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.
Even Fable hallucinates. I had it tracking down some very obscure Ancient Greek inscriptions and the response just made up a translation/context for one inscription after "looking it up." Now, it was still a very particular thing and I really had to get into the weeds to push it to that point, but who knows how many other gaps, near or far, it will happily skip over just for the sake of coherence. I think this is an issue more primarily with LLMs than sensory systems like Waymos or all the ML applied to industrial processes--that really only requires pattern recognition, often very impressive and subtle pattern recognition but its no different from an artist learning to tell the difference between Prussian blue and Navy blue or a Sommelier learning the fine distinctions between various regions of Bordeaux. Language has many more avenues and introduces inherent contradictions that do not always lend themselves to easy resolution. But there are no alternatives paths visible to the models, there is only ever the next word; stochastic, in the sense that the possibility space is open; deterministic, in the sense that the final response is always a necessary result of every token that came before it in their total sequence. Thus, any response is constantly in the work of erasing any possible alternative, slowly narrowing down what can be written. If contradictions in language necessarily involve interpretation, then the models will only ever choose one at a time, and for them, it will always be the right one. But anyone who understands the subtleties of language can tell you that when it comes to determining the truth of an indeterminate statement, there is never just one right answer; or, rather, the answer which is taken to be the "right" one depends on the possibility of its own reversal into falsehood, if any argument has to be made to justify it.
Something I've noticed is that if you run Qwen 3.6 35B-A3B (Q8) with a low temperature of 0.4, and leave default reasoning turned on, it will spend quite a lot of time in reasoning/thinking mode. But often it does figure out how to solve something on its own by correcting itself within its reasoning loop before it outputs the final 'answer'.
If you watch the progress of the reasoning in llama-server while it's doing the thinking, you can track its progress. Sometimes the dead ends it goes down or things that it considers and then disregards are themselves something useful to re-prompt it with later, and send it 'rolling downhill', to use the metaphor of another commenter here, in another direction towards the same effort.
Putting 3.6 35B-A3B into a state that lets it spend a lot of time in its reasoning mode before outputting an answer is probably not something that a web based SaaS LLM would tolerate, because it would frustrate many of the non technical end users who want a LLM to spit out an answer now.
'roll down hill' is a good way of putting it. They don't have 'will', but that's as we want it I think. I think alignment is harder if they develop will. Without will they are still tools that feel like an exoskeleton rather than something that will control us.
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.
If anything their capability of abstract inductive reasoning is way beyond the average human given how much better LLMs are at solving math problems, it's the paradox that they can do complicated reasoning before they can do intuitive peep-pe-boo.
llms are good at greedy depth first search[1]. so prompt / human needs to tell the model the breadth parts to take.
and human writing the prompt here did a lot more than that. asking to include parallel-edge 2-cycles, admitting disconnected graphs, specifying the emptyset cover, multiset counting, it all likely came from previous fake proofs.
[1]: see: ankitmaloo.com/fable - its the way most llms are trained, and is also natural owing to autoregressive nature.
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.
LLMs don't have the executive control to dynamically manage and compose mental sub-routines, when it has never seen those sub-routines in its training corpus. That requires fluid intelligence. ARAOC benchmark measures this specifically.
And all this is while they themselves proclaim that GPT 5.6 is much better at understanding intent and doesn’t need this type of guidance. Yet they themselves still do it, which I consider a more powerful signal.
Maybe in previous failed attempts that what the model landed on and they’re preemptively stopping it. Did they release the any info on the failed attempts?
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....).
This is the key to training better and better models.
You collect all of the direction prompts and anti-direction prompts, and you finetune the model to behave that way for the particular domain, under those particular conditions, as a default.
This is also why I don't believe for a second that the AI labs aren't training on our data, even when we opt out. This sort of data collection is a requirement for improving the models.
It's also why Anthropic purposefully gives you a dumb model or a corner cutting model to goad you into guiding it into the correct answers, so that you do the fine-tuning for them. OpenAI at least, does not do that. Which is why ChatGPT is far less infuriating to work with.
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.
The job of a programmer isn't to write code, but to automate things. Code itself doesn't have any value unless it solves some real problem not related to coding.
So if the work of a programmer can be automated then this means that any work can be automated. So no, it's not about software engineering only.
i think most things can be automated, but not everything
i think AI will just replace jobs with new jobs, and humans will continue with open-ended goal setting, and probably jobs where being human is necessary: responsibility and authenticity, so judges, politics, leaders, chefs, etc.
so in a way, the above commenter was correct, anything a programmer can automate, AI can do
- the extent to which solutions could be implemented as text: not sure about that. AlphaFold is basically a mechanical/geometrical/Chemical problem. There are other scientific transformer based models.
