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)
"As written, this is an explicit counterexample to the Jacobian conjecture. I checked it using exact symbolic algebra.
I do not see an algebraic catch in what you typed. Unless a term or exponent differs from the intended expression, it appears to disprove the conjecture. This deserves serious independent checking rather than casual dismissal."
Waiting for someone to write the Lean proof.
>> hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final
So where does it say that Fable "produced" the counterexample? The tweet says it was a collaboration between two people, using Fable.
The model is still useless for real data to day work, no matter how many parlor trucks it performs.
Fable often just “knows” what I want with vague instructions. It also is able to autonomously perform work that lasts an hour long from my experience. I haven’t tested further.
Without Fable included in subscriptions, I would have moved my entire team over to Codex 5.6.
It’s a strange feeling, to be overtaken by our own creation. Top dog for millions of years and then in the blink of an eye we go from “how many Rs in strawberry” to this.
(-(1+xy)^2 z - y^3(1+xy), 2x(1+xy)z + (1+xy)^3 w + y^2(7+12xy+4x^2y^2), 2x^2z + 3x(1+xy)^2w + 2y(1+10xy+6x^2y^2), 2x - 4x^2y - x^3w): C^4 → C^4 has Jacobian determinant 4, and sends (-2,0,1,0) and (-1,0,1,-2), (1,−2,−7,14), (2,−1,0,3) to (-1,-4,8,-4)
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
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.
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.
(It's gotta be a nonzero constant, right, a nonsingular matrix).
Highly relevant comment <https://np.reddit.com/r/singularity/comments/1jh9c90/why_do_...>:
>Imagine you had a frozen [large language] model that is a 1:1 copy of the average person, let’s say, an average Redditor. Literally nobody would use that model because it can’t do anything. It can’t code, can’t do math, isn’t particularly creative at writing stories. It generalizes when it’s wrong and has biases that not even fine-tuning with facts can eliminate. And it hallucinates like crazy often stating opinions as facts, or thinking it is correct when it isn't.
>The only things it can do are basic tasks nobody needs a model for, because everyone can already do them. If you are lucky you get one that is pretty good in a singular narrow task. But that's the best it can get.
>and somehow this model won't shut up and tell everyone how smart and special it is also it claims consciousness. ridiculous.
I have no idea what any of this stuff even means, but my AI thinks I’m a legend level mathematician!
I guess you can't make a career out of being a frog anymore in math.
(NGL I wanted to suggest someone to go for the JC using 5.6 after the CDC proof came out, but then on reflection felt I should neither waste people's time NOR contribute to the myth of AI :)
My prediction is that the bubble will burst in 2031 Q4, one year after the Riemann Hypothesis is expected to fall (according to Demis)
After 2031, I will suggest going for the JC for N=4 because they would (dis)prove the Dixmier conjecture for N=2 :)
https://xcancel.com/BrunsJulian1541/status/20790734625601334...
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)
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.
Open Problems Solved by LLMs? A Survey of Verifiable Mathematical Discovery [pdf] -https://news.ycombinator.com/item?id=48953756 - July 2026
Solving 20 Erdős Problems with 20 Codex Accounts Running in Parallel - https://news.ycombinator.com/item?id=48914646 - July 2026 (110 comments)
Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.
Do you have any argument why you might think this would be true?
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.)
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???
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.
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!
> 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.
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.)
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.
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?
They're still not very good at writing, but you've flipped the takeaway. The correct conclusion is writing style isn't very relevant to intelligence.
If the determinant were zero, then (for basically the same reason as why you can't divide by zero) the matrix wouldn't be invertible. You wouldn't expect the polynomial to be invertible in that case, either.
If the determinant had free variables left over, then there is some combination of variables that will make it zero (e.g., if it came out to x^2-1, then it would be 0 for x=1 or -1). So maybe inversion won't fail everywhere, but it will fail in those spots, which would also lead one to think the polynomial is not invertible.
But if it's constant and non-zero, then there is no place where you can't invert the Jacobian. So it seems very plausible that this would apply to the polynomial as well. That's essentially what motivated the conjecture in the first place (leaving out some details, of course).
Edit my bad. I was more familiar with Dixmier and doing the translation I assumed A1 -> n=2. A2 is open ofc. Maybe to fall in 2031
the ROI just isn't there. they aren't making these capital investments back in the next decade even.
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.
"be critical, but correct. I don't want affirmation, i want to actually accomplish things."
