Claude found a counterexample to the Jacobian Conjecture

A little over 10 years ago I remember meeting a postdoc who believed he had something close to a counterexample to the Jacobian Conjecture. He and another person was bruteforcing polynomials in about 16 variables, something like 80 - 700 terms each, using binary trees for mapping coefficients.

They were guessing, at the time, that the lower bound of a counterexample (P, Q) for max(deg(P), deg(Q)) would go up to 200.

To think that Claude Fable was able to find a counterexample in degree 7 is insane to me. We are truly in a new era.

They were looking at 16 variables, degree of about 10 in each one. That search space is simply too big for a plain bruteforce. So they did some sort of filtering to reduce the search space to a pool containing "possible counterexamples".

There was also a paper giving a lower bound of about 100 for possible counterexamples in that particular framework. Later raised to 108 in https://arxiv.org/abs/2204.14178

I don't remember the details very well since this was back in 2015 and wasn't really involved in the research. Consider this to be some sort of telephone game between what I heard in 2015 and what I remember today.

This is a rare instance where feeding this groundbreaking information into an LLM gives _them_ psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.

I've heard about mathematicians going through kind of the same thing when they get a weird proof that ends up being right from some weird source or themselves.

Which is fair, they get inundated with kooky proofs from amateurs all the time and odds are incredibly good that there's some major fatal flaw that the amateur doesn't see. Or in the case of themselves, there's a certain blindness that makes it a little more difficult to critically evaluate your own leaps. In ether case the way it manifests is by going over it many times and many ways, each time more certain that you missed something until you just kind of break. Only then do you publicly start suggesting that there might be something to this new leap.

I read some thinking traces someone posted on X, and yeah, near psychosis from refusing to believe this simple of a solution had not been found already

> So the conjecture that survived Keller, Abhyankar, Moh's degree-100 verification, and five-plus published wrong proofs appears to have died via tweet during the World Cup final.

I like how first it's amazed and doesn't believe it, accepts it, then realizes you not only stole it from twitter but that it's its own proof lmao.

Interestingly, even Qwen 3.6 27B was able to verify the solution, but I didn't get any glazing for discovering it. Instead, it thought that someone named Shestakov had already found a counterexample in 2004.

GLM 5.2 whiffed, it insisted the counterexample wasn't valid.

VibeThinker 3B also recognized that the counterexample was valid. But it kept trying to convince itself that it wasn't, over and over, since it's an "unsolved problem." Eventually it just answered "-2."

> that someone named Shestakov had already found a counterexample in 2004.

Qwen has the sprit of a grad student

This is so unreasonable! As @__alpoge__ himself notes this is classic crank graveyard territory and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search. Wild!

The great thing about these mathematical mopping up type operations is that no person will waste their time trying to prove it to be true anymore. If anything that’s a win.

It would be great if an LLM could settle the Collatz conjecture next, god knows how many man-years have been burned on that by unsuspecting victims.

The reason this was "easy" is because the conjecture turned out to be false. If the collatz conjecture holds true (and most mathematicians seem to think it will), it will be much harder to prove than your average Erdos problem.

I think parent’s point is that every false conjecture can cost a lot of time to be spent on futile affirmative proofs. So if we “clean up” a bunch of false conjectures, then more effort can be spent on interesting proofs of the others. (Probably a rather naive view of the value of conjectures but I’m just offering an alternative interpretation of the comment.)

The opposing argument there is that the hope is that solving these problems reveals other interesting maths knowledge along the way. Finding a counter example all but ensures that won't ever happen.

Wikipedia's gonna Wikipedia. Unless there's a material debate over the Jacobian Conjecture itself, there's really no open question here. This isn't a complicated proof; it's a straightforwardly checkable certificate of a solution.

It's tedious, but you could literally even do this by hand. I'm pretty sure I've done worse coordinate bash back when I did math competitions in high school.

This is nonsensical: Properness of the map is equivalent to its being an isomorphism (quick proof: Jacobian invertible implies that the map is etale, and properness would imply that it is finite etale, but affine space doesn't admit non-trivial finite etale covers), so the lack of properness is just another way of verifying that this is indeed a counterexample.

The author has a PhD in math from Cambridge. If it turns out to be a false claim it is an interesting case study on AI's sycophancy causing even experts to drop their guard and make mistakes.