- the extent which solutions exist online - if you have a strong verification tool, you can generate examples, you can generate feedback, i think you could start with small/smaller prior art
- the extent which solutions could be specified and checked - if you have a lot of priort art, maybe llm's can find the good "patterns" and compare against them, and at least get close to a good results - but you'd still need human verification.
> 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.
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.
The determinant trick I'm referring to is used in prop 2.4 (in the Version I found via Google).
They use the determinant of the adjugate matrix if I remember correctly... But they just hit the reader with this without any motivation or even naming it.
>(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.
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)
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?
You don't compete with a chainsaw at cutting trees. You decide what the tree is for. Then sell the rainforest for shareholder value.
https://youtu.be/UrgpZ0fUixs
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?
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.
You are a human being, one of the most wonderful thing the nature has ever created, besides all the other living beings and the wonderful earth we live in. Do not tell yourself you have a value just because some company may want to hire you or not.
Companies and industries already use tools and machinery for tasks were once done by human beings. AI is just another tool they will use and it will probably replace human beings from some intelligence related tasks.
However that may bring more disruption to the society if the government in your country do not protect and help people and leave free rein to capitalistic greed.
I'm my opinion that already happened in the US, not by using AI, but merely by using H1B visa to get intelligence worker from abroad. What happened is that the companies are doing great and getting the best smart people in the world but American people and society have been disrupted.
Live your life fully, be good to yourself and to others. Don't worry about the market.
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.
This is one reason why I can't stand hn. Edgelord nihilistic comments that say nothing matters and take everything for granted get up-voted while credible and rational statements about God get down-voted.
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."
Um. No? I'd like to see how you conjured up that one. Because evidence does indeed point exactly to things just being as they are.
"You have objective value being made in the image of a personal God."
Conjecture. Unsubstantiated.
"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."
No? Sounds like you're projecting your own belief here. "If I didn't believe in God, then what purpose is there to anything?" It's very small of a belief.
It's funny because if people get replaced and don't have money they won't pay for API tokens either so the retail AI business collapses too, there won't be enough demand because people rather buy food.
Sorry, but most of us need to work to eat. This idea that our "value" is has nothing to do with the economy is an idea rooted in deep privilege, in the ability to say -- if I don't like my job, if I'm not employable, I can just retire, and the only problem will be figuring out how to live life afterwards.
Sometimes it's easy for me to imagine a bad future like this; where most of the men are forced into military as the only org that still has use for wetware, and women are valued only for their ability to birth new people...
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).
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.
Given a new context, why couldn't the same model have a decent shot at reviewing some results? It's not like they identify whether this output is from them and then go "yeah correct", that's not how they work.
It’s the other way around. The prompt instructed a GPT-5.6 agent to try to make a proof that would survive review by a GPT-5.6 subagent. If there were some defect that would cause the reviewer subagent to accept an incorrect proof, then one might imagine that someone else asking the same model to review the same proof would give the same result. And the proof generation process might even be biased to find such an incorrect proof.
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.
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
It's a good outcome as long as the proof is valid and ubderstandable to humans and leads to the discovery of further knowledge. There has been decades of search in Theorem Proving; this is just the next step.
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.
[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 was not a fan of the writing style of the proof. There seem to be some irrelevant details: Is the mention of 8-flow at all relevant? I, at least, found the definition of L on the first line of the proof of Lemma 2.2 to be needlessly inscrutable, and my thesis advisor would have likely stopped reading there and told me to fix it.
Maybe someone should ask the model to make a more clearly written and thus easy to verify proof :)
I was confused at first when you asked if the 8-flow is relevant, when like, the 8-flow is a key input that the cycle double cover is built out of. Then I realized, oh, I guess technically they're not using the 8-flow, they're using the Z_2^3-flow. But like. The existence of an 8-flow and the existence of a Z_2^3-flow are equivalent, and I gather most graph theorists are going to talk about it in terms of the existence of an 8-flow, so noting that having a Z_2^3-flow is equivalent to having an 8-flow helps the reader to put this information in context.