I don't get any sycophancy from it. The conversations are actually quite good, and it'll push back if it thinks I'm wrong.
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 ...Yes-ish. Custom mRNA treatment used to reverse the progress of a dog's cancer.
https://www.theaustralian.com.au/business/technology/tech-bo...
Experience with Lenat's AM decades ago had it go off making all sorts of uninteresting hypotheses. That's very weak evidence, of course.
This suggests people also have role for fundung "beautiful" or "the best" proofs, since that also involves taste. More generally, perhaps the role of people is to reveal their preferences, and that requires people be in the loop somehow. Maybe "math criticism" becomes the job. And if AI is to serve people in general, it needs to know these preferences.
Of course it also contains more than enough information to learn what kind of question is interesting to humans.
The same was obviously true historically. People have had many revolutionary thoughts or even writings that nobody noticed. I think in this case it just feels worse because the AI could be the perfect tool for discerning the diamonds in the rough.
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.
That would be like saying Shakespeare isn’t creative , since grade school grammar doesn’t feel creative
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 think the solutions which rely on the least amount of theory are the most telling of the AIs being higher in intelligence than humans today already. There's almost no theory to teach someone to understand the cycle double cover conjecture as you say, yet no one finds it. I don't think the conclusion is that it was "easy", but that it was in fact irreducibly difficult in a way that proofs developed with theory are not. Theory gives the human brain abstractions to simplify complex proofs to be understandable at our capacity--I think there are many proofs which probably are not of this form.
But I think our differences hinge on how hard we perceive these solutions to be--I think they are very hard to find!
If tons of smart humans try for years and fail and then an LLM tries for a few weeks or hours and succeeds, the implications are clear. And these are by far the dumbest LLMs will ever be.
Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they were pretty good judges of it. (The last pre-AI example I can remember is the Gaussian correlation conjecture.) So thanks to AI we have learned that we were overconfident in our ability to judge difficulty.
If a truly major problem falls, like the Riemann hypothesis, and the proof turns out to be 10 pages, then the lesson will be a different one -- mathematicians are bad at math, and they should turn to more natural domains for them, like folding and putting away towels.
Whether it's 5, 10, 20, 50 years, obviously the takeaway cannot be "something had gone terribly wrong". The takeaway would be the smartest humans were never close to the theoretical intelligence and wisdom ceiling and never could've been. This will one day seem obvious in retrospect. There's no reason evolution by natural selection would've landed any species near such a ceiling.
That doesn't rule out an AI that makes a genuine breakthrough. If there's some new branch of math that no human has even imagined that answers the Riemann hypothesis, then that is exactly how I would expect it to go.
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.
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.
kind of a wild document to exist...
Moh repeatedly:
calls Zhang’s words “fake” and his claim “a lie”;
speculates, without demonstrating it, that Zhang “fooled” professors to get admitted;
says Zhang “failed miserably” and “wasted seven years of his own life and my time”;
alleges that Zhang “wanted to be famous all the time”;
publishes an irrelevant and humiliating story about Zhang not attending his father’s funeral;
says he would not touch Zhang “with a five feet pole”;
repeatedly emphasizes his own generosity, influence, mathematical work and supposed sacrifices.
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.
But in this case, as using plain brute force is already quite a common thing in searching for counterexamples, using LLMs as a sort of more advanced brute force seems to be just the right thing to do, so I struggle to understand so much hostility to this approach.
It's going to be really rough for a lot of folks as machines get better and better at domains that the human brain was exclusively useful for. I tend to view the hostility as a mix of both "unless we have proof this could all be hokum" and "we're going to lose a lot of what we consider makes humans amazing".
Compassion is going to be very, very important in the next few years. We're going to hurt, both internally and externally.
It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing.
Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them with a ton of scrutiny. And verification has taken more time than it did to make the discoveries.
Here’s my first piece if anybody is interested in number theory: https://arxiv.org/abs/2607.09793
I tried out using Claude to do some physics problem solving - mix of maths and simulation - and ended up with it getting in quite a mess. It's incredible at setting things up, suggesting approaches you might not have considered. I found it much much worse at interpreting things.
I suspect this is what happened, because the poster is coy about sharing the actual prompt / reasoning trace used to reach this result. That would be covered by an NDA until the model is properly released.
