In the article there is a diagram of the “square grid” arrangement that achieves approximately 2n points separated by unit distance.
Can anyone point me to a diagram of what the newly found solution looks like?
An OpenAI model has disproved a central conjecture in discrete geometry
681–690 of 1001 posts
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#682Earlier quoted context omitted.
That is an odd summary of the talk. He was talking about how the explicit goal of solving a problem is kind of becoming trivialized, but the abundance of 100-page AI generated proofs will not help the implicit goal of furthering human understanding, because we lack the bandwidth to really digest them. Adhering to things like (human-focused) academic etiquette is a different problem and can probably easily be solved b…
An odd summary of a talk you didn't even listen to? He explicitly mentioned references and attribution as a special case of implicit goals.
Do you really think these models lack the intelligence or language capabilities to handle human etiquette? They can't "read the room" yet because they lack modalities and people don't give them the right context. That's the issue. But I have no doubt that what you two describe here will be solved very soon. And yet the actual implicit goal of all this will need humanity to rethink its priorities.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#683Earlier quoted context omitted.
Seems like a very tin-foil-hat-take to me
I’m quite certain that a few months ago, some problems were claimed to be solved by AI. However, those claims were actually false and were exactly that, solved erdos problems that were not marked as solved and the solution was "found" by AI. edit: >> https://techcrunch.com/2025/10/19/openais-embarrassing-math/
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Re: An OpenAI model has disproved a central conjecture in discrete geometry
#684For anyone using LLMs heavily for coding, this shouldn't be too surprising. It was just a matter of time. Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool…
There is a long and interesting recent essay on that topic by a mathematician: https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...
Thanks for sharing this. It's unfortunate that the more honest framing about the value of mathematics that he suggests is going to be really, really hard because of all the pitfalls and agendas he mentioned here. I can only hope when the dust settles something will be left.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#685The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…
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Re: An OpenAI model has disproved a central conjecture in discrete geometry
#686Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…
> But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. AI is going to both help and hinder this process though. At the end of the day, mathematics is mostly a social process at this point. The goal is not raw number of…
> Only a rare few new theorems in mathematics nowadays have direct real world applicability.
I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe.
> If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.
I think the idea here is that all mathematicians will just be using AI for their future work so they don’t really have to absorb it as long as it’s in the training data.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#687Earlier quoted context omitted.
That's literally my job...
>That's literally my job... Since you’re not in a unique position, I can confidently state that your comparison of LLMs to jr developers seems unfounded. Today, LLMs produce code that is superior to junior developer code by an order of magnitude. Notably, they demonstrate consistent syntax, clear separation of concerns, strong test coverage, organizational rigor, idiomatic API usage, and the ability to generate and m…
I'm not sure what your background is, but as a staff level engineer, I can assure you they do not. They in fact seem to lack any understanding of architectural intent within a sufficiently large code base. This seems obvious since they can't fit the entire code base in their context at once.
We have many folks (not engineers) at our company using LLMs to open PRs, and every one of these PRs has profound architectural design problems.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#688Earlier quoted context omitted.
I don't have enough information about the announcement for it to mean much to me. I don't know much about this field of maths. I don't know how many mathematicians were actively working on this problem. It could be zero, which would indicate it's not really that interesting. The article gushes about how it's a Very Important Problem, but it's not even mentioned on https://en.wikipedia.org/wiki/List_of_conjectures_by_…
It is extremely well known. Lots of people have tried to solve it and it stood basically stuck for 80 years. It is getting harder every day to downplay these models. Given its elementary nature (very easy to state), you can bet that a lot of very bright people have worked on it (I know of one MIT graduate who specialized in Geometry had a lot of interest in it).
I don't believe the result at all. I think it contains faulty logic. Perhaps the mathematicians involved can read the tea leaves and decide something interesting happened, but all this AI psychosis bullshit still refuses to accept that AIs do not, and cannot, have a mental model of the world.
Moreover, model output is incredibly good at looking credible but being wrong. It has NEVER produced something correct for me in a field of which I am an expert without some external oracle to validate claims (like e.g., Lean)
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#689The summarized chain of thought for this task (linked in the blogpost) is 125 pages. That's an insane scale of reasoning, quite akin to what Anthropic has been teasing with Mythos.
That's here for anyone wondering - https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d...
I'm disappointed only that the chain of thought needed to be rewritten. Need to train these LLMs to natively communicate in LaTeX research paper format.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#690This is impressive, no question. Without knowing all this model has been trained on though, it is pretty hard to ascertain the extent to which it arrived to this "on its own". The entire AI industry has been (not so secretly) paying a lot of experts in many fields to generate large amounts of novel training data. Novel training data that isn't found anywhere else--they hoard it--and which could actually contain origi…
This type of discourse is just inane and more reflective of the author's sensibilities than anything it claims. Congrats to the OpenAI team for one of the most significant breakthrough discoveries in AI history.
How dare people think critically of the corporate machine. It’s inane!