I'm not sure why you find this proof so hard to read. I found it mostly quite readable (and the definition of L is straightforward? I wouldn't have written it quite that way but it's hardly inscrutable), although I feel like some parts are maybe lacking some exposition to explain the reason for certain things -- it doesn't feel written "in order". I also don't like that it's not cleanly separated into theorems and proofs -- some of the proof occurs in parts that aren't set off, for instance, and there isn't even a proper main theorem statement! But overall I was able to get through it without a lot of trouble and I'm not even a graph theorist...
My issues with the definition of L are mostly about the order in which things are written.
L(t, epsilon)_e breaks down the range of L onto its component values indexed by edge, but this only really makes sense when you know that t and epsilon are. They are sort of defined in the middle of a sentence in the proof of 2.1, which IMO is asking a lot of the reader, and this sort of sloppiness is a way that errors can hide in a proof. (Not that I see an error here. But a formalization in Lean or whatever would not get away with this.)
And, in the same definition of L, for some reason the e=uv part comes at the end only after u and v are used.
What would be wrong with stating, in the definition, what sorts of objects t and epsilon are and with omitting e entirely in favor of just calling the edge uv everywhere?
> 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 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 doesn't contradict anything he said though. People on HN care only because an LLM proved a very difficult conjecture, not because we are independently interested in this conjecture.
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'
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.
To be clear, I am not agreeing that people only upvote their favorite billionaires, just that if this particular thing was done by a Chinese model it would not have gotten the same attention
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.
Your comment is somewhat emotionally charged, but I choose to respond to the overall point as a mathematician. I think it could be true that utility is correlated with difficulty but it is certainly not defined by it.
In pure mathematics, we reason about a world of abstract objects which are considered interesting ab initio. It may be because they arise directly or often from extremely basic operations, they are connected to many other interesting objects, or that they present special and surprising properties. The importance is basically, there is some surprising, interesting, phenomena which occurs in our world which we don't understand and which we seek to understand. Like science but in the non-physical world.
I think if you create a simple to describe system/construction with a property which is extremely difficult to prove. You are creating an object in our world which is basic but have properties which we don't understand (because we can't prove this property). So indeed I believe it would be an interesting thing to study and be of value. I don't see any problems/issues with this. I don't think the only valuable pursuit of humans is to improve the welfare of other humans. I think understanding the world is also valuable.
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.
The ultimate goal of university education is to raise researchers, who are the people that investigate the knowledge frontier of their field and then advancing it. To do that they have to understand a large part of the existing field so they can communicate with their peers, avoid investigating things that have already been throroughly explored, and draw useful connections to other fields.
Even in more applied fields it can take decades before advances become practically relevant. Restricting teaching to topics that have immediate practical relevance would therefore do students a huge disservice and prevent them from approaching the knowledge frontier of the field.
> 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.
Many important advances in linear algebra happened in WWI to solve optimization problems for logistical and industrial planning. A lot of these applications boil down to high-dimensional systems of linear equations, which back then were solved by hand. Efficient algorithms translate to reduced labor costs.
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.
Even when this really isn't the case, the math is preferably reduced to a linear problem whenever possible since dealing with the original complexity is usually so much more painful.
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.
Teaching only the practical side risks not teaching the subject with the appropriate theoretical depth and the ability to generalize it to other applications. Courses for purely applied fields utilize calculus to solve the current problem and then move on without teaching the finer points. Basing a calculus course on physics alone might be preferable in high school, but would be of disservice to students in university.
>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.
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.
> 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?!!"
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.
There is a tension between applied and theoretical mathematics, and it's as old as the whole science itself. Mathematics arose to solve practical problems (land surveying and division, as well as trade) and recognizing the underlying principles make it possible to abstract that knowledge. That might lead to centuries of ivory tower activity and what could be regarded as a purely artistic pursuit until somebody figures out how to apply a theory to a new practical problem. Or relations to another theory are discovered, and suddenly there is a new approach to previously intractable questions. A good example is be number theory, which is the foundation of modern cryptography.
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.
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.
> 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.
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.
Sorry but I completely disagree with your statement that "every novel result is broadly perceived as a big deal". Most results certainly are not consider this way (even though the average result has difficulties that are much higher than novel computer program you may have in mind -- no offense)
"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?
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.
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.
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.
If you watch the progress of the reasoning in llama-server while it's doing the thinking, you can track its progress. Sometimes the dead ends it goes down or things that it considers and then disregards are themselves something useful to re-prompt it with later, and send it 'rolling downhill', to use the metaphor of another commenter here, in another direction towards the same effort.
