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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]

#51

Post by garethsprice »

minimaxir wrote: post #3
scrlk wrote: post #2

Announcement: https://x.com/__eknight__/status/2075643450196971805 Prompt: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...

> 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. If you ask them post-facto they'll estimate it.

This "spend at least 8 hours" trick is a new one to me, though.

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

#52

Post by thebruce87m »

not-a-llm wrote: post #16
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.

of you ask it, surely it can run a "time" in its sandbox from time to time and see how long it worked for

I wonder if the absolute value of the time result has any bearing on the subsequent analysis.

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

#53

Post by ufo »

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…

In math, the utility lies in the proof itself. A novel proof of a hard problem usually comes with new insights and abstractions that help solve even more mathematical problems.

To go with your analogy, mathematicians care more about the source code of the program than about the result of the program. But I'm afraid that we will see things change with the increase of vibecoded proof slop. A black box proof is not as useful, even if it is correct.

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

#55

Post by jvanderbot »

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…

It is very concise, and reads precisely as you suggest: to exploit properties already discovered and therefore combined in a novel way.

I'm just delighted by the prose. It reads like an old paper. The ones that were just straightforward theorems with proofs that do exactly what they say.

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

#56

Post by [deleted] »

Frost1x wrote: post #39
not-a-llm wrote: post #25

Earlier quoted context omitted.

pretty sure already millions of dollars (in inference costs) were already thrown at the Riehmann hypothesis as the models get stronger, larger amounts will be thrown at it imagine paying "just $1 bil" to go down in history as the company who's model solved the hardest/most famous open problem in mathematics. imagine the worldwide press headlines. as they say, the Riehmann Hypothesis is the hardest way to earn a milli…

I’m all for it since it’s value directly returned to humanity.

[deleted]

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

#57

Post by throw310822 »

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

What does it mean "new"? And, was it a difficult or trivial accomplishment?

A solution to a well known open math problem is both new and non-trivial- you know that many, very smart, very well trained human experts have dedicated time to the problem and haven't been able to solve it, despite good incentives.

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

#58

Post by BeetleB »

andai wrote: post #38
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 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 complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.

> I was promptly pilloried, and shunned.

Heh. In my day I may have participated in the pillorying.

I do think that there is value/merit in professors mentioning real world applications, where they exist.

What they're sensitive about are the theorems where there aren't real world applications. They don't want to (and shouldn't) justify them.

So even when there are real world applications, the posture is "Who knows if someone is making good use of this in the world somewhere? I don't care. It's not why we learn or teach this!"

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

#60

Post by Wowfunhappy »

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…

Biologists celebrate the discovery of new species of fruit fly hidden deep in the Amazon rainforest. Astronomers celebrate the discovery of new giant rocks located zillions of light years away. Neither of these things is immediately “useful” to the world, although they may turn out to be enormously beneficial in ways we can’t immediately predict. To me, these fields also feel central to the human experience—discovering new types of life, or learning more about our place in the universe. I don’t think a mathematical proof is any different.
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