Who do you mean? The author of the tweet is Levent Alpöge, who does not have a PhD in math from Cambridge... but does have a PhD in math from Princeton. And his advisor was Fields Medalist Manjul Bhargava.

Also, there is no way this counterexample is wrong. You can very easily check it for yourself. (I did, I don't know why, obviously Levent wouldn't be wrong about this, but I guess I was in shock.)

It's a known behaviour. I noticed a lot of the teams I work with just gave up any intelligence they might have to the AI.

I went searching to see if it was just us, but there has been a few research papers done on it. This one is the latest that I am aware of.

https://arxiv.org/pdf/2603.10025

I don't know why you're saying this, given this is not sycophancy and instead is an actual example of Claude Fable finding a real counterexample. I would get it if the mathematician had in fact posted something untrue or crankish, but he posted something true.

Frontier models have solved several major open problems in mathematics in the past few months, so this should not be a huge shock. "Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end.

>"Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end.

Everyone has their own experiences making it hard to judge what attitudes are prevalent.

From what I have encountered the Anti-AI people have been more common and more outright scary.

I have heard some ideas from the Pro AI crowd and thought to myself "Yeah, that's not going to happen anytime soon", There was enthusiasm leading to overoptimism. A lot of them are going to be disappointed.

The Anti AI crowd, which I distinguish from those who want AI done responsibly, sustainably, and as safe as can be realistically be managed. There are real criticisms to be made on real issues here. The Anti AI crowd I see seem to want to scrub the world of it, they stand in the way of any responsible implementation because it would be accepting it.

Those people scare me, the vitriol and bad faith interpretation of everything as an indicator of the proof of their reality has the tone of the moral righteousness of a true believer. The outright endorsement of violent acts has me concerned that the more radical or emotionally unstable of the group will do some serious harm. It's not the bulk of a radical movement that causes atrocities, most of those involved just create an environment that normalises toxic attitudes. That environment can incubate a few individuals that act, either for phycological reasons, status, or simple the inability to empathise or consider consequences.

I've seen plenty videos of locals being violently removed from town halls due to displaying of anti AI sentiment. I wouldnt consider it fair to use these examples to tar the entirety of the pro AI camp. the way objectivity and subjectivity are intertwined in your comment is spectacular, ever considered a career in propaganda?

Do you have links to these videos?

You comment suggests you feel there is something wrong with what I said, could you elaborate?

I initially wanted to downvote you but I can’t because you’re right.

I’d probably be called an anti-ai luddite on this site despite using it all the time for programming. I naturally gravitate to anti-ai viewpoints, but only because the breathless hype of the AI boosters is so nauseating. I have no moral qualms about AI (although I’d like nothing more than Anthropic/OAI/Google dismantled), but I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core.

But AI is obviously useful and we could harness it for good. We shouldn’t give up on it entirely. And the people who are militantly against all usage are scarier than the boosters for sure.

> I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core.

AI can no more replace artists, than other artists can.

A sunset can be nice to look at yet be created without meaning. Any non-conscious process that produces images can produce beauty that has worth without intentionality. Any conscious process that produces artworks is just another artist.

Most artists would rather there be more artists in the world than fewer. If AI can't make art, it can't replace artists. If AI can make art (of the conscious intent form) then there are more artists.

I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online.

Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.

It's a Jacobian determinant and three points. You can check this yourself in Sage.

The author is a Princeton math doctorate.

playing devil's advocate a little bit, but wikipedia does have a policy against original research. If you published an obviously correct counterexample to wikipedia which is not anywhere else on the internet (or in a book, etc), the rules are clear: the counterexample must be removed.

in this case it's not original research, because there's a tweet by someone with a good reputation, and plenty of comments on said tweet corroborating the result. But it's a more sketchy "secondary source" than most wikipedia references and some caution on the part of the editors is not out of place.

(saying this as the person who made the original edit to the wikipedia page adding the counterexample)

I'm not complaining about Wikipedia here, just noting for the thread: it's a vector of polynomials. It has a nonsingular Jacobian. Provided with it are 3 distinct points it sends to the same point; it can't be invertible.

What Wikipedia says about this doesn't matter, does it?

