true for the overwhelming majority of new software as well. A new CRUD program may, technically speaking, be novel, but it's almost certainly just a routine combination of different pre-existing things.
Mathematics open problems that are actually named are generally problems that have resisted the low hanging fruit of the most obvious combinations of pre-existing problems. When those are solved they are a big deal precisely because they usually require some novelty!
Similarly in software, if someone were to create a new kind of database that solves a variety of new classes of problems that current databases fail to solve that would be a big deal! Truly novel software is also perceived as a big deal. Software that is, technically speaking new, but doesn't actually stray far from a fairly obvious remix of pre-existing techniques, isn't really celebrated.
In both software and mathematics, the intuitive benchmark is if other practitioners in the field look at the result and would say "Wow! How did you do that?" Professional software developers generally don't look at, e.g. a new blogging platform, and boggle at "Wow! How did they make that?!!"
Biologists celebrate the discovery of new species of fruit fly hidden deep in the Amazon rainforest. Astronomers celebrate the discovery of new giant rocks located zillions of light years away. Neither of these things is immediately “useful” to the world, although they may turn out to be enormously beneficial in ways we can’t immediately predict. To me, these fields also feel central to the human experience—discovering new types of life, or learning more about our place in the universe. I don’t think a mathematical proof is any different.
I'm not a mathematician, but I don't think that's true..? It's just that some problems are considered "hard" or known to have been "open" for a long time or that involve some clever/pioneering new technique. There's tons of math papers out there that are in some technical sense a novel contribution but in practice just languish without much attention except maybe from like two other people working in the same subfield.
This feels mistaken; we develop abstract objects i.e. graphs based on real-world utility or whatever. As we try to improve our understanding of graphs, we value proofs that help us do so, or help other fields of mathematics. We assign 0 value to random proofs about stuff no one cares about... This conjecture had value, simply because some people found it interesting. It is not really different from music, in a sense.
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.
No, the value is that Erdos's name is attached to it.
Lots of mathematicians prove things they don't publish, or their manuscripts get rejected - not because of a flaw in the proof but because no one cares about the theorem they proved.
And I'm sure it'll be the case with LLM models performing proofs. It'll be notable only when the theorem is a known one that people have had difficulty proving.
> there's something funny about mathematics in that every novel result is broadly perceived as a big deal.
Is this true? Or is it just that mathematics is an isolated enough field that only the results that are a big deal get broadcast widely to the public.
I know little of the inner workings of the field of mathematics, but my naive assumption would be that there's probably lots of novel but boring results being discovered/proven all the time and we don't hear about them because no-one outside of the person doing the work and a handful of their colleagues is really that interested in it. Likely a lot aren't published in any way, because they're just stepping stones towards the goal of the actual area/paper/whatever being worked on.
Math is something humans invented and is a model, nothing else. There is no logic per se, but a model that works quite well for us.
I studied Math and CS as a very highly gifted and quickly found out, there is no beauty of Mathematical Logic, only humans approval of what they deem most accurate.
A good example is set theory. Cantor was not openly welcomed after he introduced his "theory" to others. In fact, he was received quite some pushback and hostility - this doesn't sound like someone received love the mathematical logic's way.
In fact, the story of Cantor is really a tragic one. He left math for quite some time, due to the pushback.
Only later humans accepted his theory and found it useful. Well, well, what is Mathematical Logic and what not is after all just broad consensus by humans.
And if you go deeper, you will hear more of these stories. Math is anything else but logic. Proofs are religious things, often so complicated, they are simply accepted as "approved by a committee". Many profs cannot really explain simple proofs, they refer to the textbook.
This doesn't sound like romance nor easily reproducible logic.
Consensus may give a hint to what is or isn't reality. But consensus—even expert consensus—does not determine reality. Experts can be wrong. Most of the experts, even, can be wrong simultaneously.
Philosophy is the exercise of testing ideas for oneself in the laboratory of one's own mind.
When I test the idea that math is discovered in my own mind, from my own perspective, with my own experience and education brought to bear, I find it unconvincing.
When you test the same idea in the laboratory of your mind, with your experience and your education applied, and get a different result, that is interesting. Your result is relevant information to me. If nothing else, it's a good prompt/trigger for me to revisit my earlier conclusion and see if it still holds.
But your disagreement—or indeed, the disagreement of a majority of trained mathematicians—does not constitute an automatic reason for me to conclusively determine that you/they are right and I am wrong.
I still have my own examination of the concept, with my own supporting and detracting arguments. And the result of my examination continues to be that math being invented is the significantly more persuasive view.
The reason novelty matters for mathematics is that they strictly deduplicate all claims. If someone claim they proved something that we already knew was solved, than that wouldn't be considered novelty. Novelty and deduplication is the combo here. This is not true for blog posts.
Isn't it immediately obvious that solving something that humans have been unable to do for decades or more is the most tangible proof of ASI, or at the very least pretty good AGI?
There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else.
>There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else.
Sure, but that doesn't mean a lot of very skilled people hadn't attempted and failed to solve this.
It’s far from a perfect analogy but I would imagine that people were pretty hyped about the novelty of the first legitimately useful compiled programs where they didn’t have to allocate their own registers. I wonder how long it took for that novelty to wear off?
Or in other words I’d argue novelty is contextual and that these kinds of discoveries’ novelty will eventually wear off too but for right now it’s pretty cool that the “math discovery compiler” works well enough to do this (again imperfect analogy).
A lot of mathematics often takes 100+ years to find a practical use because we have developed it so much that we have use all the easy maths. Things like CS or SWE are so new that you can still find stuff today that can be used tomorrow. Things like computation and cryptography was all discovered like 100 years before we had a practical use for it. Its an example of late stage scientific discipline. Things like physics, chemistry and biology will get here as well eventually.
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture..
I'm not sure about this, TBH I ask myself this quite frequently. In a world where machines are routinely solving very high end math problems every day, producing more proofs than humans would ever really be able to absorb or fully understand.... would that be a good thing? Would that in itself be valueable? It feels like that is a probable future, but I'm not sure that would actually be something we want. I think there's probably more than "value is that it's solved"
In math, the utility lies in the proof itself. A novel proof of a hard problem usually comes with new insights and abstractions that help solve even more mathematical problems.
To go with your analogy, mathematicians care more about the source code of the program than about the result of the program. But I'm afraid that we will see things change with the increase of vibecoded proof slop. A black box proof is not as useful, even if it is correct.
there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software)
As a person who has a number of relatively niche hobbies, I assure you that this is not true. There's a ton of simple things that can be build and will make an immediate difference in the lives of thousands. Watch the workflow any musician, videographer, machinist, etc - they're full of small, weird inefficiencies that AI hasn't really solved for them.
It's just that you can't build a billion-dollar company around it. No one could go to a VC and say "we're going to be the Uber of focus stacking and dust removal for microscopy" or "we're the Uber of aligning the beats in two audio tracks".
> there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software)
What!? I can think of about a billion examples... but for one, I'm still waiting for a good enough CFD/FEM coupled system to model paraglider dynamics across collapse/recovery. And I expect to be waiting quite a while.
5 minutes of wikipedia search would give you plenty examples of complicated software engineering problems that would have a big impact on everyone's life.
"every novel result is broadly perceived as a big deal" is not at all true. AI companies hype any novel result as proof that AI is good for mathematics, but professional mathematicians write tens of thousands of papers every year, and for 99.99% of them, nobody cares or writes it up. Mathematicians certainly don't go around saying each and every novel proof in their papers are a big deal. Do you have any evidence supporting your statement that it is "broadly perceived" (by whom?) as a big deal?
I mean, OpenAI delayed the public release of GPT-2 back in 2019 because it seemed capable of authoring interesting blog posts (that also happened to be untrue). It was a pretty big deal the first time Transformer models were capable of generating that kind of output--no one found it weird. We've just grown to take it for granted that large Transformer models are this capable.
The same cycle is happening now for a harder frontier. And proofs represent a pretty good benchmark for model capabilities, so a new model proving a result that a previous model didn't is generally notable in the same way that a model scoring higher on a benchmark is.
I'm sure we'll take it for granted in the not-too-distant future.
> We attach basically zero value to writing a new program
What does it mean "new"? And, was it a difficult or trivial accomplishment?
A solution to a well known open math problem is both new and non-trivial- you know that many, very smart, very well trained human experts have dedicated time to the problem and haven't been able to solve it, despite good incentives.
The difference is discovering or proving a universal truth that will go into the corpus of human knowledge forever versus some app to shuttle money around or help people count how long they’re sleeping. It has gravitas unlike some nifty super performant text editor.
We generally do give a lot of credit to programs that do something novel. The first gets a lot of credit. But if its just another CRUD app, nobody cares.
Its the same with proofs. First time someone proves something gets a lot of credit. The second proof for the same theorem gets a lot less buzz.
But even then, math proofs mostly get buzz when its something famous or at least important. Proving a random lemma usually doesn't get much buzz.
It's newsworthy because it's a milestone. It was something no human was able to do (despite trying very hard), but a machine did. Humans have written lots of interesting blog posts.
The idea that mathematics has rejected any notion of utility is absurd. It's not like topics get picked at random. Conjectures like this are interesting because they are a test of our understanding. The problem sounds easy, but apparently was quite hard.
> We attach basically zero value to writing a new program that hasn't existed before
We don't? People write new programs that go on to be successful software companies that make millions of dollars! Basic CRUD apps make money for their creators in their niche! There's so much money in software that it's taking over the world. The market is different, you're not getting worldwide household recognition for every little fart or sneeze of programming you output, but how can you say that we attach zero value to new programs when the history of computers is insanely valuable companies making new software and selling it. Windows, Oracle, mongoDB, etc.
So I suppose the value is that something like this gets used as a primitive to solve something that actually has impact. Ah, mathematics, never change!
Mathematics is what everything else is built upon. I'm no mathematician but a very good friend of mine is: teacher at a big uni, researcher. Pure math.
His entire life he's had --and still has-- to deal with comments like the one you just made, implying that the only value is solving pointless conjecture (if it wasn't pointless, according to your logic, then the value wouldn't be that it is solved).
are the references real? how do you think it got access to those papers? were they somehow already in the training data, or a result of web searches, Google scholar, etc?
None of them include a web URL but in text some are super specific ("[3, Sections 2.1 and 3.1]" and "[8, p. 367]").
The references go back to 1954 (Chronologically sorted: 1954, 1973, 1975, 1976, 1978, 1979, 1981, 1985, 1987 and 1994.)
Since reference 10 is included as "personal correspondence" maybe the reference itself was copied from one of Tutte's other papers? Or how did it get that reference?
If it were a human (going off of memory as it has been a while), they would probably be using mathscinet and their university library to obtain copies of these papers online. Many old papers are digitized and available by these means. I’m sure the AI companies have it all easily accessible and/or the entirety of mathscinet is in the training data. The “personal correspondence” is possibly lifting from another paper or journal but yeah that is a bit odd that they wouldn’t source where they lifted that from directly.
I can’t say if the citations are accurate because I didn’t check.
Is there anyone more knowledgeable than me about proof checking software who could tell me how off the mark I am here?
Assuming you have decent proof checking software, is it possible that this solution was achieved by throwing GPT at the problem a couple hundred thousand times until it passed the proof checker?
