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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]

#191

Post by TacticalCoder »

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 is what everything else is built upon. I'm no mathematician but a very good friend of mine is: teacher at a big uni, researcher. Pure math.

His entire life he's had --and still has-- to deal with comments like the one you just made, implying that the only value is solving pointless conjecture (if it wasn't pointless, according to your logic, then the value wouldn't be that it is solved).

Truth is to be found in this xkcd:

https://xkcd.com/435/

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

#192

Post by jaggederest »

dotancohen wrote: post #185
jaggederest wrote: post #142

Earlier quoted context omitted.

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?

I've used filling a tank, balloon, or bucket (rate of flow, can be subdivided to teach limits, and use weird shapes for teaching area under curve and interpolation en route to integrals), or the classic throwing a ball back and forth and trying to describe the shape, the distance it flies, peak speed vs peak height, figuring out how hard you are "actually" throwing instantaneously. Honestly as soon as you start thinking about bulk substances moving around (gravel piles! fuel tanks!) it's easier to find examples than you'd ever reckon. Rate of change is everywhere.

Seems like I start by asking "how do we know how much this tank holds?", or "how fast does this line go up on the side of the tank?" and curiosity goes from there usually.

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

#193

Post by qarl2 »

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…

> However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed.

Why is that a "however"? My reading is that it found a genuinely new solution that is both elegant and previously missed.

Seems like exactly the kind of result a human mathematician would aspire to.

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

#194

Post by HardCodedBias »

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.

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

#195

Post by brokensegue »

perching_aix wrote: post #181
moomin wrote: post #156

Earlier quoted context omitted.

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…

Fable told me

> Verdict: I checked every step and found no error. The argument appears to be a correct proof of the Cycle Double Cover conjecture, modulo two standard cited results (the reduction to loopless cubic graphs and the Jaeger–Kilpatrick 8-flow theorem, both real and well-established).

> Two caveats: this would settle a ~50-year-old open problem in three pages, so it deserves independent expert scrutiny regardless of my check; and I couldn't reach the web from here to confirm the paper's provenance or any community response, so I can't tell you its status beyond the mathematics itself.

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

#196

Post by Garlef »

qarl2 wrote: post #193
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…

> However, it seems the proof is extremely concise so it seems that it is exploiting a clever trick that somehow all the experts missed. Why is that a "however"? My reading is that it found a genuinely new solution that is both elegant and previously missed. Seems like exactly the kind of result a human mathematician would aspire to.

> a human mathematician would aspire to

Some do. But there's also the notion that a clever trick is a bad explanation.

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

#197

Post by nilkn »

varjag wrote: post #165
nilkn wrote: post #159

Earlier quoted context omitted.

You are assuming that the latter, once autonomously discovered and verified at scale, could not simply be translated into the former, also perhaps autonomously at scale (or otherwise selectively as determined by human interest, taste, and relevance).

Well we're literally discussing a human readable machine generated proof here yet you don't seem happy with that.

We're talking past each other for some reason. I'm not "unhappy" with anything. I just pointed out that (1) a result like this requires peer review by a professional human mathematician, which fundamentally bottlenecks progress in a pretty severe way; (2) such review would not be necessary if it were accompanied by a formal Lean artifact; (3) you can have both a formal proof and an informal proof together (one does not rule out the other); (4) searching for proofs formally first, then translating successful auto-verified proofs into natural language, is the most scalable approach in the near future for AI mathematics; (5) AI conjecturers would likely benefit from the results of (4) for making large leaps and connections, which can then scale into formal proofs for verification, which then feed back into the same loop ...; (6) humans guide this process through taste, judgment, and their own intuition, likely often intervening to ensure that the loop is aligned and producing a body of conceptual informal mathematics that is valuable to humanity.

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

#198

Post by djsavvy »

misrasaurabh1 wrote: post #29

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.

I did some math research in high school where the proof boiled down to dozens of cases of ugly polynomial inequalities. I can't find the PDF now, but the final paper was something like 70 pages, and several of those were full-page polynomial expressions expanded out. The actual prose was probably 5 pages or so.

It was categorically the least elegant proof of anything I've ever seen.

I'm incredibly grateful to for the opportunity to have done the research and gotten my feet wet early on, but boy do I cringe when I look back at that paper.

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

#199

Post by Miraste »

in-silico wrote: post #119
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.

The difference is that artistic sensibility is largely subjective. This means that: 1. It's hard to measure (and people can disagree about it) 2. It can't really be improved using RL without a human in the loop (which is how math is being trained)

At a certain level, yes, but AI is still so bad at writing that its failures are objective and easily measurable

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

#200

Post by paulpauper »

ves wrote: post #174
roncesvalles wrote: post #162

Earlier quoted context omitted.

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…

Writing that mathematics is a waste is such a hilariously ignorant comment to make on a programming forum.

remarkable also how it's not even greyed out either. I have been downvoted to oblivion for far more defensible claims
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