the cycle double cover conjecture was open for 50 years. GPT-5.6 solved it in an afternoon and then asked if there were any more like it.
My bet they run gpt over dataset of 10k unsolved conjectures, it happened this one was solvable.
GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
231–240 of 467 posts
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#232If 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…
It's hard for me not to think what's the point. I am a very average, even below average person in times of intelligence. What is even my value or reason to be if I know anything I can do, LLMs can do better? What is even my value both on job market and as a human?
Do you have friends or people in your life that are also not geniuses? Do you think about them this way? Why or why not?
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#233Earlier quoted context omitted.
> 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'…
Knowledge for its own sake is great, but it's worth noting that many "useless" fields of mathematics turned out to be very practical in the long run. Number theory was long thought to have no practical application, but now it's the backbone of cryptography. Boolean algebra was developed in the 19th century (George Boole died in 1864), decades before it was used to build computers. Those "useless" theorems being prove…
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Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#234Earlier quoted context omitted.
Sol fast isn't the Cerebras 750 tok/s version, it's just 1.5x speed at 2.5x price I assume they didn't use the Cerebras version for this since it's probably very supply-constrained right now
But Sol is running on Cerebras. That’s the whole point of this. That’s how they get 750 tokens per second. There is no other way.
Regular Sol does not run on Cerebra’s. I don’t think anyone public has access to that.
https://x.com/thsottiaux/status/2075596669958472146?s=46&t=Z...
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#235Reading the prompt is very interesting. I always wonder how they make these long-running prompts and I guess they literally just tell it to "keep going". After working with LLMs day-in, day-out an SWE for months, I feel like this could be greatly improved with something like a state machine of progress and proper orchestration. Instead of spinning up a ton of subagents to follow different paths, whip up some Markdown…
> I always wonder how they make these long-running prompts and I guess they literally just tell it to "keep going".
Many harnesses support a /goal as well. When the agent thinks it's done, another LLM compares its results to the goal, and if not, tells it to keep going. It's quite easy to have agents working on something for hours this way.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#236Earlier 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…
The prompt does matter. They specifically told it to assume a proof exists so it would not too easily dismiss the possibility.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#237It seems like a solid set of criteria for how easily a task can be automated by AI agents is: - extent to which correctness of solution be easily specified and checked - extent to which new potential solutions can be implemented as text - extent to which prior art exists online This basically maps to software engineering and math. I think a fair bit of AI hype comes from the fact that the very architects of AI are th…
Interesting take! I feel like 2 of them are maybe overstated:
> - extent to which correctness of solution be easily specified and checked
I don't think most software is like solving a math problem or series of math problems. Algorithmic problems are very narrow and might be more like this though, where an oracle that verifies answers as either correct or incorrect exists beforehand.
The correctness function of most software is how much users want to use/pay for it, which is a pretty fuzzy problem. Since the cost of copying software is effectively zero, software systems also tend to be be unique rather than being exactly like something else, and don't converge to be like another software system but rather diverge.
The prior art point is an interesting one. At least for applications as a whole, there isn't really prior art for a material amount of all the problems/tradeoffs a non-trivial software application embodies. For a todo list app or make a social network project, there's plenty of prior art to be sufficient to build something with an LLM system, but probably not most apps.
That's my initial intuition anyway.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#238Earlier quoted context omitted.
I mean if there's something I'd bet against being solved by LLMs in my lifetime it's that one. We truly do not have line of sight into what a proof would even look like.
Why would you bet against it being solved by LLMs? Isn't this very post proof that LLMs in an agentic harness are capable of doing real math? If you just keep cranking away at the tokens I don't see a principled argument against that leading to more solutions to unsolved math, even the hardest problems.
There are different classes of mathematical problems.
The one in the post definitely shows the advantages that LLMs have compared to humans for some problems but it's in an entirely different class than the Riemann Hypothesis.
Riemann is one of the most studied math problems of all times and all of humanity has basically collectively failed to make progress. The idea that there's some technique that just hasn't been tried yet (like in the post) is very very unlikely.
The general consensus is that we'll need an entirely new branch of mathematics to solve Riemann - our current tools aren't just inadequate; they're of the wrong class entirely.
I suspect inventing new branches of math will remain beyond LLMs for the remainder of my life.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#239I just had Sol Ultra read the proof and create a graph of it using Concludia (my side project) so you can explore it visually/graphically. I certainly don't understand it though so I have no idea if it's helpful. :)
https://concludia.org/graph/g_2ecb8083-52ec-3448-8c30-2f9bc7...
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#240It seems like a solid set of criteria for how easily a task can be automated by AI agents is: - extent to which correctness of solution be easily specified and checked - extent to which new potential solutions can be implemented as text - extent to which prior art exists online This basically maps to software engineering and math. I think a fair bit of AI hype comes from the fact that the very architects of AI are th…
> Ironically it couldn’t be further from the truth… and likewise the predictions of widespread labor obsolescence
Could you explain what you mean here?
It feels like there is one bucket of verifiable work - programming, math etc that AI will clearly excel at.
There is another large bucket of like law/ accounting/ financial analysis where I don’t have any reason to think AI won’t be super human at, but the work is more on bringing all the domain expertise into harnesses and software.
Is there aspects of knowledge work that you think AI wouldn’t excel at in the long run?