chip production and network acceleration here we come
GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
341–350 of 467 posts
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#342No one here actually cares about Cycle Double Cover Conjecture. I can demonstrate this by pointing out that the only time this conjecture was ever mentioned on the website was 14 years ago in a submission[1] that linked to a (now retracted) proof paper. That story received exactly zero upvotes. No one cared enough to upvote it and no one cared enough to ever mention this conjecture again. [1] https://news.ycombinator…
It’s the second most important problem in all of graph theory on this database of open math problems: https://www.openproblemgarden.org/category/graph_theory?sort... Obviously not an exact measurement but to give you some sense of the importance of the problem
This doesn't contradict anything he said though. People on HN care only because an LLM proved a very difficult conjecture, not because we are independently interested in this conjecture.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#343Earlier quoted context omitted.
yeah, I think what makes this post different was that AI did it. hopefully science will advance faster in the next decades with AI researchers helping
Hopefully. So far it seems to be doing more harm than good.
What harm? I think the major part of the US economy seems to be depending on AI.. without that, the economy seems bad..
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#344I am torn by these announcements. On the one hand there is the infinite potential on what we can disover, when AI prompts are solving outstanding problems. On the other, something is lost in an aesthetic sense when it wasnt a man working through this or with a novel insight. If an AI prompt runs on a data center for two weeks and then prints out p=np, it feels a little empty.
It's a good outcome as long as the proof is valid and ubderstandable to humans and leads to the discovery of further knowledge. There has been decades of search in Theorem Proving; this is just the next step.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#345Earlier quoted context omitted.
The current foundational models have basic reasoning with glimpses of brilliant reasoning.
LLMs have almost no fluid intelligence or capacity for abstract inductive reasoning, relying on crystallized intelligence for basically everything.
I think they have some fluid intelligence, if somewhat brittle
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#346This 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…
There is a tension between applied and theoretical mathematics, and it's as old as the whole science itself. Mathematics arose to solve practical problems (land surveying and division, as well as trade) and recognizing the underlying principles make it possible to abstract that knowledge. That might lead to centuries of ivory tower activity and what could be regarded as a purely artistic pursuit until somebody figures out how to apply a theory to a new practical problem. Or relations to another theory are discovered, and suddenly there is a new approach to previously intractable questions. A good example is be number theory, which is the foundation of modern cryptography.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#347It 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…
The job of a programmer isn't to write code, but to automate things. Code itself doesn't have any value unless it solves some real problem not related to coding.
So if the work of a programmer can be automated then this means that any work can be automated. So no, it's not about software engineering only.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#348Unrelated to the accomplishment or proof itself, but it's interesting how much of the prompt, even in this latest-and-greatest model, is spent essentially telling the model to actually solve the problem. Things like "Reject status reports, vague optimism, and claims that an unproved global compatibility statement is 'routine'." Also a lot prompt spent feeding it strategies, which feel like they should/will eventually…
I think a lot of this has to do with the post-training these models normally get. They are designed to answer basic questions with straightforward and short summary answers. They have the capacity to reason deeply, but they are not biased towards that unless prompted. I think it's because LLMs as they are in 2026 are both highly capable but also parlor tricks. They are not sentient, you just set them up with the cont…
'roll down hill' is a good way of putting it. They don't have 'will', but that's as we want it I think. I think alignment is harder if they develop will. Without will they are still tools that feel like an exoskeleton rather than something that will control us.
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#349If 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 don't compete with a chainsaw at cutting trees. You decide what the tree is for. Then sell the rainforest for shareholder value.
https://youtu.be/UrgpZ0fUixs
Re: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]
#350[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.]
I was not a fan of the writing style of the proof. There seem to be some irrelevant details: Is the mention of 8-flow at all relevant? I, at least, found the definition of L on the first line of the proof of Lemma 2.2 to be needlessly inscrutable, and my thesis advisor would have likely stopped reading there and told me to fix it. Maybe someone should ask the model to make a more clearly written and thus easy to veri…
I was confused at first when you asked if the 8-flow is relevant, when like, the 8-flow is a key input that the cycle double cover is built out of. Then I realized, oh, I guess technically they're not using the 8-flow, they're using the Z_2^3-flow. But like. The existence of an 8-flow and the existence of a Z_2^3-flow are equivalent, and I gather most graph theorists are going to talk about it in terms of the existence of an 8-flow, so noting that having a Z_2^3-flow is equivalent to having an 8-flow helps the reader to put this information in context.
I'm not sure why you find this proof so hard to read. I found it mostly quite readable (and the definition of L is straightforward? I wouldn't have written it quite that way but it's hardly inscrutable), although I feel like some parts are maybe lacking some exposition to explain the reason for certain things -- it doesn't feel written "in order". I also don't like that it's not cleanly separated into theorems and proofs -- some of the proof occurs in parts that aren't set off, for instance, and there isn't even a proper main theorem statement! But overall I was able to get through it without a lot of trouble and I'm not even a graph theorist...