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

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

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.

Announcement: https://x.com/__eknight__/status/2075643450196971805

Prompt: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...

It's really neat that the prompt was released!

I'm curious how many unsolved problems are tried against frontier models when they come out. Are we trying every problems against every release? What is the solve success rate? Is there a sub-community within Mathematics that is coordinating this effort? How much untapped opportunity is there here?

I just had Sol Ultra read the proof and create a graph of it using Concludia (my side project) so you can explore it visually/graphically. I certainly don't understand it though so I have no idea if it's helpful. :)

https://concludia.org/graph/g_2ecb8083-52ec-3448-8c30-2f9bc7...

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?

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.

I find it somewhat interesting only 1/5th of the prompt has to do with the actual problem, rest is just cajoling the harness into shape.

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.

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.

[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.]

the cycle double cover conjecture was open for 50 years. GPT-5.6 solved it in an afternoon and then asked if there were any more like it.

The prompt is interesting, I can’t help but wonder how many times it was run and extra instructions were added (don’t return if x, etc).

I'd love to see the failed runs too. The success is impressive, but the distribution of attempts would be just as interesting.

That's a much shorter and more elegant proof than I was expecting, especially after reading some of the earlier Erdos proofs. GPT 5.6 Sol is the real deal.

Is this the first LLM-solved problem famous enough to have been on https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...

Statement of AI use. The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol).

Clearly that sentence isn't AI generated ...

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.

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.

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?

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?

OpenAI knocked it out of the park with this one.

It did not use Lean or other proof assistant?

what's the difference between Sol Ultra and Sol pro? is pro a thing of the past now

It's great that a novel math proof was created.

But this is mostly marketing, pleasing the sneering class/the elites who believe that simply providing value for others (through sales) is repugnant and beneath them.

It seems that these tools can do real work, and people are paying for that. IMO, that is more than sufficient.

But is the proof accepted to be correct? That is what distinguishes this from being notable compared to any other AI slop proof.

"Assume for purposes of this task that a complete affirmative proof exists"

> Statement of AI use. The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol).

Quick! Someone (a human) copyright and patent it. /s

Good post, it perfectly captures the problem with AI. Here we have a claim that the double cover conjecture has a proof. Verified by… no one per the link.

Now imagine this proof is wrong. How would you know? Ok, think about the process in which you determine the correctness - why not do that initially?

And there it is. The problem laid bare. Ironically it reduces to the P and NP one.

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?