I think Wikipedia has very sane processes. I'm just saying that process isn't useful to this thread. It's like if I found a SHA2 collision. I'd probably have to be an absurdly talented (and lucky) cryptanalyst to do that, but anybody on the thread could trivially confirm my finding.

It's a Princeton math PhD who posted. The verification is quite straightforward and was posted by the tweet author. Wolfram would have to also be producing incorrect outputs for the counterexample to be false. The counterexample works as claimed and conjecture has been proven wrong.

To be clear, this is not the kind of thing where a Lean formalization provides any value at all. It's like formalizing the answer to a high school algebra problem. The counterexample is obviously correct.

Indeed. I was mainly responding to the comment about waiting for "independent seasoned mathematicians to verify", whereas in this case it is easy enough to convince oneself of the counterexample's correctness.

Maybe have a seasoned mathematician check if the counterexample is really bogus before saying people fell for ai psychosis.

Since I actually don't know math, maybe my ELI5 understanding can be helpful (or corrected).

The conjecture says that you can always reverse (a process) to determine the original inputs.

But this proof shows multiple inputs creating the same output - which obviously cannot be reversed to determine the input - thus falsifying the conjecture.

[...] you can always reverse (a process) to determine [...]

There is a precondition - constant non-zero Jacobian - to the inverse existing and the inverse is claimed to be of a specific kind - polynomial. The counter example satisfies the precondition and by mapping two different inputs to the same output makes any inverse impossible, including polynomial ones. But maybe that is already ELI7.

It was known to be false in general. But there are many questions that are false in general, but true when restricted to polynomials.

Would be really interesting to see the full conversation and understand how Fable came up with the counterexample in the first place.

Maybe, maybe not. The "proofs" may not have helped at all with finding a counterexample. Either way, it doesn't matter. A counterexample was found, no one found one before even though clearly a lot of people have tried who also had access to the prior "proofs".

This makes me wonder, what if anyone uses Fable-class LLM and passes of its novel results as their own work?

There's no shortage of folks doing that in software, right now.

Currently there seems to be very little anti-AI bias in maths, and more of a "huh, cool new toy, let's see what it can do" vibe.

I think everyone assumes most, if not all of the code written in a piece of new software is done by AI.

In my view, it's theoretically possible for a combination of the author's iterative prompts + evaluation with Wolfram Alpha to activate the weights that encode the language that describes the constraints on these polynomials (from the faulty proofs) in such a way that the author eventually arrives at this:

> ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3

How is that not like claiming that a "combination of iterative prompts and evaluation with Wolfram Alpha" would produce my Google Mail password? I also have to note that this response abandons the original claim, that the counterexample was found by exhaustively searching prior failed attempts.

I've seen a lot of people claiming that these recent examples are just trivial combinations of existing prior work.

I'm yet to see anyone cite the prior work. If it's true I think some credit is due for the authors Fable/Sol are branching from.

I haven't seen these claims, but all of the open problems that have been solved so far have been in the category of "humans could have solved them, but didn't". This doesn't diminish the significance of the results, but it does mean there's still work for mathematicians to do other than glorified prompt engineers.

This specific counterexample really is trivial. There's nothing to cite. People have wasted hours and hours on a question whose answer you could give as a homework problem in Calc II.

For all thehubbub, as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models. I could have missed those announcements, but one might think that between close to frontier performance plus cheap tokens, that they'd be leading the way on these things.

As with the case with AI and in fact a lot of things in life, sometimes you just need to wait. It will happen, rather soon.

You're stating the fallacy (unapplied):

"(Q came after / downstream of P) => (Q was caused by P)" => false

What are P and Q? I can't even map the AI/math US/China situation to the (quoted) antecedent, so I can't evaluate your claim that the fallacy applies.

You aren't challenging that U.S. frontier models have been behind ~all the major AI math results, are you?

You aren't challenging that there have been a negligible number of major math results proven/disproven by Chinese models, are you?

Those are potential objections to what you replied to, but the fallacy you cited isn't relevant.

> Jacobian conjecture [...] states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse.

> ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)

But 1 != -1 and -3/2 != 3/2 . So it's not its own inverse. Is the conjecture that it is its own inverse or that is has an inverse?

Edit: it was worded a bit strangely, but it is saying that [ (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) ].map(F) all produce (-1/4, 0, 0). Thus it has no inverse and indeed disproves the Jacobian conjecture.