Exciting times!
linked from here: https://x.com/b_shrir/status/2079094004885668003?s=20
Very short version: there’s an existing false counterexample in the literature which holds almost everywhere except at a pole. It looks like Fable used this polynomial as a base & extended it in a way that eliminated the pole whilst preserving the structure.
I honestly have no idea if it's correct lol I didn't check it (I should given I actually work in AG) but it doesn't look impossible at first sight
The issue that remains are two things, ensuring the idea of the proof is actually the thing you want to prove and the interpretation of the results you get. But besides that, everything inside of the kernel checked code is logically consistent
Could we maybe get more information about the problem from the LLM trace itself here?
I'm very excited for the impact of this effect in science and medicine and other disciplines too.
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.
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.
And the conjecture was for a specific class of processes.
And Fable found an example of one concrete* process in that class and three concrete inputs (two were enough of course) giving the same output.
*Concrete here means given by a finite string of characters
Is that true? Even restricting this to f(x,y,z) and coefficients and powers to 1 ≤ x ≤ 10, there are a lot of polynomials to check, and checking requires checking the Jacobian determinant and, if it’s a non zero constant, finding two points for which the polynomial produces the same value.
Or is there a way to generate all polynomials with a non-zero Jacobian determinant, and does that speed up things? (My intuition say it wouldn’t, because I guess those with zero determinants are rare)
Be fun to ask Fable to write a search program to find more counter examples using only early grad theory to guide the search.
> The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors.
https://en.wikipedia.org/wiki/Jacobian_conjecture#cite_note-...
For someone that use Claude Code every day, this is obvious, but for some reason many scientists refuse to accept that it's truly reasoning; perhaps not in the human sense, but in a very profound and real sense. These powerful results are devastating to their point of view.
I can sympathize, because I too called LLMs "fancy Markov chains" in the GPT 3 era. But there comes a time where you have to update your world view to match reality, or be stranded in fantasy land.
There's no shortage of folks doing that in software, right now.
> ((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
One wonders if they could turn their mechinterp work into analyzing the "thought processes" of these very special cases that turn into novel research and finally crack the creative thinking barrier.
https://gist.githubusercontent.com/pedrocr/51157b9b2eed8152e...
It seems mathematics has at least gotten a powerful new tool to automate the work to cascade results after breakthroughs are made.
> ((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.
(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])
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)
To give a metaphor that everyone here on HN would understand, reading it's responses gives me the same level of wonder as one gets learning how quicksort works for the first time. It even stretches my brain to grasp what it's even come up with. I find myself getting mentally exhausted just digesting it's brilliance.
I think the singularity will have this point where AI comes up with ideas so profound, like a Ramanujen equation, that the most brilliant among us can't even decipher the answer to our questions. The internal reasoning of the machine is at a level of complexity that's beyond human comprehension to even keep track of everything enough to integrate the understanding of what it's come up with. This will happen with any even mildly complex question about any topic.
can any serious mathematicians verify https://xcancel.com/i/article/2079135211196121363
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?
https://cdn.xcancel.com/pic/orig/EA1E99C3DAE99/media%2FHNpQX...
https://cdn.xcancel.com/pic/orig/3B5C0B867A7C3/media%2FHNpQU...
If an LLM has knowledge encoded inside it (and it's hard to argue it doesn't), then cognitive dissonance can be experienced. And once experienced, must be dealt with, especially in longer-running agentic loops.
A friend was joking the other day about sending some messages under a previously-used Slack identity for an agent (since turned off), then asking the agent about the messages.
The agent maintained it hadn't sent those messages (no memory) and then was forced to reconcile the idea that the messages indeed appeared to come from it.
Its extremely-agitated conclusion was that there had been a security breach and the entire network should be locked down.
Gemini just checks the web first it seems, and already references the news.
Kimi doesn't quite believe it.
>kimi is having a blast. i turned search back on and found this post from it’s sources cited after i suggested to check out the reaction. best thing is to go to a model with search off and plop it in the session
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.
Source?
It did the multiple verification sequence before expanding to internet search where it found this thread.
--- edit, adding an explanation:
To summarize it, the conjecture says if you have any multi-variable polynomial function that maps an input to an output in the same dimensional space (take for example: F = (x+2, y+2), which maps 2D space into another 2D space), AND that function has a constant-valued non-zero Jacobian determinant, THEN the conjecture is that the polynomial has an inverse, meaning basically you can find a polynomial that turns the output space back into the input space.
