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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]

#181

Post by perching_aix »

moomin wrote: post #156
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…

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.

Have you tried... giving it the proof?

I tried to use Sol to:

- double check the proof (provided it with the prompt and proof artifacts)

- double check some of the claims made in this comment section (no math involved newer than 30 yo, no human contribution or review, no mathematician affirmations, proof assistants not being developed enough in this area to support machine checking a proof like this)

- check for any mathematician feedbacks

It stalled out (bad first impression much? lol). I then retried with 5.5, expressing the same request and my personal skepticism, and it returned to me with cautious optimism and no obvious issues found.

I think the fact that I provided it with the actual artifacts in question vs. you simply asking it to speculate about them is a really interesting UX difference. Like certainly, a coveted 50 year old math problem having a few pager proof is not going to be very likely. But then skim reading the proof by a frontier model is not going to yield any obvious issues either. Both responses are perfectly defensible given the context (I don't necessarily think these qualify as sycophancy), but we'd walk away with entirely different impressions if we didn't know about each other's requests.

And I'm not even trying to suggest you were wrong to not approach it in the ways I did. It's a perfectly reasonable and human way to prompt it the way you describe. It's just not the way I'd do it, but I have a hard time articulating why. And it's clear that the model was never going to help with this difference either.

Half a century of computing, and we're still trying to make the machine think on the users' behalf :)

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

#182

Post by IngoBlechschmid »

roncesvalles wrote: post #162
hyperpape wrote: post #154

Earlier quoted context omitted.

> 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 unde…

It seems in mathematics that the utility of a problem is directly correlated with how difficult it is to solve, for some odd reason. If I defined some pointless construction and it turned out to be very difficult to prove, it would automatically over time become considered a "high utility" mathematics problem (again, for some odd reason). Mathematics is largely just smart people working on pointless puzzles, and only…

> If I defined some pointless construction and it turned out to be very difficult to prove, it would absolutely and automatically over time be considered a "high utility" problem (again, for some odd reason).

Yes and no.

No: There are lots of very hard open problems which are judged to be of little value by mathematicians and hence garner little attention.

Yes: If a conjecture resists proof for a long time, this can indicate that we still have a substantial gap in our understanding. We project utility into an eventual closure of this gap, not into the statement of the concrete conjecture at hand. The gain in understanding is what we actually work for. It just turns out that chasing specific results, even if they are mostly dead ends on their own, is useful for orientation.

The (by now solved) problem by Fermat (for all integers a ≥ 1, b ≥ 1, c ≥ 1, n ≥ 3, the equation aⁿ + bⁿ = cⁿ does not hold) and the (still open) Collatz conjecture are perhaps good illustrations of this situation.

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

#183

Post by esafak »

sandspar wrote: post #178
esafak wrote: post #93

Earlier quoted context omitted.

> 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.

Surely the machines will have superior conjectures soon.

I think so. Mathematics is the least of my worries. I worry about what the machines will want to do.

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

#184

Post by 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.

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

#185

Post by dotancohen »

jaggederest wrote: post #142
andai wrote: post #38

Earlier quoted context omitted.

>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 fie…

I love teaching kids and young adults calculus by socratic method. They get so mad when they figure out you were teaching them math, but they often admit it was pretty fun. Only had the chance to teach like that a few times but it's dynamite when it happens.

I did this when I taught my third grader calculus on the train, when she asked a question about the train accelerating faster sometimes than other times. She loved it, but I was just taking advantage of children's natural curiosity.

Do you have some examples that the adult could instigate, rather than waiting for the child to express curiosity?

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

#186

Post by breezybottom »

zerobees wrote: post #22

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 relea…

Mathematics isn't a scientific discipline.

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

#187

Post by [deleted] »

ToValueFunfetti wrote: post #135
npinsker wrote: post #85

Earlier quoted context omitted.

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.

I think, and I may be totally off base, that the labs are specifically avoiding art and (non-technical) writing as an endpoint. It's bad PR for them- it calls attention to the copyright question and threatens the 'human flourishing' kind of jobs- and there's no money in it because people prefer art to be human made and there's hardly any money in that anyway.

[deleted]

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

#188

Post by john-h-k »

hyperpape wrote: post #154
zerobees wrote: post #22

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 relea…

> 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 unde…

> and that can be arbitrarily far removed from practical utility

In which case it’s ~equivalent to not caring about utility

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

#189

Post by fooker »

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 is no match for rapidly shifting goalposts ;)

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

#190

Post by addaon »

not-a-llm wrote: post #36
zerobees wrote: post #22

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 relea…

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)

> 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)

What!? I can think of about a billion examples... but for one, I'm still waiting for a good enough CFD/FEM coupled system to model paraglider dynamics across collapse/recovery. And I expect to be waiting quite a while.

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