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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]

#111

Post by logicallee »

are the references real? how do you think it got access to those papers? were they somehow already in the training data, or a result of web searches, Google scholar, etc?

None of them include a web URL but in text some are super specific ("[3, Sections 2.1 and 3.1]" and "[8, p. 367]").

The references go back to 1954 (Chronologically sorted: 1954, 1973, 1975, 1976, 1978, 1979, 1981, 1985, 1987 and 1994.)

Since reference 10 is included as "personal correspondence" maybe the reference itself was copied from one of Tutte's other papers? Or how did it get that reference?

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

#112

Post by prideout »

prideout wrote: post #107
scrlk wrote: post #82

Earlier quoted context omitted.

Ultra = parallel subagents with max reasoning Pro = test-time compute (best of N responses)

Confused about how to access Ultra; I don't see it in on their plans page.

Ah, I see it as a "reasoning level" in codex after typing /model

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

#113

Post by fragmede »

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…

> We attach basically zero value to writing a new program that hasn't existed before

We don't? People write new programs that go on to be successful software companies that make millions of dollars! Basic CRUD apps make money for their creators in their niche! There's so much money in software that it's taking over the world. The market is different, you're not getting worldwide household recognition for every little fart or sneeze of programming you output, but how can you say that we attach zero value to new programs when the history of computers is insanely valuable companies making new software and selling it. Windows, Oracle, mongoDB, etc.

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

#114

Post by homeslice1234 »

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…

> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture..

I'm not sure about this, TBH I ask myself this quite frequently. In a world where machines are routinely solving very high end math problems every day, producing more proofs than humans would ever really be able to absorb or fully understand.... would that be a good thing? Would that in itself be valueable? It feels like that is a probable future, but I'm not sure that would actually be something we want. I think there's probably more than "value is that it's solved"

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

#115

Post by a_e_k »

garethsprice wrote: post #51
minimaxir wrote: post #3

Earlier quoted context omitted.

> Spend at least 8 hours on this before even thinking of returning or giving up. Do current model harnesses have concepts of amount of time spent? Sometimes the model notices if a subprocess takes too long/hangs and kills it, but I've never seen it time itself.

Many harnesses include a current date and time in their system prompt, and if there is a way for the model to call for an updated time (either a dedicated time tool or calling the OS' `date` tool) they can track time they spent doing something. If not told up-front, they can try to infer it from timestamps in their logs. Sort of like a human - if you ask them to time something and give them a stopwatch, they do it. I…

Once on a late-night session, I had Cline!Claude spontaneously point out the time to me and suggest that I get to bed and come back fresh the next day.

I don't think it's in the system prompt, but that the harnesses time-stamp each turn in the context.

And from what I've seen, they also include the current and max context, so that the model can decide whether to continue work, suggest compaction, or prefer actions that might reduce the growth of its context.

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

#116

Post by calebkaiser »

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…

I mean, OpenAI delayed the public release of GPT-2 back in 2019 because it seemed capable of authoring interesting blog posts (that also happened to be untrue). It was a pretty big deal the first time Transformer models were capable of generating that kind of output--no one found it weird. We've just grown to take it for granted that large Transformer models are this capable.

The same cycle is happening now for a harder frontier. And proofs represent a pretty good benchmark for model capabilities, so a new model proving a result that a previous model didn't is generally notable in the same way that a model scoring higher on a benchmark is.

I'm sure we'll take it for granted in the not-too-distant future.

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

#117

Post by UltraSane »

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…

Proving a novel math theroem now is incredibly hard because all the easy ones have already been proven.

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

#118

Post by sd9 »

Diogenesian wrote: post #99

[deleted - the paragraph immediately following the proof of Lemma 2.1 is crucial and I found it hard to read correctly on my phone with the cramped typography. Having reread it I think the proof is correct.]

It's just a way of breaking down the full proof into pieces.

Lemma 2.1 says 'if this assignment exists then X'

Then later in the proof you say 'here is such an assignment, so, applying lemma 2.1, therefore X'

You don't need to assume the existence of the assignment, you prove that if the assignment exists then something else follows, and then later if you can find that assignment then you get the result of lemma 2.1.

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

#119

Post by in-silico »

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.

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)

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

#120

Post by QuantumFunnel »

kridsdale1 wrote: post #46
not-a-llm wrote: post #36

Earlier quoted context omitted.

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)

This is not true. What is the perfect video game that makes the user infinitely happy? What is the perfect economy optimizing program? What algorithm can solve political strife?

As with all things, the answer is always "it depends" based on what is being optimized
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