Fable provided the example polynomial (which was very hard to do) and the coordinates which if you plug into it, results in two points being mapped to the same output point. This means that the polynomial can't be inverted, because if you have that output point, how do you know which input point it came from?
You can just plug in the two coordinates it gave into the equation and verify that you get the same output point from both. That's the contradiction of the conjecture and it takes 30 seconds.
---
Something something outsourcing of thinking something.
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."
"If you truly dreamt about that specific polynomial, you might be mathematically clairvoyant."
In the rest of the answer, it maintained a cautious skepticism about my claim, saying:
"Here is exactly why the math world is currently scrambling to verify the polynomial you "dreamt" about."
I love how it put "dreamt" in quotes.
Gemma's having trouble accepting it too. A solution?! At this time of year? At this time of day? In this part of the country? Localized entirely within my own prompt?
Qwen has the sprit of a grad student
5 minutes later: all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading.
Well, I'm not a conventional conspiracy theorist. But everyone knows that in every major LLM provider there are hell of hidden guarding systems that mark users and dialogues based on content (for topics about national security, biology, security, adult topics, etc.) - so there is a small chance a CEO of Google is now receiving a dozens of notifications about "ground-breaking results that could be attributed to Gemini, if act quick". So if any of thousands researchers have ever submitted this polynomial to Claude previously, any Anthropic employee can accidentally or intentionally "rediscover" the result of other researcher (and even hide the traces by deleting a dialogue of other user).
This happened multiple times to me with Gemini. For the most trivial of requests, like translating a video into English.
> "ground-breaking results that could be attributed to Gemini, if act quick"
This would be such a dumb thing to do, and so easy to get caught with...
> Taken literally, these two facts would make this map a counterexample to the complex Jacobian conjecture in dimension 3: scaling one output coordinate would normalize the determinant to 1 without restoring injectivity. Since the complex Jacobian conjecture is still treated as an open problem, this strongly indicates that the displayed formula has been mistranscribed or contains a subtle typographical error.
quite interesting indeed!
phatic mimicry.
"waste their time trying to prove it" is the MBA approach, where you should spit out results and articles.
Outside the MBA-thinking box, attacking hard problems, even unsuccessfully, is the way to gain deeper insight into various results and tools that you can later apply to other problems, i.e. no waste of time at all, unless you go to the extremes (like spending years on a single problem and nothing else).
Ergo: > “This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.
and
> “Collatz is a notoriously difficult problem — so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.
(from https://www.quantamagazine.org/mathematician-proves-huge-res... )
Sol is able to find the same counter example independently [1], so no reason to conclude in the existence of a benchmark destroying math beast Fable 6.
Furthermore, public GPT-5.6 pro failed six times to find a disproof to the Jacobian Conjecture [1], even with hints, which is evidence against the claim that the public GPT-5.6 Sol can solve this.
Regarding the existence of Fable 6: an internal upgraded version of Fable or Mythos almost certainly exists, given that Anthropic has been testing Mythos internally since April and previously released new models roughly every ~6 weeks.
UPD: The edit got reverted and there's this on the talk page now: https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-DaR...
UPD2: There are edit wars happening now: https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu... https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-Sea...
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.)
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.
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.
The author is a Princeton math doctorate.
A first year undergraduate can completely check this counterexample in ten minutes. The original post even linked Wolfram alpha for the calculations.
And if you genuinely try you can very quickly understand using only high school math and a bit of Wikipedia that this counterexample is vanishingly unlikely to be wrong, even if you don't do the calculations yourself.
Maybe because dishonesty is more normalized in SV business culture
16 variablese.g. "maybe the user's example is DESIGNED to be "correct in the stated facts" &c"
... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.
Many people believed the same about the Jacobian Conjecture.
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.
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.
It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.
I'm constantly surprised at how much schadenfreude there is on hacker news about LLMs. Like, do you guys (and girls) not have to work for a living?
Nobody is reading unsolicited proofs. They are like spam.
Source: personal friend of a “crank”.
Looks like this was also Fable.
For Qwen 27B, I have better luck with a Heretic-derived 8-bit quant than I did when I was trying to run the various smaller GGUFs.
Luckily there is now a New Scientist article to link to, so, the issue should now be resolved.
On top of that, I retried the same question + one simple question, and again, same behavior - second JC chat is loading forever. That's more just a funny observation over Gemini - today this is very likely some internal issue, tomorrow it can be used for plausible deniability against copyright accusations.