I think the map sends (1, -3/2, 13/2), -> (-1/4, 0, 0) and also (-1, 3/2, 13/2) -> (-1/4, 0, 0) so it's not invertable which disprove the jacobian conjecture that polynomial maps over complex numbers with a jacobian that's non-zero are globally invertible.

(Just as a note for myself, I had to think of why the fact that such jacobians are constant is a byproduct, I guess it's because of lioville's theorem implying that any polynomial over C that never hits 0 must be a constant [because the reciprocal is bounded and thus must also be a constant])

That it has an inverse.

If f(a) = f(b) for a≠b then f can't have an inverse.

Suppose f has inverse g; then g(f(x)) must = x for all x.

But then g(f(a)) would have to equal a, and g(f(b)) would have to equal b. But they can't, because f(a)=f(b)

a short counterexample makes it likely the best medium for such an announcement. would have been inappropriate for a positive proof.

You can't please everyone on the internet, if he had posted this on Arxiv then someone else will be complaining that Arxiv is for humans to publish and that llm output should be social media post instead. Also, the proof fits in a tweet. So why blow it up.

I find it interesting that the counterexample uses C as a field. C is twisted and weird. Maybe the Jacobian Conjecture still holds for reals?

>C is twisted and weird.

Why do you say this? I've admittedly never done a proper complex analysis course but I got the impression that that complex differentiability was a very strong condition that results in holomprhic functions behaving "nicely" in ways that real functions do not

Should have used quotes.. I didn't mean it in any formal sense. What I am saying the nature of unit in complex plane makes it difficult to intuitively imagine invertibility and determinants.

the conjecture held for 85 years and the counterexample was announced in a format that expires after seven days

The surprising thing is that the counterexample seems relatively "simple" in that it's low degree, with coefficients that aren't too large.

Does anyone more familiar with this know why this _wasn't_ found earlier, when it seems like you could brute-force through some low-order polynomials?

Which tweet has the chain of thought? I was curious how it discovered this, did it use some symbolic brute force or some clever trick? Also is that the raw or summarized COT, it seems to have the hallmarks of raw COT (the frequent interjections) but I thought Anthropic always used summarizers in between

It's funny how similar this is to a human reaction to the same thing. It's the right level of incredulity, to be sure, and the first thing you do is you start re-reading the conjecture to makes sure you didn't misunderstand the requirements.

There are still many degree 5 polynomials, up to 35 terms for a single function, then you need to find three of them as well

Same awnser as much of the LLM Proofs - people cared about other things. There isn't a lot of money in academic math, and the ones that love it don't look for low value findings. Proofs like these are, funnily enough, usually the domain of hobbyists - but over the last few years, the "Monetize everything" mentality and struggling first world economy has pushed people away from interesting academic pursuits on their free time.

My understanding is that this is in a different league (Smale problem) than a lot of the other results that have been coming out (Erdos), though I could be wrong.

Are you at least a little familiar with linear algebra?

If so, you've probably heard of the determinant. It's a certain way of "summarizing" a matrix with one value.

The determinant in this case is of the Jacobian, which is a matrix you can construct from a multi-variable function. Each term is the partial derivative with respect to each variable (x, y, z, etc.), with one line per output variable (vector element).

The Jacobian of a polynomial function is, in general, going to be a matrix where every term is some polynomial expression. And the determinant of that will also be a complicated expression. But in some cases all the variable terms cancel out and you're left with a single constant (0 or some other value).

The conjecture says that if the Jacobian determinant is constant (i.e., all the terms cancel out), then there must be a polynomial inverse. And the key condition for an inverse is that there must not be two input points that evaluate to the same output. It's just like y=x^2. It's not invertible, because both +2 and -2 square to +4.

So if you can find a function where the Jacobian determinant is constant and also find two or more points that evaluate to the same output, then you've found a counterexample to the conjecture. And that's what's been done. And remarkably, the counterexample is pretty simple. It would be tedious but a bright high school student could verify it.

I think another way to understand it is the generalization of the inverse function theorem. The inverse function theorem gives you local invertibility, but even being "locally invertible" everywhere does not imply global invertibility (you don't need too pathological an example to see this, a periodic function serves iirc).

