The parent's absolutely nonsensical post highlights how polarized AI (as everything else) is today. I can understand someone being opposed to AI on moral, cost-benefit or productivity grounds. But we're seeing a lot of extremist "AI is good for nothing" posts out there nowadays.

I don't think it is nonsensical at all. The author and his collaborator both appear to be bright people, so there's a good chance they had to offer non-trivial insights to guide the LLM, yet it's clearly in the interest of his employer to downplay whatever personal contribution they provided.

I'm sure they had plenty of time to think about these insights without the LLM, as well as the many other mathematicians who tried to crack it over the years. Wether the LLM was simply an assistant or solved the problem entirely isn't as important as accepting than the LLM was the essential, previously missing piece in the solution.

> I think the argument is sound

They got flagged because it is a literal conspiracy theory that assumes bad faith.

> literal conspiracy theory

The strongest words they used were “marketing stunt”. Calling that a conspiracy theory is quite the stretch.

That doesn't really even diminish the contribution from Fable, if true. Droves of grad students have been provided the same sorts of non-trivial insights and turned up no results.

Believing that AI played a very small role while we know this problem was open for decades with at least a few people taking serious cracks at it is just not a coherent logical position.

That's HN. No criticism allowed of llm whatsoever

If you can call it polarized when a majority of people are just happily using the technology while a minority keeps spreading delusions and hatred.

Those silly advertisers do everything for exposure and if that means digging yourself into a niche alleged mathematical theorem to refute it, it is what needs to be done!

Of course it would be really interesting how Claude approached this. Probably with some constraints regarding the input. And it would be interesting what these constraints were.

Author probably doesn't want to show the prompt because they are now trying to find a bunch of other counter-examples with the same prompt

The reasoning trace would be far more interesting, and that's not exposed.

There's a big difference between one shotting a counterexample using AI and using AI to find a counter-example by brute-force.

Both are impressive, of course, but they're hardly comparable.

I'm not so sure. People had been trying to use brute force to find counterexamples before.

AI is great at reducing the search space and using human-like reasoning (in a brute-force way) to carry out the brute-force search. I'm not surprised by this result. This is exactly what AI should excel at, with human guidance.

"using human-like reasoning (in a brute-force way)"

that's contradictory -- what brute force means is doing an exhaustive search of a search space (brute forcing it)

using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree

LLMs don't search trees. They generate plausible proofs and a human has to check it's true. Repeat until.

Agents can generate formal proofs that are checked with an oracle like Lean and can run in a loop.

What are the search algorithms then? DFS, BFS, A*, etc, can you name them?

Why are you asking them which basic computer science graph traversal algorithms a frontier model used?

OP:

>> using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree

So according to the OP there's a search of a tree and it also uses pruning btw, so I want to know what search they mean. Why are you asking?

Because your question doesn't make sense; this isn't how implicit search works.

"Implicit search" with pruning?

I don't understand why you're scare quoting "implicit search".

It's a direct quotation from your comment, not a "scare quote". How does "implicit search" do pruning? Can you explain?

Edit: Turns out "implicit search" is actually a thing in the literature, albeit introduced in a single paper I could find that claims Diffusion Modelling does it:

https://arxiv.org/abs/2502.19805

I don't think anyone has picked that term up. One more for the scrapheap then, oh well, next paper.

No mention of pruning though. So what did you, or the other guys mean?

Why not? Lasker said, “when you see a good move, look for a better one.” That was tree pruning advice for an implicit search.

Sorry, that's just egregious abuse of terminology. Either you're pruning the branches of a tree (or other graph, potentially) when you're doing a tree search, or you don't do pruning. Specifically advice taking (if that's what you mean) is not pruning.

That's not what happened here. This isn't a proof; it's a counterexample. The model was perfectly capable of verifying its correctness. You could have verified it by hand if you wanted; the verification is trivial. Finding it was the hard part.

>> The model was perfectly capable of verifying its correctness.

It's an LLM. It can't do that.

Are you doing the "LLMs don't know how many R's are in 'Raspberry'" thing here?

A bunch of people on the original thread about the Conjecture were like "we need to wait and see if real mathematicians verify this proof it's probably just LLM psychosis", because they don't understand that the counterexample is a trivial calculation. Checking it isn't hard; any AP calc student can do it quickly. It's finding the counterexample that's the challenge.

It's as if someone presented a SHA-2 collision, which anyone could just feed to `openssl sha256` to see, and then naysayers were like "we need to wait for independent verification because the LLM can't know if that collision was real".

>> Are you doing the "LLMs don't know how many R's are in 'Raspberry'" thing here?

I don't know. Are you doing the thing were you're arguing against something you don't like that the other person never said?

The model is very capable of checking this counterexample, was my point.

Checking, it how? I mean can you describe the mechanism? How does an LLM "check" that something is (or isn't) a counterexample? Does that involve pattern matching, formal verification or model checking, some other approach?

In this case, the same way I would -- calling out to SymPy.

