GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

> 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...

>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!)

> This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal.

This isn't true using the level of originality you're implying with your software examples.

Technically speaking, many novel mathematics proofs are written all the time (quite a few textbook exercises are actually technically novel problems that have never been posed before they were written in a textbook!) that get absolutely no fanfare. Overwhelmingly though they are not very original or difficult and really just required a fairly routine combination of different pre-existing techniques, even if technically speaking that combination didn't exist before. Those textbook problems are hence easy and therefore not given much public attention even if they are technically novel problems.

Indeed over the course of developing a new mathematical result, many many novel results are glossed over to the extent that even their proofs are left out ("as an exercise for the reader") because they are fairly trivial.

This is true for the overwhelming majority of new software as well. A new CRUD program may, technically speaking, be novel, but it's almost certainly just a routine combination of different pre-existing things.

Mathematics open problems that are actually named are generally problems that have resisted the low hanging fruit of the most obvious combinations of pre-existing problems. When those are solved they are a big deal precisely because they usually require some novelty!

Similarly in software, if someone were to create a new kind of database that solves a variety of new classes of problems that current databases fail to solve that would be a big deal! Truly novel software is also perceived as a big deal. Software that is, technically speaking new, but doesn't actually stray far from a fairly obvious remix of pre-existing techniques, isn't really celebrated.

In both software and mathematics, the intuitive benchmark is if other practitioners in the field look at the result and would say "Wow! How did you do that?" Professional software developers generally don't look at, e.g. a new blogging platform, and boggle at "Wow! How did they make that?!!"

Biologists celebrate the discovery of new species of fruit fly hidden deep in the Amazon rainforest. Astronomers celebrate the discovery of new giant rocks located zillions of light years away. Neither of these things is immediately “useful” to the world, although either may eventually turn out to be enormously useful in ways we can’t immediately predict. Both are also central to the human experience—discovering new types of life, or learning more about our place in the universe. I don’t think a mathematical proof is any different.

I'm not a mathematician, but I don't think that's true..? It's just that some problems are considered "hard" or known to have been "open" for a long time or that involve some clever/pioneering new technique. There's tons of math papers out there that are in some technical sense a novel contribution but in practice just languish without much attention except maybe from like two other people working in the same subfield.

Wow, you couldn't be more wrong here.

Math is something humans invented and is a model, nothing else. There is no logic per se, but a model that works quite well for us.

I studied Math and CS as a very highly gifted and quickly found out, there is no beauty of Mathematical Logic, only humans approval of what they deem most accurate.

A good example is set theory. Cantor was not openly welcomed after he introduced his "theory" to others. In fact, he was received quite some pushback and hostility - this doesn't sound like someone received love the mathematical logic's way.

In fact, the story of Cantor is really a tragic one. He left math for quite some time, due to the pushback.

Only later humans accepted his theory and found it useful. Well, well, what is Mathematical Logic and what not is after all just broad consensus by humans.

And if you go deeper, you will hear more of these stories. Math is anything else but logic. Proofs are religious things, often so complicated, they are simply accepted as "approved by a committee". Many profs cannot really explain simple proofs, they refer to the textbook.

This doesn't sound like romance nor easily reproducible logic.

After all, we deal with human beings.

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.

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.

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"

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?

> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.

I suspect the value is in showing the potential that LLMs have in developing new breakthroughs.

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.

In math, the utility lies in the proof itself. A novel proof of a hard problem usually comes with new insights and abstractions that help solve even more mathematical problems.

To go with your analogy, mathematicians care more about the source code of the program than about the result of the program. But I'm afraid that we will see things change with the increase of vibecoded proof slop. A black box proof is not as useful, even if it is correct.

there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software)

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!

> 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.

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 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.

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.

Proving a novel math theroem now is incredibly hard because all the easy ones have already been proven.

> 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.

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.

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.

I wonder if in each case they had parallel sessions, one trying to prove, one trying to find a counterexample

> seems that it is exploiting a clever trick that somehow all the experts missed.

Exactly, "clever". Isn't that the whole point?

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).

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.

The prompt was released, but not the cost of the result.

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

> 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.

> in just under one hour.

I wonder what the survivorship bias is though. How many other problems did they try but fail? Did they try to solve this problem but with another prompt? Still very impressive though.

[delayed]

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.

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.

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?

> What's left?

I think humans will be left to propose new conjectures while machines fill out the proofs. I don't know if there are enough interesting conjectures to go round to build new careers, though.

No there was the planar unit distance problem

(Erdős problem 90)

You seem to be suggesting that it is just as hard to understand an existing proof to a problem, than to solve it yourself? I don't follow your argument at all, what are you trying to say?

Most likely they wrote the proof in Lean and had it verified by a computer

I mean, if you've watched the past decade, this just seems like what news is today. "people are saying the double cover conjecture has a proof"

But they used LateX

…and thank God it's not Lean.

Yes, reference 10 jumped out at me as well. I thought personal correspondence references typically include one of the authors of the paper.

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.

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.

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.

Ultra = parallel subagents with max reasoning

Pro = test-time compute (best of N responses)

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.

[delayed]

I would assume/hope they had someone verify it before publishing

everybody knew the problem was impossible to solve

then one day somebody new arrived and they forgot to tell him/her, so he/she solved the problem

I've used this strategy for difficult bespoke problems and it does indeed work to incentivize the agent not to give up prematurely.

It's not gaslighting, it's motivation.