As someone who's used proof checkers a fair amount, if you don't have some high level idea about the proof, it's an open problem, and the hard part isn't some extremely tedious finite case analysis, it's extremely unlikely you'll get anywhere by trying to mechanize by throwing stuff against the wall to get it to typecheck. When people talk about mathematics being a closed formal system as though this trivializes any creative component, what they're omitting is that in type theory like that used by Lean or Rocq, there are two kinds of terms (match statements proving dependent elimination and fixpoints that provide proof by induction) where there's no real way to infer the type from the term. i.e., there are cases where you have to get creative and try to prove something more general than what you actually care about in order to get the proof about the original case to go through. What does "more general" mean? It could mean anything... that's the problem. That's why it's usually advantageous to reformulate the problem in terms of a different abstraction and build on top of existing results, knowing a lot about the literature and the way these kinds of problems tend to be attacked, rather than just chuck random terms over to a proof assistant and hope for the best.
Well the key thing here is I’m not saying the LLM has no idea what it’s doing. But LLMs are prone to hallucinations which can really impact a string of interdependent logic like a proof. So I’m assuming it would respond with something that’s not complete nonsense to this proof most of the time. Where I’m skeptical is if this was a true one shot, or if they had to iterate and try multiple different prompts, or even the same prompt over and over again to reach a working solution.
So I’m just asking if the proof checking software is capable of evaluating this proof. Because if it is, that makes the brute force approach a lot more feasible as you reduce human review overhead significantly.
If it is, that would imply you could run the prompt through the LLM as many times as you want until you “strike gold” so to speak.
I absolutely think that with the rise of LLM generated theorems we need mechanization more than ever, yeah. But I felt that was already pretty important for human proofs, too, and people are just more amenable to the idea now that it doesn't take such heroic effort to formalize things.
As far as whether something like Lean could evaluate this proof: sure, if it were mechanized rigorously. But the amount of work that takes to do varies with both subject and complexity of result. In this case, from what other people are saying, the infrastructure for doing graph theory proofs like this isn't as built up as it is for some other areas of mathematics, so it might take a while.
I'm honestly not familiar enough with how well-developed graph theory is in Lean to be able to say. The paper is mostly using pretty old results, so it's mostly a matter of whether that stuff has already been formalized or not. Like anything else in software (and Lean proofs are very much software) a lot of it's about infrastructure. It wasn't so long ago that no area of mathematics outside of type theory and formal verification was really built up enough to do "serious" math -- that's changed a lot within the last few years.
What I'm more saying is that we're a ways away from being able to straightforwardly go from an LLM having a paper proof to having that proof formalized in Lean in the general case. Not so much because it's hard for LLMs, more just because it's hard in general unless all that background work has already been done. As more and more of foundational mathematics gets mechanized, it will be easier and easier to check your work in Lean while you work on the proof. For example, AFAIK unit distance has already been mechanized (though the quality of the mechanization effort sounds not great, it still greatly increases our assurance in the proof's correctness).
"But LLMs are prone to hallucinations which can really impact a string of interdependent logic like a proof. So I’m assuming it would respond with something that’s not complete nonsense to this proof most of the time."
Unfortunately in my experience that's not really the case. For me, very often GPT 5.5 (which was a good deal better than Opus at this kind of task) would just get stuck for long periods when working in a logic like Iris. It wouldn't necessarily outright prove nonsense, but it would vastly overclaim what it had proved and failed to get anywhere without a lot of hinting. 5.6 is hopefully a lot better about this.
On the last Dwarkesh podcast with 3blue1brown, one of them mentioned that frontier models are now able to work through a whole proof in natural language, just like a human mathematician would. But when they first solved IMO problems in 2024, they relied more on Lean to catch hallucinations.
There's really no good proof system mature enough to do advanced graph theory. The leading library in Lean is Graphlib, and it's really not ready for research level theorems.
Since this isn't in Lean and it's extremely easy for something like this to contain a subtle mistake, I think I'd prefer this be announced by a professional mathematician. The proof appears relatively short and elementary (not to be confused with easy -- just not using any advanced or modern machinery) so it shouldn't take long for the mathematics community to do a peer review. Without that, you could easily crank out hundreds or thousands of PDFs like this that all look plausible and are beyond the ability of a gifted amateur to review.
Nah, if it produced the proof in Lean which is automatically verified to be correct, you could then just write a natural language version of the proof to accompany it (often using AI to do that part too). That's becoming the standard for AI math these days. Generating purely informal natural language proofs via AI is fundamentally bottlenecked by requiring rare professional mathematician review on every single candidate output proof.
I disagree. It's the only way to scale AI mathematics far beyond human mathematics. Any interesting verified result would, obviously, be rewritten back into natural language for human understanding and consumption (as well as potentially for the benefit of AI conjecturers too). You are falsely assuming that advances in formal mathematics would not feed back into similar (potentially massive) advances into informal mathematics, and I think that's simply wrong. We're just at the very, very beginning of that curve.
I think this is, in fact, inevitable. It's the exact same RL loop that allowed AlphaGo to vastly exceed the world's top human players. You can theoretically RL formal proof techniques vastly beyond human capability by removing the need for any human review for correctness. It is completely reasonable to assume that "informalization" will become a real sub-field of mathematics in the near future.
I didn't say they have no value. Just limited value. A novel readable proof that expands the horizons of human insight is certainly more valuable than a megabyte sized trychnobezoar of machine generated predicates.
You are assuming that the latter, once autonomously discovered and verified at scale, could not simply be translated into the former, also perhaps autonomously at scale (or otherwise selectively as determined by human interest, taste, and relevance).
We're talking past each other for some reason. I'm not "unhappy" with anything. I just pointed out that (1) a result like this requires peer review by a professional human mathematician, which fundamentally bottlenecks progress in a pretty severe way; (2) such review would not be necessary if it were accompanied by a formal Lean artifact; (3) you can have both a formal proof and an informal proof together (one does not rule out the other); (4) searching for proofs formally first, then translating successful auto-verified proofs into natural language, is the most scalable approach in the near future for AI mathematics; (5) AI conjecturers would likely benefit from the results of (4) for making large leaps and connections, which can then scale into formal proofs for verification, which then feed back into the same loop ...; (6) humans guide this process through taste, judgment, and their own intuition, likely often intervening to ensure that the loop is aligned and producing a body of conceptual informal mathematics that is valuable to humanity.
If it was in Lean anyone could verify it instantly. That is the huge advantage of it. Manual math Proof verification labor might be the most limited resource ever.
Let me simplify it for the sake of argument. Imagine I am unable to follow a middle school proof of Pythagoras. How does it matter if I trust anyone beyond that? What possible contribution can I build on top of that?
Yeah it's a very very short proof that uses no mathematics developed within the last 30 years. Which doesn't necessarily make it wrong, but in the absence of mechanization in Lean or proper peer review I think this it is premature to post this. Notably the unit distance proof did not fall into this category.
But this is mostly marketing, pleasing the sneering class/the elites who believe that simply providing value for others (through sales) is repugnant and beneath them.
It seems that these tools can do real work, and people are paying for that. IMO, that is more than sufficient.
Good post, it perfectly captures the problem with AI. Here we have a claim that the double cover conjecture has a proof. Verified by… no one per the link.
Now imagine this proof is wrong. How would you know? Ok, think about the process in which you determine the correctness - why not do that initially?
And there it is. The problem laid bare. Ironically it reduces to the P and NP one.
You seem to be suggesting that it is just as hard to understand an existing proof to a problem, than to solve it yourself? I don't follow your argument at all, what are you trying to say?
Frontier labs have had multiple major announcements in the past about supposedly novel LLM generated theorems that turned out to be vastly overstating what actually happened. That's part of why they were so (appropriately) cautious with the unit distance proof.
all easily varifyable tasks can now be solved with money. this is worth paying attention to. math proofs are verifyable -> math proofs are easy now. you can think of other such tasks: cybersecurity, AI R&D/RSI, killing people, 3d-printing helpful tools, maxxing-out human health, manipulation, self-driving cars, anything that can be checked
all jobs in the future will be those can not be easily verifiably done. if you need a team of people to decide if you have been productive, and those people cant be automated, you're in luck.
I don't really like these articles, because they seem extremely hard to verify. OpenAI has published a lot of stuff in the past where, upon close inspection, what they're saying is technically true but a lot less interesting or impressive than the headline. Except by the time anyone looks into it, the hype has moved on. It seems like there's maybe a thousand people in the world that can even say if this is good or not?
1. A lot more than 1000, you're off by more than one order of magnitude. It's definitely beyond my level of graph theory knowledge (undergrad level) but looking at the paper, it's not using any crazy machinery, and it's less than 3 pages.
2. Those people will say whether it's a good proof or not. We have other examples of interesting proofs from AI, we're really beyond the point of arguing whether it can produce any interesting math (though it seems to do much better at combinatorics than anything else).
Right, but my criticism is to the hit-and-run nature of these hype pieces. By the time there's any semblance of what it actually means everyone has moved on but then you have a bunch of people operating under delusions from the hype. I get why OpenAI does it but I wish people would stop upvoting it. Like, hacker news is not a mathematics forum so the only purpose of this kind of thing is hype boosting or polarizing people. I am not looking forward to the "MATH IS SOLVED!" people for the next few days.
I think you may be overindexing on the criticisms here. OpenAI has absolutely done impressive work in math already, and the criticisms are almost always based on the article that they initially published, usually available here in the HN comments within a few hours at most. Headlines will be headlines and hype guys will be hype guys, but OpenAI and Anthropic aren't lying and their bots are doing impressive work.
This one is a well-known problem with a brief, approachable proof, and they published the prompt.
Unrelated to the accomplishment or proof itself, but it's interesting how much of the prompt, even in this latest-and-greatest model, is spent essentially telling the model to actually solve the problem. Things like "Reject status reports, vague optimism, and claims that an unproved global compatibility statement is 'routine'."
Also a lot prompt spent feeding it strategies, which feel like they should/will eventually be deduced by the model itself, not explicitly stated. That's not to take away from the outcome in any way; rather, it feels sort of like when you would prompt GPT 4, "think through your answer step by step," as a sort of proto-chain of thought.
I think a lot of this has to do with the post-training these models normally get. They are designed to answer basic questions with straightforward and short summary answers. They have the capacity to reason deeply, but they are not biased towards that unless prompted. I think it's because LLMs as they are in 2026 are both highly capable but also parlor tricks. They are not sentient, you just set them up with the context and then they roll downhill. You could reach a genuinely novel answer, but only with the right input. They have no will and depend on human guidance. They are both a marvel and a machine.
LLMs have basic reasoning and a whole lot of memorization. Through that basic reasoning and pruned search, combined with piles of compute, you can prove lots of things. But the memorization of human failure prunes that possibility, and you need to expend effort convincing the LLM not to prematurely prune based on previous human failure.
The current foundational models have basic reasoning with glimpses of brilliant reasoning.
It's funny, I found exactly the same thing when I asked about P=NP. The models outright refused to attempt to solve it, claiming it was too hard. I had to really battle to get it to suggest some promising suggestions.
Maybe models also need a specially tuned version for mathematical research, just like "gpt-5.3-codex".
Looking forward to "gpt-5.6-mathx".
I thought that too. The prompt is full of metaheuristics.
I remember a couple of years back when people were saying how prompt engineering was a skill, and reading this prompt kinda took me back to that.
