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

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

#301

Post by perching_aix »

onecomment1 wrote: post #283
perching_aix wrote: post #271

Earlier quoted context omitted.

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.

[flagged]

> 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.

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

#302

Post by futureshock »

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 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.

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

#303

Post by Sniffnoy »

lquist wrote: post #294
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…

It’s the second most important problem in all of graph theory on this database of open math problems: 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

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.

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

#304

Post by onecomment1 »

scarmig wrote: post #295
fragmede wrote: post #292

Earlier quoted context omitted.

Wait, what things do we care about?

Two years from now, when a proof of the Collatz conjecture is announced and verified: "No one cares about that! What practical applications does it have? It's pretty trivial anyways. More hype from the AI psychosis crowd."

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

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

#305

Post by onecomment1 »

8note wrote: post #297
onecomment1 wrote: post #283

Earlier quoted context omitted.

[flagged]

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

I agree. We have a coordination problem, and the way our AI lords are proposing to solve this, is by cutting humans out of the equation

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

#307

Post by wbl »

Garlef wrote: post #227
qarl2 wrote: post #201

Earlier quoted context omitted.

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 p…

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.

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

#308

Post by orangecat »

npinsker wrote: post #85
WhitneyLand wrote: post #75

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 alie…

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.

AI-written stories have won several awards, e.g. https://lithub.com/a-prize-winning-story-published-in-granta...

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

#309

Post by dang »

onecomment1 wrote: post #283
perching_aix wrote: post #271

Earlier quoted context omitted.

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.

[flagged]

> What are YOU on about?

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.

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

#310

Post by riddlemethat »

scarmig wrote: post #298
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…

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.

The current foundational models have basic reasoning with glimpses of brilliant reasoning.
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