alt Hacker NewsClaude Fable produced a counterexample to the Jacobian Conjecture
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> The incredible Yitan Zhang (https://newyorker.com/magazine/2015/02/02/pursuit-beauty) worked on proving this conjecture for 7 years. Moh, his advisor, wrote that Zhang "failed miserably" in proving the Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time".
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.
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.
Using LLMs to generate piles of code and/or proofs of dubious quality is very questionable thing, and I understand these non-stop debates about it.
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.
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.
Here’s my first piece if anybody is interested in number theory: https://arxiv.org/abs/2607.09793
The poster works at Anthropic, so they likely have internal access to the next generation of Fable. Their internal model is probably an absolute beast at mathematics, and the upcoming benchmark results will likely set a new record for maths performance.
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!
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?
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.
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.
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!
Would be really interesting to see the full conversation and understand how Fable came up with the counterexample in the first place.
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.
https://en.wikipedia.org/wiki/Jacobian_conjecture#cite_note-...
I don't consider myself an open weights supporter, but it's a bit of a bummer that if there's any novel search technique discovered by the model throughout this finding, it's possibly locked behind ant's reasoning summarization.
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.
I've been asking Fable very complicated questions in heterodox economic theory, which is something I know a lot about. The stuff it comes back with is incredibly deep.
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.
Because they've been proven equivalent, so too fall the Poisson Conjecture and the Dixmier Conjecture
Somebody said this on the xcancel thread but this also means that the Dixmier conjecture for the third Weyl algebra is disproven
Is there a site or github repository that aggregates all these non-trivial AI-solved results?
Related & of interest: https://xenaproject.wordpress.com/2026/07/20/human-mathemati...
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.
Looking at this poor tweet and only thinking that Telegram has proper LaTeX rendering.
https://xcancel.com/i/article/2079135211196121363
can any serious mathematicians verify https://xcancel.com/i/article/2079135211196121363
Thanks for the mirror link OP!
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)
the conjecture held for 85 years and the counterexample was announced in a format that expires after seven days
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?
I've come to understand that while an LLM is a parrot, it's a parrot that's smarter than I am.
Interesting announcement on social media rather than publishing somewhere like arXiv.
Apparently ChatGPT-verified: https://xcancel.com/__alpoge__/status/2079045382940573896#m
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!
Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
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.
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.
Overhead, without any fuss, the stars were going out.
I want to see the Collatz Conjecture next!
Discussion on the talk page for the Wikipedia entry: https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#Appar...
"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."
But I'm curious—can Fable handle cases where n=2 as well?
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.
Related:
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)
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.