(the key being this never happened, and won't ever happen: the James Webb telescope doesn't have even 1/1000th of the resolution necessary to resolve an extraterrestrial planet)
https://www.reuters.com/technology/google-ai-chatbot-bard-of...
https://www.wired.com/story/google-openai-gemini-chatgpt-art...
https://www.popsci.com/technology/google-ai-in-paris/
I wish I could say this only happened once.
In HN terms - it's never the compiler. Yes, very occasionally it might be the compiler, but you're better off assuming it's a bug in your code.
Public GPT is certainly capable, it was able to reverse-engineer the counterexample into a short proof: https://x.com/davikrehalt/status/2079175065695035442
I still suspect Fable 6 found the counterexample, mostly because Anthropic has been silent about this achievement. This is a huge accomplishment, and companies don't usually stay quiet when they hit a breakthrough like this. The lack of a press release or blog post is very suspicious.
Similarly I've run into several "you just gotta know" type problems where LLMs still just fail. My favourite was a quirk in how async relationships work inside emberjs. Three different models gave a variation of the same incorrect answer. When I searched myself, I initially came up with nothing and eventually found what I think might be the only example of the same issue on the internet, a single stack overflow question wth two responses. The first is what Claude, gemini and chatGPT said, the second response was the OP saying it was wrong. I asked in the Ember discord and a core team member responded instantly with the answer.
There's a video I love on Youtube, where a guy uses ML to assemble a blank jigsaw. It performs amazingly, the jigsaw being blank is of no consequence and if it did have an image, it'd perform worse. That's all LLMs do. Just because the jigsaw pieces are smaller, they're still just getting assembled in whatever way fits, there's no mechanic for interpreting the image on the front.
Why do LLMs still have trouble on floating point math without forking out to a tool, but they can perform symbolic manipulation just fine?
Because symbolic manipulation is just rote work and textual stepping through symbols. A side poster commented on the number of prior attempts on this problem which were close, but not quite.
Starting from a known "close" solution (which this did), and using exploration to search around the space is exactly something an LLM would and could be good at (clearly).
The "transformer LLMs are next-token predictors with some in-GPU processing of bounded complexity with respect to token count" remains undefeated. Both because that is mathematically what they are, and also because we don't have counterexamples to that effect that don't require some higher-order tooling wrapping the systems.
Symbol manipulation is what LLMs are good at.
What still confuses people is the insane inefficiency of deep learning, and that those emergent capabilities require such an immense training corpus compared to the only other architecture that we know of.
But this already is an optimization problem. If the machine gets super human at symbolic reasoning, and at the same time, can solve the symbol grounding problem to real world data and sensors, what prevents you from saying it thinks? Can it not solve real world problems? Can it not redefine its tasks and display some form moral agency - even if a totally foreign morality for us humans? Can it not use these abilities to reproduce and expand, create ships and turn the universe into paperclips, if it finds it worthwhile?
Math is basically just a playground that is perfectly suited for these emergent capabilities, so of course we will see the first progress here; but there is no firewall separating math problems from general cognition.
And the "insane inefficiency" of deep learning is fully to be expected from how it works. As well, there are provably no—literally no—emergent properties in these models. The choice of metric was a convenient, sloppy, and embarrassing fault of the field. It should be discredited; the field should be embarrassed; expectations on messaging should have changed; and it did not.
Why? Because the industry is full of charlatans, and this is a highly profitable enterprise telling people that this would lead to AGI.
Multiply two floating point numbers without a tool call. Still can't, because it's a curve fit.
So, in summary, nothing you just said is relevant. There are no emergent capabilities, simply 1) search, + 2) the original set of learned feature vectors from throwing tons of data at this.
Remember when Anthropic wouldn't release Fable because it would be "the end of cyber security as we know it"? Yet here we are
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.
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’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.
Sorry, but if the current models haven't shown you something amazing at this point that would make this of little surprise, it because the person is an intellectual bore and completely full of shit themselves.
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)
https://en.wikipedia.org/wiki/Wikipedia:Reliable_sources#Sel...
> Self-published expert sources may be considered reliable when produced by an established expert on the subject matter, whose work in the relevant field has previously been published by reliable, independent publications.
What Wikipedia says about this doesn't matter, does it?
https://en.wikipedia.org/wiki/Wikipedia:No_original_research...
Of course given the magnitude of the statement a bit of care is warranted, but the rules do allow it.
