GPT-5.6 Sol Ultra Proves the Cycle Double Cover Conjecture
GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
I present a complete proof of the Cycle Double Cover Conjecture, a decades-old problem in graph theory. This breakthrough was achieved entirely by GPT-5.6 Sol Ultra, with the writeup assisted by Codex. By leveraging the 8-flow theorem and elementary linear algebra, we demonstrate that every finite bridgeless undirected graph possesses a collection of cycles covering each edge exactly twice.
The proof in this note is entirely due to GPT 5.6 Sol Ultra and the writeup with Codex.
- mNovak
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.
- plaidfuji
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
- ak_111
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.
- scrlk
Announcement: https://x.com/__eknight__/status/2075643450196971805
Prompt: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...
- bgirard
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?
- WhitneyLand
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?
- noname120
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.
- turzmo
Both impressive and terrifying. But as always, the methodology is buried: how many open problems were tried until they found a success?
If they tried this on 1000 problems and this is the one that succeeded, it still means that there are 999 open problems that an LLM cannot one-shot. It seems likely that this would remain the situation until the next model.
If this is the first one they tried, maybe we’re totally hosed.
The conclusions are so different in these cases that it is impossible to know what to think. Though it is reasonable, I think, to assume that a company is willing to push the maximally misleading narrative —- especially a company known for questionable ethical direction at the top, and one that is still circling an IPO, and one that is in the tech industry, where conjuring an illusion of growth and progress is sufficient for success.
- misrasaurabh1
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.
- sim04ful
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.