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.
zaik 2 hours ago [-]
Would this counterexample not be included in their search space?
koito17 1 hours ago [-]
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.
jasonfarnon 28 minutes ago [-]
based on the approach described they were probably in dimension 2, this counterexample is 3 dimensional
aizk 3 hours ago [-]
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.
minimaxir 3 hours ago [-]
Who knew DOES NOT COMPUTE would be an actual thing?
left-struck 2 hours ago [-]
The unexpected part is it does compute!
monocasa 2 hours ago [-]
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.
It did the multiple verification sequence before expanding to internet search where it found this thread.
minimaxir 3 hours ago [-]
> 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.
rvz 3 hours ago [-]
..and (you guessed it) before GTA 6.
moffkalast 2 hours ago [-]
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.
SubiculumCode 3 hours ago [-]
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
CamperBob2 3 hours ago [-]
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."
7373737373 40 seconds ago [-]
Current LLMs behave very unproductively around unsolved problems, especially if they learned that humans consider them difficult. This has many straight up preventing themselves from attempting anything...
moffkalast 1 hours ago [-]
> Matches! This is bizarre. A Jacobian counterexample has been sitting here in a prompt? Wait... is this map a known "fake" counterexample from the literature?
Many mathematicians have tried and failed. This specific map might come from a paper or a forum where it was proposed and then debunked. Or... is it actually correct?
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?
kelseyfrog 3 hours ago [-]
> that someone named Shestakov had already found a counterexample in 2004.
Qwen has the sprit of a grad student
card_zero 7 minutes ago [-]
It must mean Chestikoff. He dedicated his life to intellectual pursuits due to chronic ill health.
QuesnayJr 2 hours ago [-]
There's a certain (unfair) stereotype of Russian mathematicians that this fits.
2 hours ago [-]
wahnfrieden 2 hours ago [-]
This is also just an annoying Claude personality trait
cgio 2 hours ago [-]
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arcfour 2 hours ago [-]
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baq 1 hours ago [-]
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Davidzheng 58 minutes ago [-]
I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?
ibarrajo 45 minutes ago [-]
I’m doing this by working all logical steps into lean (formal verification) the quick feedback loop between the AI prose and the Lean verification errors and warnings ensures that its logically consistent.
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
iib 30 minutes ago [-]
In my experience, lean will show that it's correct, but does it not lose the mathematical intuition that led to the result? As far as my experience goes, that's really hard to encode in lean itself.
Could we maybe get more information about the problem from the LLM trace itself here?
olmo23 15 minutes ago [-]
As proofs become more and more complex, we will need two AI pipelines: one to generate the LEAN proof, and a second one to extract useful lessons for mathematicians from the LEAN proof.
tekacs 38 minutes ago [-]
Given how small the counterexample is... this feels like a great example of where a lot of interesting results are going to be found: not because they were super difficult, but because intelligence didn't scale, and until computers could do this for us, the number of people who seriously poked at many such things was low.
I'm very excited for the impact of this effect in science and medicine and other disciplines too.
ibarrajo 54 minutes ago [-]
I’ve been math-vibe coding a few months now.
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.
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.
free_bip 4 hours ago [-]
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.
beering 4 hours ago [-]
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.)
nommynommynom 2 hours ago [-]
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.
llelouch 3 hours ago [-]
True but Noam brown (openai researcher) said that in 2 years AI will start creating new math.
ajkjk 3 hours ago [-]
well if he said it then it must be true
1 hours ago [-]
semi-extrinsic 3 hours ago [-]
As long as they are not poisoning pigeons in the park.
Lerc 2 hours ago [-]
We'll all go together when we go.
aizk 3 hours ago [-]
Give the llms a few years, they'll be smart enough to make progress on that.
YeGoblynQueenne 9 minutes ago [-]
Actual tweet:
>> 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.
rao-v 2 hours ago [-]
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!
misiek08 49 minutes ago [-]
So IIUC no one really worked on that in „professional” or rather university world since then?