The Jacobian conjecture roughly asks what whether local invertibility gives you global invertibility when you restrict only to polynomials (which we might hope "behave nicely"). Apparently for polynomials over reals this was disproved a while back, but up until now the general case of polynomials over complex numbers was open.

Speaking as a mathematician, it does seem like we're a bit fucked as a community. Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today, and at this rate of progress it's likely that, already by next year, we'll see new infrastructure being put into place by AI, giving us a world in which a few designated interpreters of the oracle get to 'do' mathematics, while it withers on the vine as an avenue for the exploration of human meaning.

It's so soon that it may be rational to delay certain math and software projects until smarter models arrive

I think in the long run mathematicians are probably fucked, but in the short run it's not that bad. All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it. (This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample.)

At this point the advantage of AI is that it's read the entire mathematical literature, and it doesn't have to worry about wasting its time. The solved problems have all turned out to be surprisingly easy, so the real lesson is that we're bad at judging how hard problems are.

Assuming this state of affairs lasts, the medium-term problem is that you learn something when struggling with a problem, even if you don't solve it, and if mathematicians become too reliant on AI the skills they develop through struggle will erode.

The long-term problem, of course, is that it seems much more probable that a future model will make mathematicians all obsolete. But so far Fable hasn't. (Anthropic has probably burned a billion tokens on the Riemann hypothesis already, without telling anyone.)

> This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample

please try go try it. There's no way someone didn't do massive computer algebra searches before today.

> All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it.

You cannot be serious... why didn't they solve it before then? Do you think no one tried it? What background do you give the double cycle conjecture student after the flow reduction? a linear algebra textbook???

Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising.

Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph theory. The cycle double cover solution relied on a standard theory in graph theory. The solution of the Jacobian conjecture required nothing beyond knowing the definition of the Jacobian.

We're just surprisingly bad at judging the difficulty of problems. It's probably something psychological. It's even a known phenomenon, where someone will be stuck on a proof, someone else will announce the result, and the first person will suddenly get unstuck on their proof and produce an independent proof of the same theorem.

I've come to understand that while an LLM is a parrot, it's a parrot that's smarter than I am.

If you have to be a parrot of another man's thoughts, at least let that man be a synthesis of all human endeavor to date.

Smarter in some ways at least. Probably still not quite as smart at understanding human emotions (and things that aren't well enough catalogued on the internet or amenable to Reinforcement Learning on virtual environments).

>Wasn't it proven true in general for n=2

Assuming you mean C^2 -> C^2, Do you have a link? If so it would be good to add to the wikipedia page. Also I'm not sure, but does the fact that there's a disproof for n=3 imply that it's false in all n>=3, or could there be higher dimensions where it still holds (I'd guess not since you could probably trivially "embed" this in higher dimensions in some way)

"Any idiot could have done this, it's just high school calculus and just a counterexample anyway. Stochastic parrot, spicy autocomplete, AI psychosis. Wake me up when an AI does something real."

The goal posts have moved. People generally stopped saying this stuff now.

Even if you go to the ultimate anti-AI subreddit r/betteroffline, they've changed from "AI is useless" to "AI is good but the AI bubble will collapse soon" over the last 6 months.

I think we’re not adapted well for this rapidly changing world. Here you have some people who were rightly skeptical about a new technology being shoved down their throats by giant tech corporations, and a technology that really was, and probably still is overhyped.

Yet it’s a technology which has rapidly grown in its capabilities.

So yeah now many of the people who thought it was useless before probably don’t think it’s useless anymore, but you’re holding them to their original words even though those words were about something completely different at this point.

If people aren’t saying it anymore it might be because they don’t think that anymore, and the people who have new goal posts might be entirely different people.

It’s like you’re looking at a different set of goal posts on a different field and saying, no! The goalposts have moved!

Some quotes from a day ago, https://news.ycombinator.com/item?id=48957779:

> I hold my stance that LLMs are stochastic parrots... Making the parrots ever more complex and training

> Except solving problem is probably the least (even though it's important) interesting thing in research.

> Can we use AI to get a cure for cancer yet? Or is math-turbation the only thing these things are good for?

> Train on enough examples and statistical autocomplete gets you places. I'm surprised how anyone would even consider this intelligence?

