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

Post by mohamedkoubaa » Sat, Jul 11, 2026, 12:45 AM UTC

romaniv wrote: post #244

No one here actually cares about Cycle Double Cover Conjecture. I can demonstrate this by pointing out that the only time this conjecture was ever mentioned on the website was 14 years ago in a submission[1] that linked to a (now retracted) proof paper. That story received exactly zero upvotes. No one cared enough to upvote it and no one cared enough to ever mention this conjecture again. [1] https://news.ycombinator…

Post by sudo_cowsay » Sat, Jul 11, 2026, 12:54 AM UTC

kypro wrote: post #276

mNovak wrote: post #253

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…

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…

Post by koolala » Sat, Jul 11, 2026, 1:04 AM UTC

throw310822 wrote: post #62

ak_111 wrote: post #26

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 "theo…

> seems that it is exploiting a clever trick that somehow all the experts missed. Exactly, "clever". Isn't that the whole point?

Post by dominotw » Sat, Jul 11, 2026, 1:09 AM UTC

pianopatrick wrote: post #249

plaidfuji wrote: post #230

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 th…

Let's assume AI is only good at software and math. If we can use AI to write lots of good software for cheap, that software can automate away a lot of jobs. So AI does not have to take the jobs directly, software written by AI can. Especially if that works for software for robots.

Post by growfeather » Sat, Jul 11, 2026, 1:10 AM UTC

CSMastermind wrote: post #238

blovescoffee wrote: post #219

Earlier quoted context omitted.

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…