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.
qsort 2 hours ago [-]
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.
verteu 3 hours ago [-]
I agree, the conjecture never says anything about a proper map.
whimsicalism 3 hours ago [-]
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virgildotcodes 3 hours ago [-]
When you eat alphabet soup, do you take note of all the leftist messages bobbing in and out of existence in your bowl?
tacomonstrous 3 hours ago [-]
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.
Davidzheng 2 hours ago [-]
sure it does? two copies of the affine line? (I guess there's no galois group & no connected finite etale things tho)
slopinthebag 4 hours ago [-]
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.
omaranto 4 hours ago [-]
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.)
EagnaIonat 4 hours ago [-]
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.
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.
Lerc 2 hours ago [-]
>"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.
thin_carapace 2 hours ago [-]
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?
Lerc 1 hours ago [-]
Do you have links to these videos?
You comment suggests you feel there is something wrong with what I said, could you elaborate?
slopinthebag 2 hours ago [-]
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.
Lerc 1 hours ago [-]
> 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.
slopinthebag 49 minutes ago [-]
Well, I think an AI model could replace artists in a meaningful way without being conscious. So you end up with less 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.
randompartytime 4 hours ago [-]
.
rowbin 4 hours ago [-]
Anti-AI mentality claims another victim
rvz 4 hours ago [-]
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.
tptacek 4 hours ago [-]
It's a Jacobian determinant and three points. You can check this yourself in Sage.
The author is a Princeton math doctorate.
markasoftware 3 hours ago [-]
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)
endgame 2 hours ago [-]
The talk page about the conjecture clarifies WP:RSSELF:
> 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.
tptacek 3 hours ago [-]
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?
bonzini 2 hours ago [-]
Wikipedia does have a rule allowing "routine calculations"
Of course given the magnitude of the statement a bit of care is warranted, but the rules do allow it.
tptacek 2 hours ago [-]
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.
CamperBob2 3 hours ago [-]
Wikipedia's editors won't be happy until the site has joined Stack Overflow in self-owned oblivion.
urams 4 hours ago [-]
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.
sciyoshi 3 hours ago [-]
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.
rowbin 4 hours ago [-]
Maybe have a seasoned mathematician check if the counterexample is really bogus before saying people fell for ai psychosis.
tptacek 4 hours ago [-]
I'm pretty sure you can just check this for yourself. The Sage POC is like 8 lines.
Larrikin 3 hours ago [-]
HN and a good chunk of the Internet got themselves in to tizzy over super conductivity for weeks until someone with a PhD and a very regular contribution finally noticed weird Internet hype and explained it to the world. A world changing event is not going to happen on a message board
k1t 2 hours ago [-]
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.
danbruc 2 hours ago [-]
[...] 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.
2 hours ago [-]
QuesnayJr 2 hours ago [-]
It was known to be false in general. But there are many questions that are false in general, but true when restricted to polynomials.
pumuck 3 hours ago [-]
Would be really interesting to see the full conversation and understand how Fable came up with the counterexample in the first place.
hliyan 5 hours ago [-]
I suspect the LLM was able to synthesize a counterexample because of the availability of a lot of prior work:
> The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors.
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".
cornholio 17 minutes ago [-]
I think it's becoming harder and harder to argue that LLMs don't really reason and just mimicry human speech. This counterexample is clearly the result of a sequence of steps that build on previous knowledge in context and logically combine it to reach other true statements - to a degree and complexity that rivals the best human minds.
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 updated your understanding to match reality, or be stranded in fantasy land.
ignoramous 4 hours ago [-]
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.
semi-extrinsic 3 hours ago [-]
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.
IngoBlechschmid 1 hours ago [-]
Strongly depends on the subbubble of mathematics. In some parts of type theory / formal proofs for instance, there is a rather strong rejection of LLMs (for moral reasons in addition to quality reasons). The proof assistant Agda was even forked for this reason: https://types.pl/@amy/116522250630340534
aurareturn 3 hours ago [-]
I think everyone assumes most, if not all of the code written in a piece of new software is done by AI.
tptacek 4 hours ago [-]
How does that work in this case? What do those proofs do to help find this counterexample?