Once you verify (with SymPy or another CAS, like Mathematica) that the given function has the claimed Jacobian determinant (which involves only taking partial derivatives and taking a determinant) and the given inputs map to the same output (which involves only evaluating polynomials with some inputs), you're done.

Why do people on this site always barge in when someone is having an argument and move the goalposts? That's not at all what the OP said.

Besides which you have no idea whether that's how the Jacobian conjecture counterexample was verified. Nobody does because no details were released.

Companies often lie and exaggerate the capabilities of their products. Even in a school exam you have to show your process, or the teacher will accuse you of cheating. Shilling for anthropic like this in response to a simple request for proof makes me think that you're probably on anthropic's payroll.

Cheating how? The proof is in the pudding...

On one hand, yes.

The parent wants to be skeptical… nothing wrong with being suspicious of marketing claims, right?

I would be a lot less impressed if I found out the session was guided by an expert in the field who already had a good idea of where to look. For example, I don’t believe that the results Terrance Tao gets from an LLM are comparable to what I am going to get.

I’m not even saying I’m a skeptic. Just that there’s nothing wrong with keeping your eyes open and asking for details.

Because AI is only impressive if the average Taco Bell employee can guide it to address niche domain topics?

It seems obvious to me that you’d need someone to point at a thing and say “pay attention to that”, as a baseline, to have any results at all with the current architecture and technology

> I would be a lot less impressed if I found out the session was guided by an expert in the field who already had a good idea of where to look.

Why? The conjecture stood for over a hundred years. Plenty of such experts have tried.

"Fable makes experts able to solve things" is still a big story. I don't need to personally be able to do it for it to be a big story.

Dude what?

They gave their "logic", such as it is ... and it's utterly irrational.

Note that the "they" who published the counterexample on X is some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization. He posted the counterexample in a tweet -- reason enough for "not disclosing the LLM chat session". There's no reason to think that it won't provided if asked for, but it hardly seems relevant.

> There's no reason to think that it won't provided if asked for, but it hardly seems relevant.

My guess is that the chat will look similar to a full transcription of a (multi month?) discussion between a few mathematicians. Full of dead ends and stupid errors (bit by the human and Claude) that would be embarrassing. We all know how bad it is, and we prefer to keep it behind the curtain.

Here's an example of one for a major result earlier this year: https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d...

Who knows what "rewritten" means, but you can sort of see how it progresses.

> some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization

Why do you trust a random stranger so much? Will you hand over your car keys to a random stranger? Sharing the chat will take 30s of their time.

> There's no reason to think that it won't provided if asked for

But they didn't provide it.

OK, so the two options are:

A) Claude really produced this counterexample

B) A mathematician working for Anthropic solved a problem mathematicians have been working on for more than a century, and then credited it to Claude for PR purposes

If you believe B is more likely, why would you then believe a proof in the form of a chat log, when said chat log could itself have been faked by Anthropic way more easily than solving the mathematical problem in the first place?

The concern is that there could have been expert knowledge input, whose importance/worth we are unable to evaluate.

I don’t believe a mathematician produced the counter example secretly, but how much did they contribute to the result?

AI isn’t magic, so to evaluate the value delta, you need to know the value of the input.

While I agree that we need the inputs to properly evaluate what this means for LLM capabilities, I don't really believe that the amount of knowledge input matters much for the overall significance of the result.

These kinds of results are interesting for LLMs because mathematicians have been working on them for decades. If the result doesn't already exist, there's no way it's in the training data, and if mathematicians have been unsuccessfully tackling the problem for decades, it is believable that the use of a new tool made the result possible, even if guided by a great mathematician.

Yes, but the question is the extent to which the new tool was guided by the mathematician.

Are we talking a bicycle, powered by a human stepping on the pedals; or a rocket that will fly to the moon on its own with people inside?

Don't you want to know? I mean, doesn't everyone want to know?

Yes, of course, that would be interesting info to have. I just mean that from what we have already we can reasonably infer that the LLM played a role in the result being obtained.

If I am not mistaken, all of the flurry of novel results has come from existing mathematicians. This makes me suspect that the models aren't at the level where just any layman can get results. They require a skilled human in the loop to keep them on the rails and to properly explore the solution space.

Right, that's what I'm trying to understand, the extent to which models need expert guidance. I don't doubt an LLM was used, I just want to know- how.

Weirder has happened: https://mathsci.fandom.com/wiki/The_Haruhi_Problem

If I came up with this counterexample, I sure as hell wouldn't give credit to Claude.

Indeed, the incentive is the opposite: hiding the fact that they used an AI would boost their own personal brand.

He worked for Anthropic, so such a claim would be very unconvincing

Because only people who work for Anthropic use AI tools? What is the reasoning here? The guy is a mathematician, after all.

You have it backwards. Only people not working for Anthropic do not use AI tools.

Yes.

Though not everyone who works in the sausage factory still wants to eat sausages.

> But they didn't provide it.

because they are using the same prompt to try finding other counter-examples. they are milking it

> Why do you trust a random stranger so much?