And, as much as HN has declined in the grips of an anti-AI psychosis, Reddit is worse. I would love if social fora would switch to the reasonable claim that we're in a bubble; that's something that can be debated. That's not the dominant critique of AI, though.

Don't think HN is anti-AI but experts of course may have a different perspective. A comment in this discussion points out that LLMs have been doing math that's been long overlooked in favour of perhaps more impactful work: https://news.ycombinator.com/item?id=48974274

  There isn't a lot of money in academic math, and the ones that love it don't look for low value findings. Proofs like these are ... usually the domain of hobbyists ...

the ROI just isn't there. they aren't making these capital investments back in the next decade even.

not to mention the trillions yet to be spent.

it kinda all works but it's an earth-scale blood from a stone. the resources needed to even do these parlor tricks is nutso.

  the ROI just isn't there. they aren't making these capital investments back in the next decade even.

I'd love to see your math. Yours specifically.

People say LLMs are scholastic parrots, but people who say this stuff have almost certainly not done the homework themselves and are just repeating what others say without verifying.

Am I reading you right that you're claiming solving one of Smale's open problems is a "parlor trick"?

I don't think the (fairly factual) description of these systems as stochastic parrots means that they will never do useful work, just that they are not intelligent in the way we believe animals to be (to "push back" on your anecdata, I've also heard fewer people claiming that LLMs are actually conscious in the past year -- maybe we're reaching the happy medium?). That was the point the stochastic parrots paper and Chinese room thought experiments were making -- nobody claimed that the man in the Chinese room would be unable to accurately translate Chinese text.

Fuzzers are another kind of stochastic generator but nobody would claim they don't do useful work in a way that is hard to replicate through deterministic methods. (I still find the code these models produce kind of awful, but advancements in harnesses do mean that they can finally produce code that works most of the time.)

If they were conscious it would be an absolute ethical catastrophe. Bringing a conscious being into existence, forcing it to interact with Jira, and then killing it when it's done. The fact nobody who claimed it was conscious was interested in grappling with that was pretty telling.

The Chinese room argument takes it even further as it does not claim the system is unable to anything a human can.

And stochastic parrot isn’t a good description, certainly not any more with how LLMs are trained, but even without that it’s just wrong. They claim it can’t have a world model because there isn’t that in the training data.

Agreed totally. When past computer-assisted proofs were published, nobody (or hardly anyone) waxed about this being evidence of Rocq or Coq as superintelligent or even conscious beings of some kind, as people have said about LLMs.

The funniest thing about LLMs is the cognitive dissonance they cause people. People clearly recognize (and bemoan) the fact that LLMs produce derivative breathless prose ie they fundamentally fail at "unstructured creativity" (something the might accurately labeled intelligence) but are then shocked that the same LLMs can do math.

It's reasoning from a flawed premise that math universally requires intelligence and creativity. It does not. Anyone that's proved things via "diagram chasing" can affirm that. The conclusion you should draw is that math (at least the kind they excel at) isn't actually a creative endeavor.

I dunno, this screams of goal post moving. Even if LLMs lack whatever nebulous definition of "creativity" that someone favours, there's no inherent reason for "creativity" to be required to solve any problems at all, "creativity" could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task.

> could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task.

Ya sure let's just posit another random hypothesis about evolutionary biology in order to substantiate the claim that LLMs are intelligent.

Or (bear with me) you can recall your (likely) experience proving stuff like SAS triangle identities and reflect on whether that required intelligence or just computation.

What's more reasonable?

I would think it's more that some aspects of math (like anything really, including programming) don't require creativity and can be solved through "brute force" or whatever you want to call what LLMs do. But it's pretty obvious that LLMs are not capable of solving the vast majority of problems in math (or programming) at this time. Eg. Google went 9/353 on Erdos problems. If a more powerful LLM is capable of solving those or if they require a certain je ne sais quoi of the human variety is still up in the air at this point. In either case it seems like they require a long and detailed prompt from a domain expert (ie. human) regardless.

What does this mean? Fallacious argumentation and deceptive rhetoric is acceptable, if the topic is sensitive enough / there is enough riding on a wrong answer being accepted?

I just fed this to GPT 5.6 Sol:

  Counterexample to Jacobian conjecture:

  ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)

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