There are "sovereign" clouds like StackIT, OVH, OTC or Scaleway. What they all have in common is that they are lightyears behind hyperscalers in terms of performance, reliability and security. Needless to say, these alternatives can only serve open-source models because they are not capable of training a SOTA equivalent by themselves. The cherry on top is that you have to pay a sovereignty premium for inferior performance.
If you wanted a Russian puppet in the oval office, it would do exactly what Epstein's BFF Donald is doing. There are zero reasons to support anything Putin is doing (read: daily war crimes), however, the destruction from within in the US and EU is executed amazingly well.
On the one-hand side, it's really impressive how LLMs drive mathematics forward, and this pace is only accelerating very quickly.
At the same time, most of the proofs I've looked at appear super messy and chaotic to me (while still being correct of course, so it doesn't matter). LLMs do not care about "elegance" the way human beings do, which is a big advantage. LLMs for mathematics is such a great fit on many levels. Can't wait for a significant breakthrough, prove P=NP and all hell breaks loose.
> LLMs do not care about "elegance" the way human beings do, which is a big advantage.
It's just a matter of time before you can post train it for elegance too. Mathematical proofs in particular can be formally verified automatically which is a big advantage.
Why is it “just a matter of time”? Why do we assume and say this?
The amount of times humanity has said this and time itself was not enough of an ingredient to achieve some anticipated outcome are legion. But we filter those out and go back to making more predictions based on the current linear derivative we’re observing.
I'm not sure that elegance will be so easy to train for, the same way that writing skill has plateaued (or arguably declined) since earlier models. "Have you solved the problem" is verifiable, but questions of taste are harder to pin down.
You can select for 'short proof', or 'elementary proof', or assign the 'cost' of the proof as a some combination of its length, the number and complexity of the new terms it needs to define, and so on.
This might not help you with finding the proof, but once you have a machine that can produce several different proofs, you can select among them and incrementally polish the best one.
I'm sure you could select for shorter proofs, but then that might be confounding in its own way. I think it's a general problem for LLMs that taste is both subjective and hard to pin down to a single metric. There's a reason mathematicians talk about elegance rather than brevity. Sometimes a long geometric proof with a simple algebraic alternative is still elegant, or elucidates the problem in a new way.
Well, there are not that many proofs from 'The Book'.
We are a bit ahead of time, currently I would settle for 'as easy to understand as possible' proof. Not a long, complicated, inpenetrable, mess, that Lean says is correct, but reading it provides no insight.
I've actually been involved in annotation projects doing RLHF to train LLMs to do exactly that. It's not a matter of time, it's already happening - it's just seemingly lower priority than "profitable" projects like post-training LLMs to replace white collar workers.
Before LLMs, empire building was a very large incentive to hire. Teams tended to become larger than they needed to be so the boss feels good about their life choices.
LLMs do not fix this problem, they make it worse. Instead of the team being oversized, they’re now way oversized. It is still in everyone’s best interest to look busy anyways and LLMs do help a lot with that.
The mathematics is to a great extent about understanding of abstract structures.
As humans, we prefer simple structures/proofs (I suspect that is to a great extent because those are easier to understand), and as such find elegance in simplicity.
In fact, the capability of the human brain to understand complex structures and proofs is rather limited.
LLMs (hmm, I would prefer to use 'AI solver', as LLM is nowadays just a part of it) finding a complex proof can mean several things:
1) AI by its nature/construction does not have preference for simple stuff (it 'thinks' differently than human: a human will, in its search for a proof, start by exploring the 'simpler' parts of the proof space, and hence more likely find a 'simple' proof, while a AI might be more target oriented and descend deeply in depth-first-search manner to recursively solve sub-tasks, without much regard about the overall simplicity of the proof). This can be eventually solved, by subsequent 'polishing' passes, similarly as things work in human science.
2) there might simply not exist a simple/elegant proof of a given problem. The world is a complex beast. Its just our brains trying to find simple/elegant meaning/structure, even in places where there is none.
That very much does not describe what happened. Someone found the bug and used it to disprove the Collatz conjecture as a demonstration of the bug. Nobody ever claimed it as an LLM proof.
Isn't it a bit Catch 22 anyway? If someone finds a algorithm to reduce some NP task X to class P, then that just means X wasn't a true NP task and P!=NP is still undecided?
If it’s an NP-complete [0] problem like SAT, as many NP problems are, then we are done, because all NP problems can be reduced to it (in polynomial time).
You can prove something is in NP by providing a (polynomial) reduction from a known NP hard task, and vice versa. All the known NP problems (Knapsack, SAT, etc) are mutually reducable in this way, so solving one lets you solve the others. So if X was shown to be NP, then given a polynomial time solution to X, you can stack the polynomial time reduction from X to SAT to solve SAT in polynomial time too.