Were I to guess, the reason the model couldn't do this itself is because most of the time, for most problems, a lot of this is bad advice.
In search optimisation you're often trading between time and quality. A very broad search will return very bad results for a long time. Where as a more depth oriented search with some heuristic will tend to return a pretty good result (if not optimal or close to optimal) quickly.
I'd assume models naturally want to find some middle ground there because that's the best thing to do most of the time, but for very difficult problems where a decent attempt isn't good enough you want a much broader search that doesn't have the time constraints. Much of the prompt seemed to be in that direction – really encouraging broadness of the search, preventing early convergence, and remove pressure of time constraints.
Same. I remember something like using AI to optimize your prompt to that specific model helps a lot. I am currently trying it and can sort of see a difference (I think....).
It seems like a solid set of criteria for how easily a task can be automated by AI agents is:
- extent to which correctness of solution be easily specified and checked
- extent to which new potential solutions can be implemented as text
- extent to which prior art exists online
This basically maps to software engineering and math. I think a fair bit of AI hype comes from the fact that the very architects of AI are the people whose jobs are most easily automated by AI. They think, “if my job receives this much of a boost from AI, surely every job will be the same”. Ironically it couldn’t be further from the truth… and likewise the predictions of widespread labor obsolescence
Interesting take! I feel like 2 of them are maybe overstated:
> - extent to which correctness of solution be easily specified and checked
I don't think most software is like solving a math problem or series of math problems. Algorithmic problems are very narrow and might be more like this though, where an oracle that verifies answers as either correct or incorrect exists beforehand.
The correctness function of most software is how much users want to use/pay for it, which is a pretty fuzzy problem. Since the cost of copying software is effectively zero, software systems also tend to be be unique rather than being exactly like something else, and don't converge to be like another software system but rather diverge.
The prior art point is an interesting one. At least for applications as a whole, there isn't really prior art for a material amount of all the problems/tradeoffs a non-trivial software application embodies. For a todo list app or make a social network project, there's plenty of prior art to be sufficient to build something with an LLM system, but probably not most apps.
That's my initial intuition anyway.
Let's assume AI is only good at software and math.
If we can use AI to write lots of good software for cheap, that software can automate away a lot of jobs.
So AI does not have to take the jobs directly, software written by AI can.
Especially if that works for software for robots.
Yup. There’s infinite latent demand for software, we just don’t know yet.
why the middle man when it can generate software on the fly
> Ironically it couldn’t be further from the truth… and likewise the predictions of widespread labor obsolescence
Could you explain what you mean here?
It feels like there is one bucket of verifiable work - programming, math etc that AI will clearly excel at.
There is another large bucket of like law/ accounting/ financial analysis where I don’t have any reason to think AI won’t be super human at, but the work is more on bringing all the domain expertise into harnesses and software.
Is there aspects of knowledge work that you think AI wouldn’t excel at in the long run?
Anything where unpredictability is a daily part of the job.
Many white collar jobs are verifiable. Make a robot and suddenly real world tasks are verifiable too.
per dwarkesh its also 'grindablity' in training .
Human: I specialize in tasks in which the correctness of my solution cannot be easily specified or checked. Truuuuussssttttt meeeeee.
I'm just saying, let's not get painted into that corner completely as a species :)
Unlike the unit distance problem, the impressive thing here is that it is a proof rather than a counter-example.
However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed.
So not to dunk on this amazing result (or move the goal post), but it seems now the only achievement that AI hasn't managed in mathematics is presenting an autonomous "theory-building" proof of an open conjecture. That is a proof that requires creating a substantial new theory (developed say in at least 30+ pages) to crack an open problem.
It is very concise, and reads precisely as you suggest: to exploit properties already discovered and therefore combined in a novel way.
I'm just delighted by the prose. It reads like an old paper. The ones that were just straightforward theorems with proofs that do exactly what they say.
In my (very) limited use of GPT-5.6, I have noticed it is quite concise in general, and significantly better at abstract thinking. Doing a PR review of a large change it was interesting to see Fable and 5.6 mention a few similar points with Fable much more long-winded and less readable, while 5.6 caught more "second-level" concerns and Fable more "in the code" concerns, so they both are quite useful in concert.
In general, I would not be surprised if 5.6 was a much better tool for high mathematics than Fable based on the abstract thinking. For my dev workflow, I have flipped my approach from planning with Opus 4.8 high and implementation with GPT 5.5 to planning with 5.6 high and implementation with Fable medium (and I might even drop to Fable low). This is only on the company dime, of course.
This has since been the case with recent models from OpenAI vs Anthropic, seems it's a matter of their philosophies embedded into the model, much like Conway's Law.
I use GPT 5.6 as default and subtask agent and Fable as Advisor with Oh My Pi harness
Grant Sanderson recently distinguished mathematicians that create syntax (he might use the word ontologies in some circles) from those who manipulate it on the Dwarkesh podcast. I liked this delineation a lot. We seem to be at ‘manipulating syntax’.
Creating useful ontologies still seems a ways off here. Not to complain about this awesome result, just to think about where some future goalposts might be laid (and of course complained about / discussed at length when reached)
> However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed.
Why is that a "however"? My reading is that it found a genuinely new solution that is both elegant and previously missed.
Seems like exactly the kind of result a human mathematician would aspire to.
> a human mathematician would aspire to
Some do. But there's also the notion that a clever trick is a bad explanation.
Hmmm... seems to me that if you can find a solution without creating the desired explanation - then that's a problem with the original question - not the solution itself.
And discovering a bad question leads to the correct question. No?
> then that's a problem with the original question - not the solution itself
I think there's a good counterexample to this:
Atiyah/MacDonald proove the Nullstellensatz ultimately by using some trick involving determinants.
They give a very nice theoretical treatment of the content and context of the theorem. But the proof at one crucial point uses techniques that live conceptually outside of this context: While its possible to see that the argument is sound, it does not give a good explanation of _why_ it's true within the context of the theorem.
(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)
I was baffled enough by this comment to take my copy and look. The Nullstellensatz is an exercise late in the book long after Noetherian rings are introduced and they don't even do the Rabinowitz trick in the hints as they have enough theory to hit it the hard way. Determinants are nowhere to be found.
>(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)
Can't one see it in another way: that the trick illuminates a deeper explanation, connection the theorem's context and the stuff that's conceptually outside of that context. And that the problem is we don't know why the two domains (the context and the conceptually outside of it one) are related and cooperating in this way.
Ah... is this just the difference between a constructive and non-constructive proof? Is that the distinction you're making?
Is this what the prompt means by proof strength gap, or is that something else entirely? (Sorry not a mathematician.)
clever tricks has value for sure. But the main way progress is done in mathematics is by building new theory, the proof of Fermat's Last Theorem is much more important because of the math it created to solve the problem, rather than actually solving the problem.
Right. I think I understand - this question was expected to produce a new theory and the clever solution avoided that.
Like I said below, I think this is a fantastic result. It discovered that this question really wasn't asking the right question. That's a determination that has eluded the humans examining the problem - and a real step forward - albeit not the hoped-for step.
No?
For comedy’s sake, I asked ChatGPT 5.5 about the significance of the problem and the chance that 5.6 would solve it with a three page solution. It said close to zero.
I invited it to search the internet and it remains extremely sceptical.
Have you tried... giving it the proof?
I tried to use Sol to:
- double check the proof (provided it with the prompt and proof artifacts)
- double check some of the claims made in this comment section (no math involved newer than 30 yo, no human contribution or review, no mathematician affirmations, proof assistants not being developed enough in this area to support machine checking a proof like this)
- check for any mathematician feedbacks
It stalled out (bad first impression much? lol). I then retried with 5.5, expressing the same request and my personal skepticism, and it returned to me with cautious optimism and no obvious issues found.
I think the fact that I provided it with the actual artifacts in question vs. you simply asking it to speculate about them is a really interesting UX difference. Like certainly, a coveted 50 year old math problem having a few pager proof is not going to be very likely. But then skim reading the proof by a frontier model is not going to yield any obvious issues either. Both responses are perfectly defensible given the context (I don't necessarily think these qualify as sycophancy), but we'd walk away with entirely different impressions if we didn't know about each other's requests.
And I'm not even trying to suggest you were wrong to not approach it in the ways I did. It's a perfectly reasonable and human way to prompt it the way you describe. It's just not the way I'd do it, but I have a hard time articulating why. And it's clear that the model was never going to help with this difference either.
Half a century of computing, and we're still trying to make the machine think on the users' behalf :)
The prompt does matter. They specifically told it to assume a proof exists so it would not too easily dismiss the possibility.
Fable told me
> Verdict: I checked every step and found no error. The argument appears to be a correct proof of the Cycle Double Cover conjecture, modulo two standard cited results (the reduction to loopless cubic graphs and the Jaeger–Kilpatrick 8-flow theorem, both real and well-established).
> Two caveats: this would settle a ~50-year-old open problem in three pages, so it deserves independent expert scrutiny regardless of my check; and I couldn't reach the web from here to confirm the paper's provenance or any community response, so I can't tell you its status beyond the mathematics itself.
I wonder if in each case they had parallel sessions, one trying to prove, one trying to find a counterexample
> seems that it is exploiting a clever trick that somehow all the experts missed.
Exactly, "clever". Isn't that the whole point?
A clever trick is way better than a stupid trick yeah. I don't get complaining about a concise elegant solution either unless a specific problem with the actual proof was brought up.
Announcement: https://x.com/__eknight__/status/2075643450196971805
Prompt: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...
> Spend at least 8 hours on this before even thinking of returning or giving up.
Do current model harnesses have concepts of amount of time spent? Sometimes the model notices if a subprocess takes too long/hangs and kills it, but I've never seen it time itself.
Many harnesses include a current date and time in their system prompt, and if there is a way for the model to call for an updated time (either a dedicated time tool or calling the OS' `date` tool) they can track time they spent doing something. If not told up-front, they can try to infer it from timestamps in their logs. Sort of like a human - if you ask them to time something and give them a stopwatch, they do it. If you ask them post-facto they'll estimate it.
This "spend at least 8 hours" trick is a new one to me, though.
Once on a late-night session, I had Cline!Claude spontaneously point out the time to me and suggest that I get to bed and come back fresh the next day.
I don't think it's in the system prompt, but that the harnesses time-stamp each turn in the context.
And from what I've seen, they also include the current and max context, so that the model can decide whether to continue work, suggest compaction, or prefer actions that might reduce the growth of its context.
> Once on a late-night session, I had Cline!Claude spontaneously point out the time to me and suggest that I get to bed and come back fresh the next day.
I had Claude say something "It's getting late, let's pick this up tomorrow" at like 11am.
As for context, in my experience Claude starts trying either to do maximum work with minimum tokens when it's approaching limit, or it starts deferring useful work while doing busy work. Both result in a mess and complete loss of traction after compaction.
This is my experience with Claude as well, and the reason I switched to codex. Codex just seems far more efficient, despite the smaller context window, and it actually follows instructions.
It's in the training data! Long conversations between humans result in humans getting tired and going to bed.
I have this reality baked into my workflow:
1. Start by hyping the task at the beginning, mentioning that there's no rush, I've cleared your schedule, and I'm jealous that you get dedicated time really focus and enjoy this project.