4 hours ago [-]
hliyan 4 hours ago [-]
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
tptacek 4 hours ago [-]
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.
TOMDM 4 hours ago [-]
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.
QuesnayJr 2 hours ago [-]
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.
dragonmage65535 37 minutes ago [-]
Can someone validate this counterexample from gpt-5.6 sol
(-(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)
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.
Jweb_Guru 2 hours ago [-]
"Anthropic and OpenAI" it's basically been all ChatGPT outside of this one.
bcstyle 2 hours ago [-]
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.
charlieyu1 3 hours ago [-]
I mean of course an employee has cheap tokens
trentor 3 hours ago [-]
Post hoc ergo propter hoc.
SubiculumCode 3 hours ago [-]
Are you sure that this applies?
harshreality 52 minutes ago [-]
I don't think it's likely, but I think trentor might be suggesting that people simply aren't attempting to solve open math problems with Chinese models, even if they're roughly as capable on a cost-efficiency basis. If the people interested in paying to solve open theoretical math problems are almost entirely employed by entities that have large allowances from the U.S. frontier labs, they're going to use those labs' models, right?
I think it's false because there are many entities, especially outside the U.S., interested in math research, entities that would not have such Anthropic/OpenAI allowances. But it's at least a plausible application of the fallacy.
trentor 2 hours ago [-]
Yes, why shouldn't it?
1 hours ago [-]
23 minutes ago [-]
mdurana 51 minutes ago [-]
Somebody said this on the xcancel thread but this also means that the Dixmier conjecture for the third Weyl algebra is disproven
> 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.
krackers 4 hours ago [-]
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])
jameshart 4 hours ago [-]
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)
sam_lowry_ 2 hours ago [-]
Looking at this poor tweet and only thinking that Telegram has proper LaTeX rendering.
Athanase000 2 hours ago [-]
Is there a site or github repository that aggregates all these non-trivial AI-solved results?
charlieyu1 3 hours ago [-]
Interesting announcement on social media rather than publishing somewhere like arXiv.
teravor 3 hours ago [-]
a short counterexample makes it likely the best medium for such an announcement. would have been inappropriate for a positive proof.
se4u 3 hours ago [-]
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.
js8 2 hours ago [-]
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?
yorwba 1 hours ago [-]
All the coefficients and evaluation points are rational, so it's a counterexample in all fields where 2 ≠ 0 and 3 ≠ 0, doesn't matter whether that field is the complex numbers, real numbers, rational numbers or even a finite field.
js8 31 minutes ago [-]
Ah you're right, thanks.
krackers 2 hours ago [-]
>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
js8 2 hours ago [-]
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.
Barbing 5 hours ago [-]
Thanks for the mirror link OP!
luciana1u 5 hours ago [-]
the conjecture held for 85 years and the counterexample was announced in a format that expires after seven days
krackers 5 hours ago [-]
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?
tyre 3 hours ago [-]
Reading the chain of thought for Fable, it is incredulous as well. It keeps thinking that it must be missing something or that this is a trick, because it can't have just found a counterexample to a famous conjecture. It's just like us!
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
ajkjk 3 hours ago [-]
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.
sawjet 3 hours ago [-]
Neither of those links work for me
charlieyu1 3 hours ago [-]
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
nullsanity 4 hours ago [-]
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.
h-jones 4 hours ago [-]
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.
4 hours ago [-]
gammarator 5 hours ago [-]
he just...he tweeted it out
markasoftware 5 hours ago [-]
I almost felt bad adding {{Cite tweet ...}} to the wikipedia article
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.
krackers 4 hours ago [-]
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.
spuz 1 hours ago [-]
Thanks that's a very nice summary.
SyzygyRhythm 26 minutes ago [-]
Thanks, though I realize (thanks to tptacek) that I didn't properly emphasize that the determinant must be constant but non-zero.
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).
tptacek 4 hours ago [-]
Nice!
(It's gotta be a nonzero constant, right, a nonsingular matrix).
SyzygyRhythm 4 hours ago [-]
Yeah, the only possible counterexamples would have a constant but non-zero determinant (in this case it was -2).
tacomonstrous 2 hours ago [-]
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.
