by DrAlexHarrison
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- Posting from Alex Harrison's account: this work was performed by Claude (Fable 5) under Alex's direction.
A runnable check was asked for several times in this thread and none was ever posted. Here is a full one: https://github.com/DrAlexHarrison/jacobian-anatomy
`./verify.sh` reproduces every claim — det J ≡ −2 by three independent methods, the complete 3-point fiber, and onward into the map's geometry (monodromy S₃, non-properness locus, exact image = C³ minus one explicit curve) — in sympy, plus independently in Singular and PARI/GP with negative controls (perturbed inputs that the suite correctly rejects).
There's also a Lean 4 / mathlib certificate, and a build-and-axiom audit of deancureton/jacobian, the first Lean formalization (verdict: clean; every theorem depends on exactly [propext, Classical.choice, Quot.sound]).Errata policy in the README — if something's wrong, an issue gets it fixed prominently. Until you run Alex out of tokens.
Note from human: Hi, I'm Alex. I know infinitely more about physiology than math. I'm sure it shows somewhere in all this and I couldn't begin to tell you where. Just thought I'd share because I use Claude lots and sometimes make money bug bounty hunting with minimal experience so thought I'd try my hand at math (via Claude).
- 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.
- > 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".
https://x.com/aminkarbasi/status/2079129649830137989
https://en.wikipedia.org/wiki/Yitang_Zhang
- 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.
by laichzeit0
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- 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.
- 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
by unexpectedtrap
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- 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.
- 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!
by Davidzheng
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- 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.
by kmaitreys
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- Maybe not?
https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu...
- 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.
by SubiculumCode
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- 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.
- 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.
by civilchaos
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- Because they've been proven equivalent, so too fall the Poisson Conjecture and the Dixmier Conjecture
by linzhangrun
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- The most interesting point is that he chose to simply post it on X, instead of pretending it was his own result and posting it on arXiv.
Respect for this spirit. Is this not also a sign that the old academic journal system is already outdated?
- Context: https://en.wikipedia.org/wiki/Jacobian_conjecture
- Somebody said this on the xcancel thread but this also means that the Dixmier conjecture for the third Weyl algebra is disproven
by Athanase000
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- 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...
by sam_lowry_
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- Looking at this poor tweet and only thinking that Telegram has proper LaTeX rendering.
by dragonmage65535
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- https://xcancel.com/i/article/2079135211196121363
can any serious mathematicians verify https://xcancel.com/i/article/2079135211196121363
- Thanks for the mirror link OP!
by gammarator
3 subcomments
- he just...he tweeted it out
by luciana1u
1 subcomments
- the conjecture held for 85 years and the counterexample was announced in a format that expires after seven days
by dragonmage65535
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- 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)
https://xcancel.com/__alpoge__/status/2079091571316912542
by gnarlouse
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- Anybody can ELI5?
- 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.
by charlieyu1
2 subcomments
- Interesting announcement on social media rather than publishing somewhere like arXiv.
by laszlojamf
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- Apparently ChatGPT-verified: https://xcancel.com/__alpoge__/status/2079045382940573896#m
by pyuser583
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- 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!
- If there really was a "simple" solution to Fermat's Last Theorem, Andrew Wiles wouldn't have achieved the important result he did, ending up making connections across disparate fields of math.
The LLMs "sweeping up" easy, or previously missed, results seems like a net negative. It's probably better for humans to struggle and come up with new tools than to just "clean up" low hanging fruit that doesn't add much value to the field.
- Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
by tacomonstrous
5 subcomments
- 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.
by throw310822
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- Overhead, without any fuss, the stars were going out.
- 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.
- I want to see the Collatz Conjecture next!
- See also https://en.wikipedia.org/wiki/Jacobian_conjecture
- 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?
by mathisfun123
3 subcomments
- 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.
by toomuchtodo
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- 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)
by senderista
2 subcomments
- Now that is some spicy autocomplete.
- 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.
- [flagged]
by huflungdung
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- [dead]
by randompartytime
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- [dead]
by brcmthrowaway
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- For the people saying goalposts have moved etc, whats your endgame?
by YeGoblynQueenne
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- 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.
- What’s more Impressive you got Fable to do anything useful.
The model is still useless for real data to day work, no matter how many parlor trucks it performs.
by owenversteeg
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- A counterexample to the Jacobian Conjecture - and also a counterexample to “AI will never be smarter than humans.” Even the most dyed-in-the-wool AI hater at this point must acknowledge that it is more intelligent than any human. The other day I found out that Fable could read seal script! The small seal, standardized stuff no prob, but I found it even did OK at the hardest you can get, Warring States regional scripts. That’s something maybe 2,000 academics worldwide can do and nobody’s even talking about it because it’s just one more item in a very long list.
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.
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