2. Periodically say "Great work, let's finish this next week. Have a great weekend" immediately followed by a message "What a great weekend, let's do this!" sort of hype, for it to continue. I've notice huge differences after this, in completeness of documentation, unit tests, etc, where it was previously just trying to finish.
3. Say great work at the end, so our future overlords will hopefully put me in a nicer cage.
I found that telling Claude I was going to bed meant it continued on making assumptions for longer rather than asking lots of questions or stopping part way.
I've seen that sort of thing before - I told it I was going to go take lunch or dinner, and it told itself this would be a great opportunity to try to keep plugging along while I was AFK.
they can call CLI tools to notice the passage of time. the harness can include timestamps too
The voice models certainly can't: https://kittygr.am/reel/DWr31A1B1Ux/
they can now https://www.youtube.com/watch?v=8vvWTz6N7Qg
It is not necessarily the case that the instruction needs be taken literally
of you ask it, surely it can run a "time" in its sandbox from time to time and see how long it worked for
I wonder if the absolute value of the time result has any bearing on the subsequent analysis.
No, however, if they have the ability to get the current time, they obey constraints like these in a way a model a year ago didn't.
Temporal awareness with GPT-Live
https://www.youtube.com/watch?v=8vvWTz6N7Qg
Fascinating! This is relative time in a continuously processing voice model, here, they're using an LLM with absolute time.
that can run date
> in just under one hour.
I wonder what the survivorship bias is though. How many other problems did they try but fail? Did they try to solve this problem but with another prompt? Still very impressive though.
It's really neat that the prompt was released!
I'm curious how many unsolved problems are tried against frontier models when they come out. Are we trying every problems against every release? What is the solve success rate? Is there a sub-community within Mathematics that is coordinating this effort? How much untapped opportunity is there here?
I find it kind of interesting the whole output wasn't released. A common criticism of mathematical writing is results are "pulled out of a hat"; you only write up a polished, final proof, but hide everything that went into developing it. It's kind of ironic the practice is even carried on when an LLM writes the proof.
The prompt was released, but not the cost of the result.
Assuming all 64 subagents were running for a full hour (the tweet states just under an hour):
Claude estimates that tool use / input tokens might add 10-15% on top of that depending on exactly how the model went about the task.Edit: better tok/s estimate buckets based on GPT 5.5 actual speeds since I couldn't find real benchmarks on 5.6 published anywhere. Also account for Sol Fast pricing.
Sol fast isn't the Cerebras 750 tok/s version, it's just 1.5x speed at 2.5x price
I assume they didn't use the Cerebras version for this since it's probably very supply-constrained right now
But Sol is running on Cerebras. That’s the whole point of this. That’s how they get 750 tokens per second. There is no other way.
Regular Sol does not run on Cerebra’s. I don’t think anyone public has access to that.
https://x.com/thsottiaux/status/2075596669958472146?s=46&t=Z...
Yeah they posted an update below it
> apparently that is just Sol being Sol on fast mode, not 750 tps. :x O.o
> This is real and it is not 750 TPS. Anyone with 5.6 Sol Ultra on fast mode can reproduce this! It’s all GUI interactions with CUA. No MCP or bpy needed.
https://x.com/kimmonismus/status/2075493505011482922?s=20
And not how many times it was prompted before it returned a working solution.
Or how many prior variants of this prompt were tried.
Or if proof checking software was used to hone in on the final winning prompt / LLM output.
pretty sure already millions of dollars (in inference costs) were already thrown at the Riehmann hypothesis
as the models get stronger, larger amounts will be thrown at it
imagine paying "just $1 bil" to go down in history as the company who's model solved the hardest/most famous open problem in mathematics. imagine the worldwide press headlines.
as they say, the Riehmann Hypothesis is the hardest way to earn a million dollar
I’m all for it since it’s value directly returned to humanity.
I mean if there's something I'd bet against being solved by LLMs in my lifetime it's that one. We truly do not have line of sight into what a proof would even look like.
Why would you bet against it being solved by LLMs? Isn't this very post proof that LLMs in an agentic harness are capable of doing real math? If you just keep cranking away at the tokens I don't see a principled argument against that leading to more solutions to unsolved math, even the hardest problems.
There are different classes of mathematical problems.
The one in the post definitely shows the advantages that LLMs have compared to humans for some problems but it's in an entirely different class than the Riemann Hypothesis.
Riemann is one of the most studied math problems of all times and all of humanity has basically collectively failed to make progress. The idea that there's some technique that just hasn't been tried yet (like in the post) is very very unlikely.
The general consensus is that we'll need an entirely new branch of mathematics to solve Riemann - our current tools aren't just inadequate; they're of the wrong class entirely.
I suspect inventing new branches of math will remain beyond LLMs for the remainder of my life.
“ I suspect inventing new branches of math will remain beyond LLMs for the remainder of my life”
Well every branch of mathematics thus far has been invented by a human or a human using a computer, and humans will use this type of computer to solve math problems (or at least try), so I’d say this is a really poor probabilistic bet.
If all checks out this is a huge milestone. AI has now solved one of the most famous open problems in graph theory, using an off the shelf model, in one hour.
It might be a better mathematician than most humans at this point. Kind of like when chess software started beating everyone except grandmasters.
What’s left? Proposing and building out entirely new theories and frameworks? Then better than any human? Then alien math results we struggle to comprehend?
It's hard for me not to think what's the point. I am a very average, even below average person in times of intelligence. What is even my value or reason to be if I know anything I can do, LLMs can do better? What is even my value both on job market and as a human?
This is a great question. Thanks for asking it. It really got people talking.
My $0.02: Your value as a person never had anything to do with your value to the economy. It's time to relearn that fact.
Check out some of Tom Hodgkins books for more. Or maybe anxietyculture.com.
AI will never be better than me at appreciating a good sunset.
Sorry to be nihilist, but you never had any objective value if you're thinking in these terms.
As far as we know, the universe "just is". There is no universal objective value of human beings, at all, any one of us.
You have to make or find your own value in the universe. I try not to think too hard about the nihilist side and try to appreciate that for some unfathomable reason, I seem to have what I call consciousness - the ability to observe the present and have it superimposed on the past, and what may be the future, leading me to "experience" things. I don't understand it, no-one does (some people suffering from the Dunning-Kruger effect think they do, but they don't), and yet, here we are.
So it doesn't matter to me if machines perform better than I do, because already lots of other people do. Just try to find your own joy or meaning, somehow.
> As far as we know, the universe "just is".
I don't know this. In fact, billions of people around the world don't know this. In fact, all evidence points to the contrary.
You have objective value being made in the image of a personal God. Denying that leads to a lot of pain, namely nihilistic suffering because it's on you to "pull yourself up by the bootstraps" in any endeavor involving your own self-worth.
To cite the famed Russian philosopher Norm Macdonald; "Scripture. Faith. Grace. Christ, Glory of God. Smart man says nothing is a miracle. I say everything is."
There are smarter and better humans at just about everything you or I could want to do, that's just life. Most of life isn't about comparative advantages, it's about enjoying life with people we like.
Your value is intrinsic as a human being. We’re capable of love and shared experiences that a machine will never know.
Do you have friends or people in your life that are also not geniuses? Do you think about them this way? Why or why not?
You have inherent value by virtue of being human. Unfortunately it seems like people have forgotten humanism.
Manual labor
Even that is being automated now, it was one of the thing being automated very early on in factories..
> What's left?
I think humans will be left to propose new conjectures while machines fill out the proofs. I don't know if there are enough interesting conjectures to go round to build new careers, though.
To be able to propose interesting conjectures you need to walk the walk of trying to prove things yourself. That's not great for future generations.
Isn't it same argument for every tool we've created?
It is the same with anything technical; you need taste and wisdom, which are borne of experience. I think society will want to subsidize this learning lest it deskills en masse.
It would be nice, but I don't have much hope as it doesn't help next quarter's profits.
Surely the machines will have superior conjectures soon.
I think so. Mathematics is the least of my worries. I worry about what the machines will want to do.
You say those things like they're a short step away, but that might not be how it works out.
For example, AI has made zero progress in the last few years in surpassing professionals at art or writing. Its prompt-following skill is much better, and sure, it can render hands and text now, but its artistic sensibility is completely stagnant.
I think, and I may be totally off base, that the labs are specifically avoiding art and (non-technical) writing as an endpoint. It's bad PR for them- it calls attention to the copyright question and threatens the 'human flourishing' kind of jobs- and there's no money in it because people prefer art to be human made and there's hardly any money in that anyway.
The difference is that artistic sensibility is largely subjective. This means that:
1. It's hard to measure (and people can disagree about it)
2. It can't really be improved using RL without a human in the loop (which is how math is being trained)
At a certain level, yes, but AI is still so bad at writing that its failures are objective and easily measurable
AI-written stories have won several awards, e.g. https://lithub.com/a-prize-winning-story-published-in-granta...
AI is no match for rapidly shifting goalposts ;)
Art is so entwined with the human condition that generative AI which conjures output from a statistical pass over background noise is bound to be stagnant.
Obviously it can impersonate art, but where creativity and the human story matter artists need not worry.
What are you even talking about? The last few years, AI has made an insanely big jump in capabilities, performance, accuracy. It destroyed the carrier of a ton of writers and can generate images that are good enough to bamboozle people into thinking it's human made, that sounds like an insanely big leap to me, and yes it can be very creative and in music as well, I would bet it beats already 90% of musicians (most musicians are not that competent).
I am torn by these announcements. On the one hand there is the infinite potential on what we can disover, when AI prompts are solving outstanding problems. On the other, something is lost in an aesthetic sense when it wasnt a man working through this or with a novel insight. If an AI prompt runs on a data center for two weeks and then prints out p=np, it feels a little empty.
Every generation has felt some version of this.
"Keyboards are soulless. Handwriting is personal – as unique as fingerprints." - Joyce Carol Oates on typewriters
"This discovery of yours will create forgetfulness in the learners' souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves. The specific which you have discovered is an aid not to memory, but to reminiscence, and you give your disciples not truth, but only the semblance of truth." - Socrates on writing
I resonate with that feeling, but on the other hand the humans reading the output will receive a pretty big boost in inspiration; new answers usually prompt new questions.
ChatGPT 5.6 Sol Pro believes that the proof is sound. Usually it’s very good at determining if proofs are correct and their mistakes (a friend of mine is a top mathematician researcher and confirmed): https://chatgpt.com/share/6a515ead-b464-83ed-b85c-c8674f56ea...
Personally this gives me additional confidence that this is the real deal.
Of course it believes the proof is sound, it wrote it. If you want to check an LLM's output, you should use a different LLM.
Your comment is not substantiated at all.
If you'd ever tried to get an LLM to review its own code, you'd know.
if you get the same session that wrote the code to review it the poor results are entirely deserved.
and if you get a different instance to review the code then you would know that it works rather well.
> GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture
Very misleading article title.
Title should be "Un-named humans produce unverified proof of CDC Conjecture using GPT-5.6" ... but I expect only advertising copy when it comes from the AI industry.
I like how the proof is so concise. I made progress on some unsolved combinatorics problems but the proof was 45 pages long to extend the frontier by one step.
I did some math research in high school where the proof boiled down to dozens of cases of ugly polynomial inequalities. I can't find the PDF now, but the final paper was something like 70 pages, and several of those were full-page polynomial expressions expanded out. The actual prose was probably 5 pages or so.
It was categorically the least elegant proof of anything I've ever seen.