Davidzheng 2 hours ago [-]
next year is a long time away friend
meowface 1 hours ago [-]
It's so soon that it may be rational to delay certain math and software projects until smarter models arrive
jasonfarnon 38 minutes ago [-]
no, he's right, we're fucked
Davidzheng 29 minutes ago [-]
? did you take my comment to mean that math will not be affected? I meant like it'll come much sooner--like in <5 months
QuesnayJr 2 hours ago [-]
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.)
Davidzheng 2 hours ago [-]
> 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???
QuesnayJr 1 hours ago [-]
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.
Davidzheng 41 minutes ago [-]
Sorry last comment was a bit emotional from me, but I do not think it's findable like you say--during my PhD I tried to find some ideals I knew existed in char 2 in 5 variables and low degree and I didn't think I ever got close. 3 variables, 7 degree, coefficients up to 6 is like 6^100 possibilities. You've got to narrow it down somewhat no? Even sparse is intractable I would guess.
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!
Davidzheng 11 minutes ago [-]
But actually my feeling is that the final solutions of these last two problems probably is hiding how the AI came up with them! For all we know it used a LOT of theory! As Dolly Parton says "it takes a lot of money to look this cheap" and it takes a lot of intelligence for the proofs to look this dumb. [In high school, I knew of this competition math kid joke where after you derive an inequality with various methods, you use standard results to write the original equations as just a sum-of-squares---like in a "are you stupid, it's >=0 bc it's a sum of squares" sort of way]
meowface 1 hours ago [-]
You can use that retroactive logic about any hard problem though. Unsolved murder cases, math, theoretical physics.
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.
aizk 3 hours ago [-]
I want to see the Collatz Conjecture next!
5 hours ago [-]
dnhkng 1 hours ago [-]
Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
jdw64 4 hours ago [-]
I've come to understand that while an LLM is a parrot, it's a parrot that's smarter than I am.
meowface 1 hours ago [-]
Hot take: even GPT-3 was not a parrot. Skeptics have never properly internalized that the fundamental operation is basically irrelevant to the gestalt. Humans are not parrots yet neurons likely also largely operate via predictive processing.
sobellian 4 hours ago [-]
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.
userbinator 4 hours ago [-]
"If I have seen further it is by standing on the shoulders of Giants." -Isaac Newton
4 hours ago [-]
thinking_cactus 3 hours ago [-]
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).
neuroticnews25 2 hours ago [-]
Probably not as smart as highest EQ humans but smarter than me, I use LLMs to help pin down what emotion I'm feeling.
kolinko 39 minutes ago [-]
Same here
vatsachak 21 minutes ago [-]
And yet it can't even replace a call center employee.
I guess you can't make a career out of being a frog anymore in math.
mathisfun123 4 hours ago [-]
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.
slopinthebag 4 hours ago [-]
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.
ViscountPenguin 4 hours ago [-]
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.
mathisfun123 4 hours ago [-]
> 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?
gmadsen 4 hours ago [-]
Why is your example SAS triangles and not something like the modularity theorem?
mathisfun123 4 hours ago [-]
Because SAS triangle identity proofs are relatable and I'm trying to make a point...?
meowface 1 hours ago [-]
Hilariously absurd cope.
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.
senderista 3 hours ago [-]
Now that is some spicy autocomplete.
baq 6 minutes ago [-]
"the Riemann Hypothesis is " <tab>
throw310822 6 hours ago [-]
Overhead, without any fuss, the stars were going out.
jdw64 4 hours ago [-]
But I'm curious—can Fable handle cases where n=2 as well?
oliculipolicula 3 hours ago [-]
Wasn't it proven true in general for n=2
(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 :)
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)
For the people saying goalposts have moved etc, whats your endgame?
3 hours ago [-]
perching_aix 2 hours ago [-]
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?
wenc 5 hours ago [-]
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)
GPT wrote some SymPy code to check it. The response?