Putting 3.6 35B-A3B into a state that lets it spend a lot of time in its reasoning mode before outputting an answer is probably not something that a web based SaaS LLM would tolerate, because it would frustrate many of the non technical end users who want a LLM to spit out an answer now.
and human writing the prompt here did a lot more than that. asking to include parallel-edge 2-cycles, admitting disconnected graphs, specifying the emptyset cover, multiset counting, it all likely came from previous fake proofs.
[1]: see: ankitmaloo.com/fable - its the way most llms are trained, and is also natural owing to autoregressive nature.
Looking forward to "gpt-5.6-mathx".
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.
You collect all of the direction prompts and anti-direction prompts, and you finetune the model to behave that way for the particular domain, under those particular conditions, as a default.
This is also why I don't believe for a second that the AI labs aren't training on our data, even when we opt out. This sort of data collection is a requirement for improving the models.
It's also why Anthropic purposefully gives you a dumb model or a corner cutting model to goad you into guiding it into the correct answers, so that you do the fine-tuning for them. OpenAI at least, does not do that. Which is why ChatGPT is far less infuriating to work with.
- 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
> - 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.
Isn’t this quantifiable by revenue?
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.
So if the work of a programmer can be automated then this means that any work can be automated. So no, it's not about software engineering only.
Compilers, programming languages, IDEs, toolings all solve problems related to coding and are valuable.
This is a very typical programmer thing to say.
i think AI will just replace jobs with new jobs, and humans will continue with open-ended goal setting, and probably jobs where being human is necessary: responsibility and authenticity, so judges, politics, leaders, chefs, etc.
so in a way, the above commenter was correct, anything a programmer can automate, AI can do
If it was so valuable then AI companies would directly create valuable things, not just sell API tokens.
A proof like this is a good effort at trying to provide value directly, but still far away from real use.
When OpenAI cures cancer I'll accept the fate, but till then I'm still seeing a lot of gambling going on.
- the extent which solutions exist online - if you have a strong verification tool, you can generate examples, you can generate feedback, i think you could start with small/smaller prior art
- the extent which solutions could be specified and checked - if you have a lot of priort art, maybe llm's can find the good "patterns" and compare against them, and at least get close to a good results - but you'd still need human verification.
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?
Think very hard about what this implies for the future pace of AI R&D
I'm just saying, let's not get painted into that corner completely as a species :)
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.
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 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.
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.
Some do. But there's also the notion that a clever trick is a bad explanation.
And discovering a bad question leads to the correct question. No?
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)
The determinant trick I'm referring to is used in prop 2.4 (in the Version I found via Google).
They use the determinant of the adjugate matrix if I remember correctly... But they just hit the reader with this without any motivation or even naming it.
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.
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?
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)
I invited it to search the internet and it remains extremely sceptical.
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 :)
> 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.
Exactly, "clever". Isn't that the whole point?
Prompt: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...
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.
This "spend at least 8 hours" trick is a new one to me, though.
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.
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.
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.
https://www.youtube.com/watch?v=8vvWTz6N7Qg
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?
https://github.com/teorth/erdosproblems
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.
I assume they didn't use the Cerebras version for this since it's probably very supply-constrained right now
https://x.com/thsottiaux/status/2075596669958472146?s=46&t=Z...
> 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
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.
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
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.
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.
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?
Companies and industries already use tools and machinery for tasks were once done by human beings. AI is just another tool they will use and it will probably replace human beings from some intelligence related tasks.
However that may bring more disruption to the society if the government in your country do not protect and help people and leave free rein to capitalistic greed.
I'm my opinion that already happened in the US, not by using AI, but merely by using H1B visa to get intelligence worker from abroad. What happened is that the companies are doing great and getting the best smart people in the world but American people and society have been disrupted.
Live your life fully, be good to yourself and to others. Don't worry about the market.
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.
Literally nothing is rational about that. It's also just flat out off-topic.
Um. No? I'd like to see how you conjured up that one. Because evidence does indeed point exactly to things just being as they are.
"You have objective value being made in the image of a personal God."
Conjecture. Unsubstantiated.
"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."
No? Sounds like you're projecting your own belief here. "If I didn't believe in God, then what purpose is there to anything?" It's very small of a belief.
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.
Sometimes it's easy for me to imagine a bad future like this; where most of the men are forced into military as the only org that still has use for wetware, and women are valued only for their ability to birth new people...
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.
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.
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)
Obviously it can impersonate art, but where creativity and the human story matter artists need not worry.
Would, say, Fable or GLM 5.2 solve this given infinite amount of time?
Personally this gives me additional confidence that this is the real deal.
and if you get a different instance to review the code then you would know that it works rather well.