That’s not a counterexample to P != NP, it’s a proof that P = NP. You can’t prove that two sets are the same by counterexample. What you could do is disprove P = NP by counterexample, by showing that some problem is in NP but not in P.
At best, a polynomial algorithm for SAT would be a counterexample to the claim that no NP-complete problem is in P.
Sorry, it seems like nitpicking to me. You haven't shown my usage of the word counterexample is wrong, at all.
I think a proven counterexample to Q is always a proof of not Q.
> You can’t prove that two sets are the same by counterexample.
You can in this case.
> What you could do is disprove P = NP by counterexample, by showing that some problem is in NP but not in P.
You could argue that counterexample is defined in one direction only, by convention, as to which hypothesis is more believed. In that case, my usage would be more valid, because the general consensus is P!=NP.
You could also argue that a counterexample should be some finite, constructible object. But that's actually also in favor of my usage - a difficult class is an infinite set, while an algorithm has a finite description.
Also note that AI can still find the counterexample (the actual algorithm), without proving it is a counterexample. Again, my usage of the word counterexample favors that definition of what counterexample is.
But honestly I think it would be more productive to spend this effort on thinking about actual counterexample to P!=NP.
Sure they care about elegance, or at least brevity. Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems. It doesn't mean they are perfect at it though.
"They" don't "care" about anything. It is a stateless computational run across thousands of semiconductors. There is no objective this software has other than the computational function completing. To care would mean the model would have a level of discernment that goes along with sentience.
The supposed "magicality" of human consciousness is hard to reconcile with evolution -- where on that long path from amoeba to homo sapiens did God (PBUH) implant our connections to the soul realm?
Even evolution-deniers run into difficult-to-justify self-contradictions when trying to explain how the supposed supernatural part of our consciousness is both affecting the natural world and affected by the natural world, but not part of it.
> Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems.
First time I heard that, and I doubt it. Don’t customers pay for output tokens? If so, why would a company specifically spend time training their LLM to generate fewer?
Even if you specifically train the model on producing shorter answers, I would think producing good short answers would require more resources just as it does for humans (https://quoteinvestigator.com/2012/04/28/shorter-letter/: “If I Had More Time, I Would Have Written a Shorter Letter”)
If so, charging per output token is the wrong incentive.
All of these are lightyears behind US/China LLM offerings. None of them offer any model close to open-source SOTA. Runtime and reliability is a disaster and sure enough you have to pay a "sovereignty" mark up.
I live in the EU and want to live in a world where my children actually CAN read the Bible in class. However that’s not really possible anymore…
(Also no, modern science is (still) not incompatible with (the Christian) religion. I want my children to have both, as do I, or at least be taught both, so they can decide for themselves what they want when they are grown enough to do so.)
This went to market horribly (if you can even call it that), just look at the comments. Mistral played themselves big time over the past ~18 months. Non-competitive products and models combined with bad marketing and GTM...Oh Europe
Anthropic's run on the model and product side of things is highly impressive. They got Sam A. punching the air consistently, which is well-deserved and self-inflicted above all.
Wdym? They've been knocking it out of the park on marketing, but Claude Code is still a meme, and Opus is getting trashed by GPT5.5 meanwhile you can't even use their "dominant" model, and anecdotal reports from when people could use Fable, when they weren't getting silently poisoned, was that it was only marginally better than GPT 5.5 in terms of SWE smarts, mostly being better in terms of pleasantness to interact with and design taste.
Like I said, Anthropic's marketing is killing it, they've got people freely(?) shilling for them on public forums so even if they have shit developer relations and community relations and a model that's mostly worse while being more expensive, they can ride a wave of misinformation.
Look at the way that Anthropic has legally threatened people who do stuff they don't like around Claude Code and their subs, and compare that to how OpenAI has acted. Look at how mixed up and unstable their communication is on policies is relative to OpenAI. Don't take my word for it, Theo/Primeagen have a whole back catalog outlining how shitty Anthropic is.
Look at the cost per intelligence of Opus vs GPT 5.5.
Anthropic is the Taylor Swift of frontier labs... Not bad, but massively, MASSIVELY stan'd for inexplicable reasons, in violation with reality.
If I was an investor I'd be very worried, to say the least. These guys had all the chances to be established across the relevant LLM players and are blowing it at this very moment....remarkable.
With all the attention on and hunger for "sovereignty" one would think they would be the ones capitalizing the most....nope, not at all
There are "sovereign" clouds like StackIT, OVH, OTC or Scaleway. What they all have in common is that they are lightyears behind hyperscalers in terms of performance, reliability and security. Needless to say, these alternatives can only serve open-source models because they are not capable of training a SOTA equivalent by themselves. The cherry on top is that you have to pay a sovereignty premium for inferior performance.
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