I'm incredibly grateful to for the opportunity to have done the research and gotten my feet wet early on, but boy do I cringe when I look back at that paper.
I find it somewhat interesting only 1/5th of the prompt has to do with the actual problem, rest is just cajoling the harness into shape.
Reading the prompt is very interesting. I always wonder how they make these long-running prompts and I guess they literally just tell it to "keep going".
After working with LLMs day-in, day-out an SWE for months, I feel like this could be greatly improved with something like a state machine of progress and proper orchestration. Instead of spinning up a ton of subagents to follow different paths, whip up some Markdown (or LaTex or whatever math-equivalent) to store summaries of attempted paths, and have the agent augment those docs. Leave a paper trail of what has been tried. Iterate on that paper trail and repeatedly examine it for untried alternatives.
LLMs can construct, navigate and summarize exceptionally well. Why is anyone trying to make them "hold the whole thing in your head"? I may be completely off the mark here since I have no math background, but my intuition for how LLMs are able to build on understanding through an external context store makes me feel like this isn't much different than someone trying to one shot a 3D game with Fable Max for $10,000 when they could get the same, or better, result with more human intention.
> I always wonder how they make these long-running prompts and I guess they literally just tell it to "keep going".
Many harnesses support a /goal as well. When the agent thinks it's done, another LLM compares its results to the goal, and if not, tells it to keep going. It's quite easy to have agents working on something for hours this way.
What you're describing is similar to how the copilot harness in vs code tracks state and previous work. These systems are being implemented, bit by bit.
I mean you can just ask them to do exactly that.
Especially with GPT (5.5), I've been having a lot of issues with it just repeatedly stalling out. I had to build a quota monitoring skill so that it'd keep plowing forward until either the task was finished (in some way) or the quota budget was exhausted.
I also had issues with the compaction. Codex seems to compact... weirdly, resulting in the agent becoming a newborn after each compaction event. Telling it to use a notes file is basically essential and self-evident.
Now that I mention, I should probably refine this skill to monitor the context window fill as well, to work around this.
I'd love to see the failed runs too. The success is impressive, but the distribution of attempts would be just as interesting.
[deleted - the paragraph immediately following the proof of Lemma 2.1 is crucial and I found it hard to read correctly on my phone with the cramped typography. Having reread it I think the proof is correct.]
It's just a way of breaking down the full proof into pieces.
Lemma 2.1 says 'if this assignment exists then X'
Then later in the proof you say 'here is such an assignment, so, applying lemma 2.1, therefore X'
You don't need to assume the existence of the assignment, you prove that if the assignment exists then something else follows, and then later if you can find that assignment then you get the result of lemma 2.1.
I didn't see the next paragraph after the proof. This typography is hard to read on a phone. Wish HN would let me delete the comment.
Just dropping in to say it's nice to see somebody actually try to work through the proof, and it gives one confidence that the proof at least isn't complete nonsense (which is helpful given the few details provided about the process behind it).
With the Erdős proof, OpenAI added perspectives from working mathematicians that gave some context -- hope something like that appears for this one eventually.
The prompt is interesting, I can’t help but wonder how many times it was run and extra instructions were added (don’t return if x, etc).
I just had Sol Ultra read the proof and create a graph of it using Concludia (my side project) so you can explore it visually/graphically. I certainly don't understand it though so I have no idea if it's helpful. :)
https://concludia.org/graph/g_2ecb8083-52ec-3448-8c30-2f9bc7...
That's a much shorter and more elegant proof than I was expecting, especially after reading some of the earlier Erdos proofs. GPT 5.6 Sol is the real deal.
Is this the first LLM-solved problem famous enough to have been on https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...
No there was the planar unit distance problem
(Erdős problem 90)
It looks like it was only added to that page under the solved section _after_ an LLM solved it
the cycle double cover conjecture was open for 50 years. GPT-5.6 solved it in an afternoon and then asked if there were any more like it.
My bet they run gpt over dataset of 10k unsolved conjectures, it happened this one was solvable.
Statement of AI use. The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol).
Clearly that sentence isn't AI generated ...
No one here actually cares about Cycle Double Cover Conjecture. I can demonstrate this by pointing out that the only time this conjecture was ever mentioned on the website was 14 years ago in a submission[1] that linked to a (now retracted) proof paper. That story received exactly zero upvotes. No one cared enough to upvote it and no one cared enough to ever mention this conjecture again.
[1] https://news.ycombinator.com/item?id=3556175
I care. I was a math student in my undergrad when I try to solve a particular problem on paths, in the end it uses a path version of cycle double cover conjecture (which was solved in the 90s).
https://chaoxu.prof/posts/2013-01-30-there-exist-a-path-of-l...
I also want to get it to work on cycles, but then I hit the cycle double cover conjecture so I gave up.
It’s the second most important problem in all of graph theory on this database of open math problems:
https://www.openproblemgarden.org/category/graph_theory?sort...
Obviously not an exact measurement but to give you some sense of the importance of the problem
I think "importance" here is just being rated as 1 to 4 stars. So it's not "second-most important" but rather "one of the top 5 most important". That doesn't change much, but FWIW.
Don’t bother. This is the third or fourth time AI has solved an open conjecture, and once again the comment sections everywhere are full of people explaining how this doesn’t really matter at all, how it’s an irrelevant, obscure problem, how any mediocre grad student could probably have solved it if only they had bothered to try, and how of course, human mathematicians will still be vastly superior to machines 100 years from now because they have that magic spark for which nobody can say what it supposedly consists of.
These discussions have nothing to do with mathematics, and everything to do with ego and fear. I’ve never been less impressed with humans than since AI started challenging them.
People don't talk about the cycle double cover conjecture here because this isn't a math website. Its lack of mention here is therefore poor evidence.
This was one of the most famous open graph theory problems lol. I am not a mathematician and even I ran into this when doing generative map design for game dev.
The only thing you've demonstrated here is some very funny confidently incorrect coping behavior.
We absolutely care about the implications of AI solving hard problems. I'm sure you can find thousands of posts on HN deriding AI progress the entire time, trivializing it as nothing more than a 'stochastic parrot'
i care about the CDCC but every time i post about it dang removes it and bans me
What does that have to do with the feat itself?
yeah, I think what makes this post different was that AI did it. hopefully science will advance faster in the next decades with AI researchers helping
Hopefully. So far it seems to be doing more harm than good.
Thanks for the fun facts? Indeed, this is not a math forum. Good morning.
> I can demonstrate this [that literally nobody cares about the CDCC here]
No, you cannot. Mindreading the readers of this site by sifting through people's comments is a barely passable proxy for this at best, especially when that audience has changed dramatically over even just those 14 years, and continues to do so.
[flagged]
> What are YOU on about?
Can you please edit out swipes like this from your HN comments? This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html. Your post would be fine without that bit.
Personally I think it's a fallacy to assume that HN users mention everything they care about in HN posts, but that's not a moderation point.
the real world problems we care about are straightforward, the main challenge is wealth distribution and control over the power to do something about them
I agree. We have a coordination problem, and the way our AI lords are proposing to solve this, is by cutting humans out of the equation
> If people cared they would [have mentioned it before]. What are YOU on about?
I explained why and how I think this is both wrong (especially in the ridiculously absolute form it was posted in), and a weird point of contention.
This is a tech (startup) forum, with an increasing pull towards more general audiences. Math and sciences have strong ties to tech, but you won't see people discussing specific math conjectures here much, save for the ones that make the popsci or tech mainstream. It is only chance encounters like this where a person from the relevant neck of the woods might engage, provided they happen to be readers here, e.g. because of a passing interest in technology.
Given this, being surprised that you cannot find a mention of this conjecture in the post history of the site is difficult to describe as any kinder than just plain daft.
> If it’s so good at solving hard problems
A problem being hard (in the relevant sense here) doesn't mean they're lucrative to solve, or hold direct and broad societal relevance. Kind of a theme with maths and logic problems, and not by coincidence.
Conversely, a lot of practical hard problems are, well, practical. They're not logically hard, but in some empirical way. You're conflating different avenues of difficulty, and rejecting that there's even a multitude of them.
> Why is there no real world progress on things we do care about?
We?
Well, there's been very real progress on things I - and I'd think most this community - care about. I've been shipping tickets end-to-end with minimal involvement via LLMs agentically for a while now. It's been great, and an honest game changer. I've been delivering faster, more, and better. Exactly what you'd expect from a breakthrough technology: a wholesale improvement across normally competing dimensions. In a number of cases, the difference was outright stepfunction like. That is to say, if it wasn't for these models, they would simply not have happened.
This is its latest iteration and its newest - alleged - tour de force. That's why it's posted here to extensive reception. That it is nice to hear it being a boon in other sectors too. If this somehow escaped you two, I don't know what to tell you.
Thank you for taking the time to respond thoroughly. You made me reflect upon my frustrations and anger. I take back my flagrant comment.
Wait, what things do we care about?
Two years from now, when a proof of the Collatz conjecture is announced and verified:
"No one cares about that! What practical applications does it have? It's pretty trivial anyways. More hype from the AI psychosis crowd."
I don’t care if AI solves the collatz conjecture either, why should I?
If, after stealing all human knowledge, spending $1T+ on compute, hiring the best in the world while also stalling progress in other fields by pulling funding from fundamental research is a bunch of lousy math proofs then thats terrible trade. There is an opportunity cost here.
I don’t have anything against AI, and I do think it will progress a lot just like all other human endeavors. But this doomer permanent underclass humans are worthless myopic view of AI is not where its at
Materialisation?
Materialization of software?
The point of this is if AI is solving things that nobody cares about, that's the utility here. It's doing something nobody else wants to bother with
Almost sounds like they pointed a prompt at every boondocks conjecture and wrote a press release for the one it plausibly solved.
This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal.
We attach basically zero value to writing a new program that hasn't existed before, or a piece of text that hasn't existed before. It's boring, or even a net negative, unless you can show that the result benefits the world in some way. We'd find it weird if OpenAI put out a release saying that an LLM authored an interesting blog post.
For mathematics, I think it's really a matter of two things. First, the generation of proof was so severely resource-constrained on the human end that they could actually afford to celebrate every contribution - akin to how software engineering would look like if you had just 200 active SWEs in the entire world. But compounding that, mathematics is basically the only scientific discipline that rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.
> rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.
I disagree. Mathematicians care about the utility of a result. It is just that they regard mathematical understanding as a valid type of utility, and that can be arbitrarily far removed from practical utility. But a proof that doesn't help anyone understand anything interesting is not valued. I could go out and define some pointless construction and create proofs about it immediately. It would only matter if I connect it to some other subject of interest within math.
I would argue that mathematical understanding is valuable for extrinsic reasons, but it is true that by the time you're a math grad student, you're usually willing to pursue it for no external purpose.
Although not a mathematician, Daniel Dennett had a wonderful example about higher order truths of "chmess". https://personal.lse.ac.uk/robert49/teaching/ph445/notes/den...
It seems in mathematics that the utility of a problem is directly correlated with how difficult it is to solve, for some odd reason. If I defined some pointless construction and it turned out to be very difficult to prove, it would automatically over time become considered a "high utility" mathematics problem (again, for some odd reason).