Only people can really verify clankers.
Can't trust anything LLM since it will confidently lie too.
It can't take responsibility for verification so it can't verify.
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.
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.
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.
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.
"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
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.
With the Erdős proof, OpenAI added perspectives from working mathematicians that gave some context -- hope something like that appears for this one eventually.
Maybe someone should ask the model to make a more clearly written and thus easy to verify proof :)
I'm not sure why you find this proof so hard to read. I found it mostly quite readable (and the definition of L is straightforward? I wouldn't have written it quite that way but it's hardly inscrutable), although I feel like some parts are maybe lacking some exposition to explain the reason for certain things -- it doesn't feel written "in order". I also don't like that it's not cleanly separated into theorems and proofs -- some of the proof occurs in parts that aren't set off, for instance, and there isn't even a proper main theorem statement! But overall I was able to get through it without a lot of trouble and I'm not even a graph theorist...
L(t, epsilon)_e breaks down the range of L onto its component values indexed by edge, but this only really makes sense when you know that t and epsilon are. They are sort of defined in the middle of a sentence in the proof of 2.1, which IMO is asking a lot of the reader, and this sort of sloppiness is a way that errors can hide in a proof. (Not that I see an error here. But a formalization in Lean or whatever would not get away with this.)
And, in the same definition of L, for some reason the e=uv part comes at the end only after u and v are used.
What would be wrong with stating, in the definition, what sorts of objects t and epsilon are and with omitting e entirely in favor of just calling the edge uv everywhere?
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.
https://concludia.org/graph/g_2ecb8083-52ec-3448-8c30-2f9bc7...
(Erdős problem 90)
Clearly that sentence isn't AI generated ...
[1] https://news.ycombinator.com/item?id=3556175
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.
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
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.
The only thing you've demonstrated here is some very funny confidently incorrect coping behavior.
The idea that all this enthusiasm is for certain Silicon Valley billionaires and not from genuine interest in AI technology is a baffling take.
Well, you would be wrong.
How many people do you honestly think even know who owns or leads Z.ai?
I certainly don’t.
> 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.
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.
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.
"No one cares about that! What practical applications does it have? It's pretty trivial anyways. More hype from the AI psychosis crowd."
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
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.
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...
In which case it’s ~equivalent to not caring about utility
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.
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.
In pure mathematics, we reason about a world of abstract objects which are considered interesting ab initio. It may be because they arise directly or often from extremely basic operations, they are connected to many other interesting objects, or that they present special and surprising properties. The importance is basically, there is some surprising, interesting, phenomena which occurs in our world which we don't understand and which we seek to understand. Like science but in the non-physical world.
I think if you create a simple to describe system/construction with a property which is extremely difficult to prove. You are creating an object in our world which is basic but have properties which we don't understand (because we can't prove this property). So indeed I believe it would be an interesting thing to study and be of value. I don't see any problems/issues with this. I don't think the only valuable pursuit of humans is to improve the welfare of other humans. I think understanding the world is also valuable.
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 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!"
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.
Even in more applied fields it can take decades before advances become practically relevant. Restricting teaching to topics that have immediate practical relevance would therefore do students a huge disservice and prevent them from approaching the knowledge frontier of the field.
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.
Do you have some examples that the adult could instigate, rather than waiting for the child to express curiosity?
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.
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.
Yes, the math department.
In any case linear algebra, stochastics, calculus; plenty of engineering and science applications for all these.
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.
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.
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.
If it was the professor, then that would be very embarassing on his or her part.
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?!!"
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.
That's unnecessarily reductive. you could have said "most of the value is that erdos' name is attached to it"
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.
"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...
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.
It's almost like a twisted mirror of Conway's law.
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.
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.
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.
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).
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"
I suspect the value is in showing the potential that LLMs have in developing new breakthroughs.
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.
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".
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.
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?
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.
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.
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.
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 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.
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/
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?
I can’t say if the citations are accurate because I didn’t check.
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?
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.
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.
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).
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.
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.
There can't be too many people working in that corner of graph theory, and I expect the result to them being eminently straightforward.
Pro = test-time compute (best of N responses)
then one day somebody new arrived and they forgot to tell him/her, so he/she solved the problem
It's not gaslighting, it's motivation.
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.
Quick! Someone (a human) copyright and patent it. /s
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.
Why wouldn't they verify it, knowing that any shenanigans would certainly come to light?
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.
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).
This one is a well-known problem with a brief, approachable proof, and they published the prompt.