I mean, lol.
Additionally, posting this during the World Cup would be the most inefficient way to do marketing.
Is the marketing stunt that Anthropic has secretly built a world-class mathematics research group?
I'm also not going to live all that much longer, most likely, so it's kind of annoying I'm not really going to see any upsides to an AI world either. I'm discounting the possibility of our AI overlords figuring out a miracle like reversing aging.
But it is really f-ing cool that automated math is now a thing and we are seeing it. Eat your heart out, past me.
I don't think the "permanent underclass" will see many upsides either. They certainly won't have the money to pay for it. I'm excluding myself in that statement because luckily I have access to large amounts of barbiturates.
You may be seeing a loss of privilege due at least in part to the inevitable reversion to the mean of US economic dominance. But on a global scale, things are going well. AI will lead to a much larger pie.
You comment suggests you feel there is something wrong with what I said, could you elaborate?
I am intellectually curious whether "intelligence" and "consciousness" are simply layers that can be separated from the hardware they operate on or first emerged from, but I never had the urge or saw the point to practically replace human beings by machines.
Not everything that can be done should be done.
When I go to a supermarket, I always avoid self check-out. Why? First, I enjoy human interaction, even if machines may be more efficient (not yet the case). Second, I don't want the people working at the checkout to lose their jobs.
When we purchased a GPU cluster, I felt bad when I learned that the electricity bill was going to be north of 60 000 EUR every year (that number apparently assumes no jobs are running - so 100% idle time).
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.
Art is about human expression, it's not about the reflection of light off a surface. But economically I could see a world in which human expression is replaced by a facsimile and the world would be worse off for it. And that is what scares me.
Brave new world indeed.
edit: whoever downvoted the person replying to me, shame on you. they make good points and makes for good discussion.
People will make things they don't care about to survive. This is how we get reality television.
Irrespective of what the short term future of AI is, sooner or later there will be technical means of sustaining yourself without needing to work. If we aim for a society that values working on something because they believe in it, then we could have a better world. Right now it seems the only thing a lot of people are aiming for is to undermine whatever their adversaries want.
Edit: Now the OP is flagged/dead for some reason. You could disagree on their take (calling it a marketing stunt is maybe a bit much), but I think the argument is sound, so flagging seems counterproductive to the discussion.
They got flagged because it is a literal conspiracy theory that assumes bad faith.
Why not be skeptical about that?
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.
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.
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.
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.
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?
because they are using the same prompt to try finding other counter-examples. they are milking it
Both are impressive, of course, but they're hardly comparable.
I never claimed they were. The term "permanent underclass" refers to the preferred world order of AI techbro billionaires[1] where human labour is worthless and in consequence normal people no longer have any leverage. Societies where most of the value is dug out of the ground and where human labour is comparatively worthless already exist and living there is not necessarily fun[2].
> World wealth is increasing, the actual global underclass is seeing rapidly improving lives.
The global underclass has been seeing improving lives well before LLMs were a thing. The question is whether they will _continue_ to see their lives improve if their labour is no longer required.
> You may be seeing a loss of privilege due at least in part to the inevitable reversion to the mean of US economic dominance.
I'm not from the US, I don't live in the US, and in fact I've never been to the US.
> AI will lead to a much larger pie.
Which will only benefit people if their piece of the pie doesn't shrink because they don't have any leverage. There is also more than one pie. The real estate pie for example won't grow, instead every piece of the pie will become even more expensive. Most people only own pieces of the pie that will decrease in value (human labour) not pieces of pies that will increase in value.
[1]: https://www.nytimes.com/2026/04/30/opinion/ai-labor-work-for...
Absent a world dictatorship there's always competition here. And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.
Because they're billionaires. They don't actually work for a living. They own stuff for a living.
> And fully superhuman AI eliminates the main objection to socialism, that you need markets with smart people to make an economy function. Full automated luxury communism (like Ian Banks) is a possible outcome.
Luxury communism is also a possible outcome in countries with large oil reserves (Venezuela for example), it's just not a likely outcome because common people have no leverage and billionaires didn't become billionaires by giving stuff away.
Even if "fully automated luxury communism" were a likely outcome, the transition there would be very rough and would take years. If software engineering is automated away in 2028 it could still take years or decades before the first general purpose robot can take over plumbing. (Former) software engineers will have a rough transition.
that's self-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
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.
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.
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?