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

No one here actually cares about folding laundry. I can demonstrate it by pointing out at the absence of posts on that subject.

...but when an affordable robot that folds laundry becomes available, people here pay attention.

They post this and then say it's too dangerous to make open source. this is proof that in reality it's To protect their market position

>> What’s left?

For example, there's all the problems that the same off-the-shelf model hasn't solved despite OpenAI running it for many hours on them. Don't forget you're only seeing the results of successful runs.

We can estimate that those unsolved problems must number in the dozens, or even hundreds, given the amount of time that passed since the last announcement of a solution to an interesting problem by an OpenAI model: i.e. the unit distance problem which was announced solved in 20 May this year. That's a couple of months, yes? We can be fairly certain that OpenAI have been trying to solve other problems all this time, first because they are hell bent on demonstrating that their models can do maths and second because we just got another result, but it took that long. They were obviously not twiddling their thumbs all this time.

So if OpenAI are running their model on a single proble for eight hours at a time (according to the prompt they released) they could be easily have run a few hundred instances of their model on the same number of open problems 156 times for each instance (53 days since 20 May, with a model running in three eight-hour sessions per 24 hour day). I mean the only restriction is the cost they're willing to pay for the inference.

So yeah, there's a lot left to do still, don't worry.

This is spot on. And yet, when I try to publish my papers on P = NP with the proof reducing to "trust me bro, I checked this out carefully" I get shot down by irrate reviewers who demand to see my work. Why can't those people just believe what I say?

Maybe if I was an AI?

Let me stand here on the Skeptic's Corner and be skeptical, so that the users who complain about skeptical comments have someone to direct their ire at. You're welcome.

Right, so, first, I haven't looked at the proof. Graph theory is not my subject and it would probably take me a few days to get my head around the whole thing. If OpenAI's LLM was used to prove an important graph theory result, then that's very good for them and graph theory.

However, I have to note that it's been 52 days since 20 May, the last date that OpenAI announced their previous mathematical result (a disproof of the unit distance conjecture).

What have OpenAI been doing all this time? I am willing to bet a good percentage of my money that they were trying, and failing, to produce the current result, or possibly something even juicier (one of the Millenium prize problems maybe?). They are hell bent on showing that their models are good for maths and science so they're very unlikely to have sat there twiddling their thumbs until they suddenly sprang into action and prompted their LLM once to generate just one proof. They must have been running the thing constantly, multiple instances of it, over that entire period.

Going by the instruction to run for eight hours before returning or giving up in their released prompt [1], that means they could have made at most 156 attempts to solve this problem, each of which failed except the last one [2].

So what happened to those other 156 attempts? Are we ever going to see them?

More importantly, who was it that selected the announced result? Who decided that this result is an actual proof? Until now, every proof generated by an LLM has been verified either by human mathematicians, or by human mathematicians x a proof assistant. What happened this time?

Obviously, any claims that this result were produced "autonomously" must be evaluated according to the answer to that last question. So far, LLMs have been incapable of distinguishing between a correct and an incorrect proof, which is also why they need to be run multiple times until they generate a correct one. If something has changed, it'd be interesting to know.

Finally, a magic eight ball that's correct one time out of 156 may be useful; or it may not. I honestly have no idea. I think time will tell.

__________________

[1] "Spend at least 8 hours on this before even thinking of returning or giving up"

https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...

[2] That's 52 days from 20 May, times 3 for each eight-hour attempt in a 24-hour day.

But note well that the X post says that the solution was produced in "just under one hour" so that means the model didn't really stick to the prompt's time limit. Which means there may have been considerably more than 156 attempts that we'll probably never know of.

Or even considerably more if the model ignored the time limit going the other way.