"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.
baq 5 minutes ago [-]
...no need for any lean here
radokirov 3 hours ago [-]
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huflungdung 58 minutes ago [-]
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randompartytime 3 hours ago [-]
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scarmig 5 hours ago [-]
"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."
aurareturn 5 hours ago [-]
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.
tptacek 5 hours ago [-]
People have definitely not stopped saying this stuff.
aurareturn 5 hours ago [-]
I said "generally". It's declined sharply over the last few months.
rowbin 4 hours ago [-]
as can be seen in this comment section
CamperBob2 3 hours ago [-]
Morons still say it, but then they would, wouldn't they?
left-struck 2 hours ago [-]
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!
> 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.
endgame 30 minutes ago [-]
>> Can we use AI to get a cure for cancer yet?
Yes-ish. Custom mRNA treatment used to reverse the progress of a dog's cancer.
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 ...
4 hours ago [-]
cyphar 4 hours ago [-]
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.)
arcticbull 2 hours ago [-]
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.
IanCal 1 hours ago [-]
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.
tomjakubowski 4 hours ago [-]
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.
aurareturn 3 hours ago [-]
I don't think LLM labs ever claimed that their LLMs are intelligent the way we believe animals are.
simianwords 2 hours ago [-]
Classic case of Motte and Bailey applied.
whateveracct 4 hours ago [-]
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.
aurareturn 3 hours ago [-]
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.
tptacek 4 hours ago [-]
Am I reading you right that you're claiming solving one of Smale's open problems is a "parlor trick"?
pyuser583 4 hours ago [-]
I asked Fable to verify and it absolutely freaked out!
I have no idea what any of this stuff even means, but my AI thinks I’m a legend level mathematician!
To be fair to the sycophancy tendencies, this was an open conjecture that held for 85 years, and not for lack of trying. So, maybe a bit warranted here? :)
tptacek 4 hours ago [-]
I told GPT5.6 I came up with this walking down the street. "Jesus Christ" was its response.
SubiculumCode 3 hours ago [-]
Lol, you must have some custom instructions on that beast, lol
s1artibartfast 3 hours ago [-]
[flagged]
sosodev 4 hours ago [-]
That’s the worst thing about sycophancy. You can never tell when it’s warranted or not. I imagine that many AI chats have had real gold in them and yet we’ll never know.
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.
meowface 4 hours ago [-]
The interesting thing about using Claude Fable 5 is it's nearly as irritatingly sycophantic as past Claudes while genuinely being smarter than the previous models. So you get a kind of yo-yoing of it glazing you as a creative genius and disappointedly revealing to you that your ideas are bad and dumb.
shepardrtc 4 hours ago [-]
That is easily fixable with the "Instructions for Claude" in your settings. Just put:
"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.
baq 4 minutes ago [-]
gotta try with "think in russian"
krackers 4 hours ago [-]
I mean if someone came to you with a value of n that disproved collatz, wouldn't you go crazy as well?
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.
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.
It did the multiple verification sequence before expanding to internet search where it found this thread.
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."
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
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.
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
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.
>> 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.
https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu...
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.
https://arxiv.org/pdf/2603.10025
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.
You comment suggests you feel there is something wrong with what I said, could you elaborate?
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.
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.
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.
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.
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.
> 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 updated your understanding 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
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.
(-(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)
https://xcancel.com/__alpoge__/status/2079091571316912542
I think it's false because there are many entities, especially outside the U.S., interested in math research, entities that would not have such Anthropic/OpenAI allowances. But it's at least a plausible application of the fallacy.
> ((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)
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
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...
https://en.wikipedia.org/wiki/History_of_quaternions
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.
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).
(It's gotta be a nonzero constant, right, a nonsingular matrix).
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???
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.
I guess you can't make a career out of being a frog anymore in 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.
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.
(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...
https://arxiv.org/abs/2410.06959
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)
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)
"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.
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.
Yes-ish. Custom mRNA treatment used to reverse the progress of a dog's cancer.
https://www.theaustralian.com.au/business/technology/tech-bo...
https://www.youtube.com/watch?v=COYSRbF1F-Y
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.)
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.
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.
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.
I have no idea what any of this stuff even means, but my AI thinks I’m a legend level mathematician!
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.
"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.