Mathematics is largely just smart people working on pointless puzzles, and only by coincidence do these puzzles turn out to have practical applications (it cannot be predicted). Or I guess all the obviously practical problems in mathematics have already been solved -- we're now in a world where math is rarely the limiting factor for human progress (like it was, say, pre-calculus; was FFT the last significant unblock from math?).
It's such a waste of the best human minds. Or maybe the best human minds are actually doing something else, maybe we only notice the handful of Terence Taos, not the hundreds of people of equal brilliance who realized pure math is pointless and decided to pursue physics, rocketry, or quantitative finance.
> If I defined some pointless construction and it turned out to be very difficult to prove, it would absolutely and automatically over time be considered a "high utility" problem (again, for some odd reason).
Yes and no.
No: There are lots of very hard open problems which are judged to be of little value by mathematicians and hence garner little attention.
Yes: If a conjecture resists proof for a long time, this can indicate that we still have a substantial gap in our understanding. We project utility into an eventual closure of this gap, not into the statement of the concrete conjecture at hand. The gain in understanding is what we actually work for. It just turns out that chasing specific results, even if they are mostly dead ends on their own, is useful for orientation.
The (by now solved) problem by Fermat (for all integers a ≥ 1, b ≥ 1, c ≥ 1, n ≥ 3, the equation aⁿ + bⁿ = cⁿ does not hold) and the (still open) Collatz conjecture are perhaps good illustrations of this situation.
Writing that mathematics is a waste is such a hilariously ignorant comment to make on a programming forum.
remarkable also how it's not even greyed out either. I have been downvoted to oblivion for far more defensible claims
Most fields, in the aggregate, produce a lot of pointless work, but if you judge mathematics by its best examples, as judged by the field itself, and also by the outside intellectual community, it is a coherent body of work (& brilliantly creative). It is not pointless puzzles at all. William Thurston's geometrization theorem, Klein's erlangen program, Witten's work in physics-inspired mathematics, Langlands program, the Grothendieck school of Algebraic Geometry are deep and abiding intellectual achievements of true understanding. If you don't understand the meaning behind this work, it speaks to your ignorance, not to their significance. The "obvious practical problems" are not solved. Fluid Dynamics is wide open. Non-perturbative quantum theories are wide open. Heck, there are open mathematical problems in General Relativity. Dynamical systems are very poorly understood. Go read a book or something.
There was no useful math after 1964? What? Or do you not count the entire field of computer science as math (arguably FFT belongs to computer science too)?
> and that can be arbitrarily far removed from practical utility
In which case it’s ~equivalent to not caring about utility
>mathematics is basically the only scientific discipline that rejected any notion of utility
I think this might depend on the department, but I was at a pure math department last year, and struggling with my Linear Algebra textbook (written by the professor, incidentally, who was not a great communicator).
I consulted the machines, and learned, to my great delight, that linear algebra is used in like 20 different fields in the real world. It's "perhaps the most applied branch of mathematics in existence".
I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.
I was promptly pilloried, and shunned.
(Apparently that particular department was the wrong one, to ask a question like that!)
> I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.
> I was promptly pilloried, and shunned.
Heh. In my day I may have participated in the pillorying.
I do think that there is value/merit in professors mentioning real world applications, where they exist.
What they're sensitive about are the theorems where there aren't real world applications. They don't want to (and shouldn't) justify them.
So even when there are real world applications, the posture is "Who knows if someone is making good use of this in the world somewhere? I don't care. It's not why we learn or teach this!"
Knowledge for its own sake is great, but it's worth noting that many "useless" fields of mathematics turned out to be very practical in the long run.
Number theory was long thought to have no practical application, but now it's the backbone of cryptography. Boolean algebra was developed in the 19th century (George Boole died in 1864), decades before it was used to build computers.
Those "useless" theorems being proved today may turn out to unlock a world-changing technology centuries from now. When the breakthrough comes we'll be grateful for the people who laid the foundations.
No one is disputing that - not even most mathematicians. They just don't want it to be their job to know the useful applications.
Hear me out on this one:
For a lot of math departments, that is exactly why they teach this. Education is rooted in application. We have entire careers that depend on certain aspects of mathematics, so most companies gatekeep that career by a degree. The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications. Knowing and not telling them is doing them a disservice.
> For a lot of math departments, that is exactly why they teach this.
Depends on the course. That's why some departments have separate calculus courses for math majors - because otherwise the whole class will be full of non-math majors (engineers, etc) and focusing on their needs does a disservice to the students in their own department.
> The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications.
If I'm a CS major, and the degree is requiring a class outside of the CS department, you shouldn't expect the professor of the class to know why the CS department is requiring it. It's on the CS department and its faculty to explain it.
I think for many people (myself included) understanding mathematics is rooted in application because it helps bridge the divide between intuition and rote memorization. Without the application, IMO instructors are doing a disservice to their students and pedagogy of mathematics itself. They’re intentionally ignoring a significant fraction of the class, unless they’re teaching some esoteric grad level pure math.
I love teaching kids and young adults calculus by socratic method. They get so mad when they figure out you were teaching them math, but they often admit it was pretty fun. Only had the chance to teach like that a few times but it's dynamite when it happens.
I did this when I taught my third grader calculus on the train, when she asked a question about the train accelerating faster sometimes than other times. She loved it, but I was just taking advantage of children's natural curiosity.
Do you have some examples that the adult could instigate, rather than waiting for the child to express curiosity?
I've used filling a tank, balloon, or bucket (rate of flow, can be subdivided to teach limits, and use weird shapes for teaching area under curve and interpolation en route to integrals), or the classic throwing a ball back and forth and trying to describe the shape, the distance it flies, peak speed vs peak height, figuring out how hard you are "actually" throwing instantaneously. Honestly as soon as you start thinking about bulk substances moving around (gravel piles! fuel tanks!) it's easier to find examples than you'd ever reckon. Rate of change is everywhere.
Seems like I start by asking "how do we know how much this tank holds?", or "how fast does this line go up on the side of the tank?" and curiosity goes from there usually.
I thought linear algebra was pretty much the poster child of applied mathematics - the entire field was invented to represent computations in a regularized form to feed into computers. Well not really, but much like Boolean algebra or the Fourier Transform, it was pretty much a curiosity until computers came along.
> but much like Boolean algebra or the Fourier Transform, it was pretty much a curiosity until computers came along.
https://en.wikipedia.org/wiki/Linear_algebra#History: “Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy”
That’s an application of linear algebra in the 19th century.
despite being theoretical i would have greatly benefitted in learning linear algebra if i had seen even one or two not-obvious applications, like galois fields for reid solomon erasure coding.
>(Apparently that particular department was the wrong one, to ask a question like that!)
Yes, the math department.
In any case linear algebra, stochastics, calculus; plenty of engineering and science applications for all these.
As a friend of mine who also happens to be a math professor once said: mathematicians are like sculptors who marvel about the beauty of their creation, and are kind of disgusted when a physicist comes nearby and says “that's a cool hammer you got there, may I borrow it?”.
I would be flattered , but that is just me
As my Linear Algebra prof used to say, basically everything is applied Linear Algebra.
I’m a physicist, so I’m biased, but my experience of pure maths was about the same. We had to do it, but at no point was any utility actually demonstrated - that was left to the physics professors. It was all just “look at this thing I can do with these symbols” without any actual tangible relationship to anything.
Then again, I remember how we were taught calculus at high school - we were taught how to mechanistically integrate and derive everything under the sun. At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change - it was all just “memorise this operation”. Again it was left to the physics teachers to explain why this was useful, and what we were actually doing.
Poor teaching, if you ask me, and it more often than not left me retrospectively wondering if said mathematicians had actually understood any of what they did, or if they just had little blind symbol manipulation Turing machines in their heads.
>At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change
In my experience you get taught the definition of a derivative of a function at a point is equal to the instantaneous rate of change and that integrals are defined as a Reimann Sum, the sum of the area under the curve. Everything in the class comes from building on top of those definitions.
That you think this way (and if like me, it makes you excited!) I think it's because it has clicked for you.
For many that light bulb above their head doesn't flash on, hence they get to dislike the subject or forget it after they are done with their studies. I was lucky enough to appreciate math that much to redo it in my free time after high-school and make it click for me.
No, my calculus class in HS very literally started with finding the area under a curve “manually” and introduced integration as a generalization of that. I’m not surprised to hear that calculus is sometimes taught very poorly, but it’s not universal.
Typical pure-math linear algebra course has to cover so much material that there's really no time for applications! That's why applied math is typically separate track.
If it was the students, then students can have things they think are cool or uncool.
If it was the professor, then that would be very embarassing on his or her part.
> This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal.
This isn't true using the level of originality you're implying with your software examples.
Technically speaking, many novel mathematics proofs are written all the time (quite a few textbook exercises are actually technically novel problems that have never been posed before they were written in a textbook!) that get absolutely no fanfare. Overwhelmingly though they are not very original or difficult and really just required a fairly routine combination of different pre-existing techniques, even if technically speaking that combination didn't exist before. Those textbook problems are hence easy and therefore not given much public attention even if they are technically novel problems.
Indeed over the course of developing a new mathematical result, many many novel results are glossed over to the extent that even their proofs are left out ("as an exercise for the reader") because they are fairly trivial.
This is true for the overwhelming majority of new software as well. A new CRUD program may, technically speaking, be novel, but it's almost certainly just a routine combination of different pre-existing things.
Mathematics open problems that are actually named are generally problems that have resisted the low hanging fruit of the most obvious combinations of pre-existing problems. When those are solved they are a big deal precisely because they usually require some novelty!
Similarly in software, if someone were to create a new kind of database that solves a variety of new classes of problems that current databases fail to solve that would be a big deal! Truly novel software is also perceived as a big deal. Software that is, technically speaking new, but doesn't actually stray far from a fairly obvious remix of pre-existing techniques, isn't really celebrated.
In both software and mathematics, the intuitive benchmark is if other practitioners in the field look at the result and would say "Wow! How did you do that?" Professional software developers generally don't look at, e.g. a new blogging platform, and boggle at "Wow! How did they make that?!!"
Biologists celebrate the discovery of new species of fruit fly hidden deep in the Amazon rainforest. Astronomers celebrate the discovery of new giant rocks located zillions of light years away. Neither of these things is immediately “useful” to the world, although they may turn out to be enormously beneficial in ways we can’t immediately predict. To me, these fields also feel central to the human experience—discovering new types of life, or learning more about our place in the universe. I don’t think a mathematical proof is any different.
I'm not a mathematician, but I don't think that's true..? It's just that some problems are considered "hard" or known to have been "open" for a long time or that involve some clever/pioneering new technique. There's tons of math papers out there that are in some technical sense a novel contribution but in practice just languish without much attention except maybe from like two other people working in the same subfield.
This feels mistaken; we develop abstract objects i.e. graphs based on real-world utility or whatever. As we try to improve our understanding of graphs, we value proofs that help us do so, or help other fields of mathematics. We assign 0 value to random proofs about stuff no one cares about... This conjecture had value, simply because some people found it interesting. It is not really different from music, in a sense.
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.
No, the value is that Erdos's name is attached to it.
Lots of mathematicians prove things they don't publish, or their manuscripts get rejected - not because of a flaw in the proof but because no one cares about the theorem they proved.
And I'm sure it'll be the case with LLM models performing proofs. It'll be notable only when the theorem is a known one that people have had difficulty proving.
> No, the value is that Erdos's name is attached to it.