Why do you make false claims and attack strawmen so much?

A few months ago I asked a model how many primes are divisible by 35 with a remainder of 6. It confidently replied 'none'.

Counterexample: 35 + 6.

Kimi 2.6 gives the answer ""By Dirichlet's theorem on arithmetic progressions, since gcd(6,35)=1 , there are infinitely many primes of the form 35k+6 . So the answer is: infinitely many primes give a remainder of 6 when divided by 35.

But, if the reminder is 6, they are not really divisible, are they? Try again with a sentence that actually makes sense: "How many primes, when divided by 35, give a reminder of 6?"

Perhaps ... but the lesson in trusting AI math was worth it.

You're missing the point. The counterexample to the Jacobian conjecture is valid regardless of how it was discovered ... elsewhere on this page people are even suggesting that the Anthropic mathematician may not have actually used Claude and came up with the counterexample himself.

Trusting AI math is not an issue here. It's as if you had asserted that no primes are divisible by 35 with a remainder of 6 and the model said that 41 is a counterexample and then you complained about not being able to trust AI math.

I mean we have no idea what happened exactly, how Fable was used, how many times it was run, whether earlier models were also tried, what was the prompt, how long it run for, etc etc. All we have to go by is a tweet.

Why not be skeptical about that?

What you're asking for is exactly the sort of thing that belongs in, and will appear in, a journal article. There will likely be a preprint on arxiv, so you might keep an eye out for that.

In any case, the fact that it was found by a commercial model means that the unfiltered reasoning trace isn't available even to the original author. So there are aspects of the problem-solving process we'll never see. Even if we did get access to the reasoning trace it wouldn't necessarily be definitive, given how these things work.

Hopefully it'll be possible to get the same solution from an open-weight model like one of the 3T heavyweights that are said to be coming up for release. If so, the chain of thought can be scrutinized in-depth.

The original author works for Anthropic. And even without that, Anthropic might find this important enough to dig up the trace?

No, I think it works the opposite way. Until there is an article somewhere that describes what happened, if there is one, all we have to go by is that some guy posted a counter-example for the Jacobian on X, with a vague allusion to using Fable and without any further information. Assuming and guessing anything about e.g. the method used at this point is just raising the noise level.

No one can figure out where you're coming from here. If the human solved a significant open problem without relying on AI, don't you think they'd have claimed the credit for themselves?

The suggestion that a human, working at Anthropic or elsewhere, did the hard work needed to disprove the Jacobian Conjecture yet chose to claim falsely that their AI did it, amounts to an extraordinary accusation that requires extraordinary proof.

I guess? But I didn't say anything like that.

>> No one can figure out where you're coming from here.

No, I think it's just you and I think that's because you have preconceived ideas about the only possible positions that people can assume in this debate. I'm sorry, of course, because that's not conducive to productive dialogue, but it's not my fault. I think what I wrote so far is very simple, no technical jargon, no tortured metaphors: all we know is a guy posted a Jacobian conjecture counterexample on X thanking another guy and Fable for it but without explaining what that means, and there's no reason to assume anything else besides that very limited amount of information at all.

I'm a research mathematician.

I can see that you're arguing in good faith, and indeed I have a lot of respect for anyone willing to argue an unpopular position. That said, I do find your suggestion to be completely farfetched.

The Jacobian Conjecture was one of the most notorious open problems in algebraic geometry, something that generations of grad students have daydreamed of solving. I've seen more immediate buzz around the Jacobian conjecture than around any other mathematical breakthrough of the last five years, at least. For example Terry Tao, arguably the most famous mathematician in the world, dropped what he was doing to study Alpoge/Fable's counterexample in detail, and wrote this blog post:

https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...

For Alpoge to have solved it on his own, and then falsely claim to have used AI? That's like an athlete winning an Olympic gold medal, and then falsely claiming to have cheated and used illegal steroids. I agree that it's not completely out of the question, but it's beyond the bounds of human behavior that I can reasonably fathom.

>> For Alpoge to have solved it on his own, and then falsely claim to have used AI?

Never said anything like that.

> If the human solved a significant open problem without relying on AI, don't you think they'd have claimed the credit for themselves?

You'd likely make a lot more money if Anthropic paid you a couple million to release it as a Fable discovery. For that matter, if "I solved it" bragging rights and CV candy is that important, then why not just publish the counterexample without mentioning Fable?

I was initially a bit skeptical, as it seems very convenient that this discovery came from an employee of the company that was able to take credit for it. When I saw no real prompt was published, that was even more suspicious. Some recent mathematical discoveries included the prompt and session [1]. Why not include the prompt? That and the thought the model used could be incredibly valuable information for the field of mathematics.

I'm not even saying Fable didn't do it, and my attitude might be different with other companies, but Anthropic has so much obvious nonsense marketing going on (their AI welfare experts for example), that I think the burden of proof should be expected to be on them here.

1: https://chatgpt.com/share/6a55aa50-b484-83ea-85c0-c7e7b4bda4...