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Claude Fable produced a counterexample to the Jacobian Conjecture

Claude Fable produced a counterexample to the Jacobian Conjecture

loubbrad

Comments (243)

  • koito17
    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.
  • aizk
    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.
  • Davidzheng
    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?
  • tekacs
    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.
  • laichzeit0
    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.
  • rao-v
    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!
  • ibarrajo
    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
  • k1t
    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.
  • kmaitreys
  • pumuck
    Would be really interesting to see the full conversation and understand how Fable came up with the counterexample in the first place.
  • hliyan
    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-...
  • dragonmage65535
  • civilchaos
    Because they've been proven equivalent, so too fall the Poisson Conjecture and the Dixmier Conjecture
  • SubiculumCode
    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.
  • mdurana
    Somebody said this on the xcancel thread but this also means that the Dixmier conjecture for the third Weyl algebra is disproven
  • loubbrad
  • dragonmage65535
    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
  • charlieyu1
    Interesting announcement on social media rather than publishing somewhere like arXiv.
  • sam_lowry_
    Looking at this poor tweet and only thinking that Telegram has proper LaTeX rendering.
  • Athanase000
    Is there a site or github repository that aggregates all these non-trivial AI-solved results?
  • luciana1u
    the conjecture held for 85 years and the counterexample was announced in a format that expires after seven days
  • Barbing
    Thanks for the mirror link OP!
  • js8
    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?
  • anon
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  • anon
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  • gammarator
    he just...he tweeted it out
  • YeGoblynQueenne
    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 finalSo where does it say that Fable "produced" the counterexample? The tweet says it was a collaboration between two people, using Fable.
  • aizk
    I want to see the Collatz Conjecture next!
  • anon
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  • anon
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  • gnarlouse
    Anybody can ELI5?
  • jdw64
    I've come to understand that while an LLM is a parrot, it's a parrot that's smarter than I am.
  • anon
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  • tacomonstrous
    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.
  • dnhkng
    Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
  • throw310822
    Overhead, without any fuss, the stars were going out.
  • senderista
    Now that is some spicy autocomplete.
  • vatsachak
    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
    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.
  • jdw64
    But I'm curious—can Fable handle cases where n=2 as well?
  • toomuchtodo
    Related:Open Problems Solved by LLMs? A Survey of Verifiable Mathematical Discovery [pdf] -https://news.ycombinator.com/item?id=48953756 - July 2026Solving 20 Erdős Problems with 20 Codex Accounts Running in Parallel - https://news.ycombinator.com/item?id=48914646 - July 2026 (110 comments)
  • scarmig
    "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."
  • claud_ia
    [flagged]
  • brcmthrowaway
    For the people saying goalposts have moved etc, whats your endgame?
  • wenc
    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.
  • huflungdung
    [dead]
  • randompartytime
    [dead]
  • pyuser583
    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!
  • laszlojamf