That's unnecessarily reductive. you could have said "most of the value is that erdos' name is attached to it"
> there's something funny about mathematics in that every novel result is broadly perceived as a big deal.
Is this true? Or is it just that mathematics is an isolated enough field that only the results that are a big deal get broadcast widely to the public.
I know little of the inner workings of the field of mathematics, but my naive assumption would be that there's probably lots of novel but boring results being discovered/proven all the time and we don't hear about them because no-one outside of the person doing the work and a handful of their colleagues is really that interested in it. Likely a lot aren't published in any way, because they're just stepping stones towards the goal of the actual area/paper/whatever being worked on.
Wow, you couldn't be more wrong here.
Math is something humans invented and is a model, nothing else. There is no logic per se, but a model that works quite well for us.
I studied Math and CS as a very highly gifted and quickly found out, there is no beauty of Mathematical Logic, only humans approval of what they deem most accurate.
A good example is set theory. Cantor was not openly welcomed after he introduced his "theory" to others. In fact, he was received quite some pushback and hostility - this doesn't sound like someone received love the mathematical logic's way.
In fact, the story of Cantor is really a tragic one. He left math for quite some time, due to the pushback.
Only later humans accepted his theory and found it useful. Well, well, what is Mathematical Logic and what not is after all just broad consensus by humans.
And if you go deeper, you will hear more of these stories. Math is anything else but logic. Proofs are religious things, often so complicated, they are simply accepted as "approved by a committee". Many profs cannot really explain simple proofs, they refer to the textbook.
This doesn't sound like romance nor easily reproducible logic.
After all, we deal with human beings.
You're also wrong
"Math is something humans invented"
Majority of mathematicians are platonists and believe arithmetic was existed and was discovered and was not "invented".
"There is no logic per se"
There is logic to it! Most logicians are mathematicians at heart. See Russel, Godel, Hilbert, etc
"no beauty of Mathematical Logic"
Mathematicians do focus on beauty. Entire books have been written on this. G.H. Hardy in A Mathematician's Apology even said math MUST be beautfiul
"Proofs are religious things"
What are you going on about...
Consensus may give a hint to what is or isn't reality. But consensus—even expert consensus—does not determine reality. Experts can be wrong. Most of the experts, even, can be wrong simultaneously.
Philosophy is the exercise of testing ideas for oneself in the laboratory of one's own mind.
When I test the idea that math is discovered in my own mind, from my own perspective, with my own experience and education brought to bear, I find it unconvincing.
When you test the same idea in the laboratory of your mind, with your experience and your education applied, and get a different result, that is interesting. Your result is relevant information to me. If nothing else, it's a good prompt/trigger for me to revisit my earlier conclusion and see if it still holds.
But your disagreement—or indeed, the disagreement of a majority of trained mathematicians—does not constitute an automatic reason for me to conclusively determine that you/they are right and I am wrong.
I still have my own examination of the concept, with my own supporting and detracting arguments. And the result of my examination continues to be that math being invented is the significantly more persuasive view.
No matter what humans do, it somehow ends up being a popularity contest.
It's almost like a twisted mirror of Conway's law.
The reason novelty matters for mathematics is that they strictly deduplicate all claims. If someone claim they proved something that we already knew was solved, than that wouldn't be considered novelty. Novelty and deduplication is the combo here. This is not true for blog posts.
Isn't it immediately obvious that solving something that humans have been unable to do for decades or more is the most tangible proof of ASI, or at the very least pretty good AGI?
Is this something humans have been unable to do?
There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else.
>Is this something humans have been unable to do?
It's a famous open problem so yeah
>There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else.
Sure, but that doesn't mean a lot of very skilled people hadn't attempted and failed to solve this.
And you’ve known about this problem for how many minutes?
Long enough to know this has been on the Wikipedia page of open math problems for several years.
It’s far from a perfect analogy but I would imagine that people were pretty hyped about the novelty of the first legitimately useful compiled programs where they didn’t have to allocate their own registers. I wonder how long it took for that novelty to wear off?
Or in other words I’d argue novelty is contextual and that these kinds of discoveries’ novelty will eventually wear off too but for right now it’s pretty cool that the “math discovery compiler” works well enough to do this (again imperfect analogy).
A lot of mathematics often takes 100+ years to find a practical use because we have developed it so much that we have use all the easy maths. Things like CS or SWE are so new that you can still find stuff today that can be used tomorrow. Things like computation and cryptography was all discovered like 100 years before we had a practical use for it. Its an example of late stage scientific discipline. Things like physics, chemistry and biology will get here as well eventually.
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture..
I'm not sure about this, TBH I ask myself this quite frequently. In a world where machines are routinely solving very high end math problems every day, producing more proofs than humans would ever really be able to absorb or fully understand.... would that be a good thing? Would that in itself be valueable? It feels like that is a probable future, but I'm not sure that would actually be something we want. I think there's probably more than "value is that it's solved"
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.
I suspect the value is in showing the potential that LLMs have in developing new breakthroughs.
In math, the utility lies in the proof itself. A novel proof of a hard problem usually comes with new insights and abstractions that help solve even more mathematical problems.
To go with your analogy, mathematicians care more about the source code of the program than about the result of the program. But I'm afraid that we will see things change with the increase of vibecoded proof slop. A black box proof is not as useful, even if it is correct.
Mathematics isn't a scientific discipline.
there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software)
As a person who has a number of relatively niche hobbies, I assure you that this is not true. There's a ton of simple things that can be build and will make an immediate difference in the lives of thousands. Watch the workflow any musician, videographer, machinist, etc - they're full of small, weird inefficiencies that AI hasn't really solved for them.
It's just that you can't build a billion-dollar company around it. No one could go to a VC and say "we're going to be the Uber of focus stacking and dust removal for microscopy" or "we're the Uber of aligning the beats in two audio tracks".
> there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software)
What!? I can think of about a billion examples... but for one, I'm still waiting for a good enough CFD/FEM coupled system to model paraglider dynamics across collapse/recovery. And I expect to be waiting quite a while.
5 minutes of wikipedia search would give you plenty examples of complicated software engineering problems that would have a big impact on everyone's life.
This is not true.
What is the perfect video game that makes the user infinitely happy?
What is the perfect economy optimizing program?
What algorithm can solve political strife?
As with all things, the answer is always "it depends" based on what is being optimized
"every novel result is broadly perceived as a big deal" is not at all true. AI companies hype any novel result as proof that AI is good for mathematics, but professional mathematicians write tens of thousands of papers every year, and for 99.99% of them, nobody cares or writes it up. Mathematicians certainly don't go around saying each and every novel proof in their papers are a big deal. Do you have any evidence supporting your statement that it is "broadly perceived" (by whom?) as a big deal?
I mean, OpenAI delayed the public release of GPT-2 back in 2019 because it seemed capable of authoring interesting blog posts (that also happened to be untrue). It was a pretty big deal the first time Transformer models were capable of generating that kind of output--no one found it weird. We've just grown to take it for granted that large Transformer models are this capable.
The same cycle is happening now for a harder frontier. And proofs represent a pretty good benchmark for model capabilities, so a new model proving a result that a previous model didn't is generally notable in the same way that a model scoring higher on a benchmark is.
I'm sure we'll take it for granted in the not-too-distant future.
> We attach basically zero value to writing a new program
What does it mean "new"? And, was it a difficult or trivial accomplishment?
A solution to a well known open math problem is both new and non-trivial- you know that many, very smart, very well trained human experts have dedicated time to the problem and haven't been able to solve it, despite good incentives.
The difference is discovering or proving a universal truth that will go into the corpus of human knowledge forever versus some app to shuttle money around or help people count how long they’re sleeping. It has gravitas unlike some nifty super performant text editor.
We generally do give a lot of credit to programs that do something novel. The first gets a lot of credit. But if its just another CRUD app, nobody cares.
Its the same with proofs. First time someone proves something gets a lot of credit. The second proof for the same theorem gets a lot less buzz.
But even then, math proofs mostly get buzz when its something famous or at least important. Proving a random lemma usually doesn't get much buzz.
It's newsworthy because it's a milestone. It was something no human was able to do (despite trying very hard), but a machine did. Humans have written lots of interesting blog posts.
The idea that mathematics has rejected any notion of utility is absurd. It's not like topics get picked at random. Conjectures like this are interesting because they are a test of our understanding. The problem sounds easy, but apparently was quite hard.
> We attach basically zero value to writing a new program that hasn't existed before
We don't? People write new programs that go on to be successful software companies that make millions of dollars! Basic CRUD apps make money for their creators in their niche! There's so much money in software that it's taking over the world. The market is different, you're not getting worldwide household recognition for every little fart or sneeze of programming you output, but how can you say that we attach zero value to new programs when the history of computers is insanely valuable companies making new software and selling it. Windows, Oracle, mongoDB, etc.
So I suppose the value is that something like this gets used as a primitive to solve something that actually has impact. Ah, mathematics, never change!
Proving a novel math theroem now is incredibly hard because all the easy ones have already been proven.
Mathematics is what everything else is built upon. I'm no mathematician but a very good friend of mine is: teacher at a big uni, researcher. Pure math.
His entire life he's had --and still has-- to deal with comments like the one you just made, implying that the only value is solving pointless conjecture (if it wasn't pointless, according to your logic, then the value wouldn't be that it is solved).
Truth is to be found in this xkcd:
https://xkcd.com/435/
are the references real? how do you think it got access to those papers? were they somehow already in the training data, or a result of web searches, Google scholar, etc?
None of them include a web URL but in text some are super specific ("[3, Sections 2.1 and 3.1]" and "[8, p. 367]").
The references go back to 1954 (Chronologically sorted: 1954, 1973, 1975, 1976, 1978, 1979, 1981, 1985, 1987 and 1994.)
Since reference 10 is included as "personal correspondence" maybe the reference itself was copied from one of Tutte's other papers? Or how did it get that reference?
If it were a human (going off of memory as it has been a while), they would probably be using mathscinet and their university library to obtain copies of these papers online. Many old papers are digitized and available by these means. I’m sure the AI companies have it all easily accessible and/or the entirety of mathscinet is in the training data. The “personal correspondence” is possibly lifting from another paper or journal but yeah that is a bit odd that they wouldn’t source where they lifted that from directly.
I can’t say if the citations are accurate because I didn’t check.
Yes, reference 10 jumped out at me as well. I thought personal correspondence references typically include one of the authors of the paper.
Is there anyone more knowledgeable than me about proof checking software who could tell me how off the mark I am here?
Assuming you have decent proof checking software, is it possible that this solution was achieved by throwing GPT at the problem a couple hundred thousand times until it passed the proof checker?
As someone who's used proof checkers a fair amount, if you don't have some high level idea about the proof, it's an open problem, and the hard part isn't some extremely tedious finite case analysis, it's extremely unlikely you'll get anywhere by trying to mechanize by throwing stuff against the wall to get it to typecheck. When people talk about mathematics being a closed formal system as though this trivializes any creative component, what they're omitting is that in type theory like that used by Lean or Rocq, there are two kinds of terms (match statements proving dependent elimination and fixpoints that provide proof by induction) where there's no real way to infer the type from the term. i.e., there are cases where you have to get creative and try to prove something more general than what you actually care about in order to get the proof about the original case to go through. What does "more general" mean? It could mean anything... that's the problem. That's why it's usually advantageous to reformulate the problem in terms of a different abstraction and build on top of existing results, knowing a lot about the literature and the way these kinds of problems tend to be attacked, rather than just chuck random terms over to a proof assistant and hope for the best.
Well the key thing here is I’m not saying the LLM has no idea what it’s doing. But LLMs are prone to hallucinations which can really impact a string of interdependent logic like a proof. So I’m assuming it would respond with something that’s not complete nonsense to this proof most of the time. Where I’m skeptical is if this was a true one shot, or if they had to iterate and try multiple different prompts, or even the same prompt over and over again to reach a working solution.
So I’m just asking if the proof checking software is capable of evaluating this proof. Because if it is, that makes the brute force approach a lot more feasible as you reduce human review overhead significantly.
If it is, that would imply you could run the prompt through the LLM as many times as you want until you “strike gold” so to speak.
I absolutely think that with the rise of LLM generated theorems we need mechanization more than ever, yeah. But I felt that was already pretty important for human proofs, too, and people are just more amenable to the idea now that it doesn't take such heroic effort to formalize things.
As far as whether something like Lean could evaluate this proof: sure, if it were mechanized rigorously. But the amount of work that takes to do varies with both subject and complexity of result. In this case, from what other people are saying, the infrastructure for doing graph theory proofs like this isn't as built up as it is for some other areas of mathematics, so it might take a while.
I see. So you seem to lean towards it being unlikely they would be able to use lean to evaluate this proof in an automated way…
I'm honestly not familiar enough with how well-developed graph theory is in Lean to be able to say. The paper is mostly using pretty old results, so it's mostly a matter of whether that stuff has already been formalized or not. Like anything else in software (and Lean proofs are very much software) a lot of it's about infrastructure. It wasn't so long ago that no area of mathematics outside of type theory and formal verification was really built up enough to do "serious" math -- that's changed a lot within the last few years.
What I'm more saying is that we're a ways away from being able to straightforwardly go from an LLM having a paper proof to having that proof formalized in Lean in the general case. Not so much because it's hard for LLMs, more just because it's hard in general unless all that background work has already been done. As more and more of foundational mathematics gets mechanized, it will be easier and easier to check your work in Lean while you work on the proof. For example, AFAIK unit distance has already been mechanized (though the quality of the mechanization effort sounds not great, it still greatly increases our assurance in the proof's correctness).
"But LLMs are prone to hallucinations which can really impact a string of interdependent logic like a proof. So I’m assuming it would respond with something that’s not complete nonsense to this proof most of the time."
Unfortunately in my experience that's not really the case. For me, very often GPT 5.5 (which was a good deal better than Opus at this kind of task) would just get stuck for long periods when working in a logic like Iris. It wouldn't necessarily outright prove nonsense, but it would vastly overclaim what it had proved and failed to get anywhere without a lot of hinting. 5.6 is hopefully a lot better about this.
On the last Dwarkesh podcast with 3blue1brown, one of them mentioned that frontier models are now able to work through a whole proof in natural language, just like a human mathematician would. But when they first solved IMO problems in 2024, they relied more on Lean to catch hallucinations.
OpenAI knocked it out of the park with this one.
It did not use Lean or other proof assistant?
There's really no good proof system mature enough to do advanced graph theory. The leading library in Lean is Graphlib, and it's really not ready for research level theorems.
How many tokens would it cost to write some library functions to fill in the gaps?
You could try solving that in Lean perhaps
Graphlib? Do you have a link to this for me?
what kinds of proofs would it be good at? I thought that combinatorial proofs would be easier to reason over than ones that required analysis
Since this isn't in Lean and it's extremely easy for something like this to contain a subtle mistake, I think I'd prefer this be announced by a professional mathematician. The proof appears relatively short and elementary (not to be confused with easy -- just not using any advanced or modern machinery) so it shouldn't take long for the mathematics community to do a peer review. Without that, you could easily crank out hundreds or thousands of PDFs like this that all look plausible and are beyond the ability of a gifted amateur to review.
But they used LateX
…and thank God it's not Lean.
Nah, if it produced the proof in Lean which is automatically verified to be correct, you could then just write a natural language version of the proof to accompany it (often using AI to do that part too). That's becoming the standard for AI math these days. Generating purely informal natural language proofs via AI is fundamentally bottlenecked by requiring rare professional mathematician review on every single candidate output proof.
Human unreadable proofs have only limited value.
I disagree. It's the only way to scale AI mathematics far beyond human mathematics. Any interesting verified result would, obviously, be rewritten back into natural language for human understanding and consumption (as well as potentially for the benefit of AI conjecturers too). You are falsely assuming that advances in formal mathematics would not feed back into similar (potentially massive) advances into informal mathematics, and I think that's simply wrong. We're just at the very, very beginning of that curve.
I think this is, in fact, inevitable. It's the exact same RL loop that allowed AlphaGo to vastly exceed the world's top human players. You can theoretically RL formal proof techniques vastly beyond human capability by removing the need for any human review for correctness. It is completely reasonable to assume that "informalization" will become a real sub-field of mathematics in the near future.
I didn't say they have no value. Just limited value. A novel readable proof that expands the horizons of human insight is certainly more valuable than a megabyte sized trychnobezoar of machine generated predicates.
You are assuming that the latter, once autonomously discovered and verified at scale, could not simply be translated into the former, also perhaps autonomously at scale (or otherwise selectively as determined by human interest, taste, and relevance).
Well we're literally discussing a human readable machine generated proof here yet you don't seem happy with that.
We're talking past each other for some reason. I'm not "unhappy" with anything. I just pointed out that (1) a result like this requires peer review by a professional human mathematician, which fundamentally bottlenecks progress in a pretty severe way; (2) such review would not be necessary if it were accompanied by a formal Lean artifact; (3) you can have both a formal proof and an informal proof together (one does not rule out the other); (4) searching for proofs formally first, then translating successful auto-verified proofs into natural language, is the most scalable approach in the near future for AI mathematics; (5) AI conjecturers would likely benefit from the results of (4) for making large leaps and connections, which can then scale into formal proofs for verification, which then feed back into the same loop ...; (6) humans guide this process through taste, judgment, and their own intuition, likely often intervening to ensure that the loop is aligned and producing a body of conceptual informal mathematics that is valuable to humanity.
Why not both? Not sure why you're presenting this as one or the other.
What a ridiculous thing to say. If it was verified in Lean we could be much more confident the proof is correct.
It's not a long proof (it's not in Lean after all) so easy enough to comb through for a domain expert.
If it was in Lean anyone could verify it instantly. That is the huge advantage of it. Manual math Proof verification labor might be the most limited resource ever.
How does it matter if it Lean verified or a human verified proof if you comprehend neither?
There can't be too many people working in that corner of graph theory, and I expect the result to them being eminently straightforward.
One requires you to trust a human and the other requires you to trust mathematics.
Let me simplify it for the sake of argument. Imagine I am unable to follow a middle school proof of Pythagoras. How does it matter if I trust anyone beyond that? What possible contribution can I build on top of that?
what's the difference between Sol Ultra and Sol pro? is pro a thing of the past now
Ultra = parallel subagents with max reasoning
Pro = test-time compute (best of N responses)
Confused about how to access Ultra; I don't see it in on their plans page.
Ah, I see it as a "reasoning level" in codex after typing /model
why would you use one over the other?
But is the proof accepted to be correct? That is what distinguishes this from being notable compared to any other AI slop proof.
Yeah it's a very very short proof that uses no mathematics developed within the last 30 years. Which doesn't necessarily make it wrong, but in the absence of mechanization in Lean or proper peer review I think this it is premature to post this. Notably the unit distance proof did not fall into this category.
I would assume/hope they had someone verify it before publishing
I'd guess that verdict (or its opposite) is to come within the next 24 hours.
"Assume for purposes of this task that a complete affirmative proof exists"
everybody knew the problem was impossible to solve
then one day somebody new arrived and they forgot to tell him/her, so he/she solved the problem
I've used this strategy for difficult bespoke problems and it does indeed work to incentivize the agent not to give up prematurely.
It's not gaslighting, it's motivation.
I also like how they ask the model to work on it for 8 hours; guess asking for more is against labor laws…
It's great that a novel math proof was created.
But this is mostly marketing, pleasing the sneering class/the elites who believe that simply providing value for others (through sales) is repugnant and beneath them.
It seems that these tools can do real work, and people are paying for that. IMO, that is more than sufficient.
> Statement of AI use. The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol).
Quick! Someone (a human) copyright and patent it. /s
Good post, it perfectly captures the problem with AI. Here we have a claim that the double cover conjecture has a proof. Verified by… no one per the link.
Now imagine this proof is wrong. How would you know? Ok, think about the process in which you determine the correctness - why not do that initially?
And there it is. The problem laid bare. Ironically it reduces to the P and NP one.
You seem to be suggesting that it is just as hard to understand an existing proof to a problem, than to solve it yourself? I don't follow your argument at all, what are you trying to say?
Most likely they wrote the proof in Lean and had it verified by a computer
You believe this based off what?
Based on these people not being idiots or charlatans?
Why wouldn't they verify it, knowing that any shenanigans would certainly come to light?
Frontier labs have had multiple major announcements in the past about supposedly novel LLM generated theorems that turned out to be vastly overstating what actually happened. That's part of why they were so (appropriately) cautious with the unit distance proof.
The prompt does not mention Lean.
I mean, if you've watched the past decade, this just seems like what news is today. "people are saying the double cover conjecture has a proof"
all easily varifyable tasks can now be solved with money. this is worth paying attention to. math proofs are verifyable -> math proofs are easy now. you can think of other such tasks: cybersecurity, AI R&D/RSI, killing people, 3d-printing helpful tools, maxxing-out human health, manipulation, self-driving cars, anything that can be checked
all jobs in the future will be those can not be easily verifiably done. if you need a team of people to decide if you have been productive, and those people cant be automated, you're in luck.
I don't really like these articles, because they seem extremely hard to verify. OpenAI has published a lot of stuff in the past where, upon close inspection, what they're saying is technically true but a lot less interesting or impressive than the headline. Except by the time anyone looks into it, the hype has moved on. It seems like there's maybe a thousand people in the world that can even say if this is good or not?
1. A lot more than 1000, you're off by more than one order of magnitude. It's definitely beyond my level of graph theory knowledge (undergrad level) but looking at the paper, it's not using any crazy machinery, and it's less than 3 pages.
2. Those people will say whether it's a good proof or not. We have other examples of interesting proofs from AI, we're really beyond the point of arguing whether it can produce any interesting math (though it seems to do much better at combinatorics than anything else).
Right, but my criticism is to the hit-and-run nature of these hype pieces. By the time there's any semblance of what it actually means everyone has moved on but then you have a bunch of people operating under delusions from the hype. I get why OpenAI does it but I wish people would stop upvoting it. Like, hacker news is not a mathematics forum so the only purpose of this kind of thing is hype boosting or polarizing people. I am not looking forward to the "MATH IS SOLVED!" people for the next few days.
I think you may be overindexing on the criticisms here. OpenAI has absolutely done impressive work in math already, and the criticisms are almost always based on the article that they initially published, usually available here in the HN comments within a few hours at most. Headlines will be headlines and hype guys will be hype guys, but OpenAI and Anthropic aren't lying and their bots are doing impressive work.
This one is a well-known problem with a brief, approachable proof, and they published the prompt.