12 ms·

Claude Fable produced a counterexample to the Jacobian Conjecture

• loubbrad 2mo ago
  Context: https://en.wikipedia.org/wiki/Jacobian_conjecture https://en.wikipedia.org/wiki/Jacobian_conjecture
  • hyperhello 2mo ago
    > 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 2mo 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 2mo 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)
• throw310822 2mo ago
  Overhead, without any fuss, the stars were going out.
• meowface 2mo ago
  The tweeter, Levent Alpöge, is a professional mathematician who works at Anthropic.
• gammarator 2mo ago
  he just...he tweeted it out
  • markasoftware 2mo ago
    I almost felt bad adding {{Cite tweet ...}} to the wikipedia article
    • mitthrowaway2 2mo ago
      At least it's not {{Cite bridge graffiti}} https://en.wikipedia.org/wiki/History_of_quaternions https://en.wikipedia.org/wiki/History_of_quaternions
  • tptacek 2mo ago
    The "thanx" is one for the ages.
  • deleted 2mo ago
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• toomuchtodo 2mo ago
  Related: Open Problems Solved by LLMs? A Survey of Verifiable Mathematical Discovery [pdf] -https://news.ycombinator.com/item?id=48953756 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 https://news.ycombinator.com/item?id=48914646 - July 2026 (110 comments)
• Barbing 2mo ago
  Thanks for the mirror link OP!
• wenc 2mo 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.
  • radokirov 2mo ago
    [dead]
  • baq 2mo ago
    ...no need for any lean here
  • rirze 2mo ago
    Here's your Lean proof https://github.com/google-deepmind/formal-conjectures/pull/4474/changes https://github.com/google-deepmind/formal-conjectures/pull/4...
• luciana1u 2mo ago
  [flagged]
  • krackers 2mo 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?
    • nullsanity 2mo 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 2mo 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.
        • pfdietz 2mo ago
          The Erdos problem solutions have been accelerating, including a $250 problem and a $100 problem. It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.
          • HappyPanacea 2mo ago
            This is slow?
          • 2mo ago
• scarmig 2mo 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 2mo 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 2mo ago
      People have definitely not stopped saying this stuff.
      • aurareturn 2mo ago
        I said "generally". It's declined sharply over the last few months.
      • rowbin 2mo ago
        as can be seen in this comment section
      • CamperBob2 2mo ago
        Morons still say it, but then they would, wouldn't they?
    • scarmig 2mo ago
      Some quotes from a day ago, https://news.ycombinator.com/item?id=48957779 https://news.ycombinator.com/item?id=48957779: > 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.
• hliyan 2mo 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. https://en.wikipedia.org/wiki/Jacobian_conjecture#cite_note-vdE1997-4 https://en.wikipedia.org/wiki/Jacobian_conjecture#cite_note-...
  • tptacek 2mo ago
    How does that work in this case? What do those proofs do to help find this counterexample?
    • hliyan 2mo 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 2mo 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 2mo 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.
          • 2mo ago
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• pyuser583 2mo 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!
  • bigyabai 2mo ago
    https://en.wikipedia.org/wiki/Sycophancy_(artificial_intelligence) https://en.wikipedia.org/wiki/Sycophancy_(artificial_intelli...
    • meowface 2mo 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 2mo 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 2mo ago
          gotta try with "think in russian"
        • pyuser583 2mo ago
          The I do that it starts acting like a jerk.
    • krackers 2mo ago
      I mean if someone came to you with a value of n that disproved collatz, wouldn't you go crazy as well?
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• gnarlouse 2mo ago
  Anybody can ELI5?
  • SyzygyRhythm 2mo ago
    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.
    • tptacek 2mo ago
      Nice! (It's gotta be a nonzero constant, right, a nonsingular matrix).
      • SyzygyRhythm 2mo ago
        Yeah, the only possible counterexamples would have a constant but non-zero determinant (in this case it was -2).
    • krackers 2mo 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.
• laichzeit0 2mo ago
  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 2mo 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 2mo 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 2mo 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.
        • c32c33429009ed6 2mo ago
          No, it opens up a whole new suite of questions. Now we can ask, for example: what conditions do we need for the result to hold? What dimensions does it hold in? And many more...
        • pegasus 2mo ago
          I guess the question of why counterexamples are so rare and/or hard to find is still left to be investigated.
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• kmaitreys 2mo ago
  Maybe not? https://en.wikipedia.org/w/index.php?title=Jacobian_conjecture&diff=prev&oldid=1365082008 https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu...
  • Tiberium 2mo ago
    I think the editor themselves misunderstood the conjecture. UPD: The edit got reverted and there's this on the talk page now: https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-DaRealPoopster2-20260720044500-~2026-40618-57-20260720042600 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_conjecture&action=history https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu... https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-SeaHaircutSoilReplace-20260720050600-NOR/RS_dispute https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-Sea...
    • tptacek 2mo ago
      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 2mo 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.
      • jibal 2mo ago
        I removed the "claimed" weasel wording, and now others have followed up. The editors that don't understand math and don't realize how easily this counterexample can be confirmed have lost the debate -- not that there really ever was one.
        • kzrdude 2mo ago
• mathisfun123 2mo 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.
  • ViscountPenguin 2mo 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 2mo 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 2mo ago
        Why is your example SAS triangles and not something like the modularity theorem?
        • mathisfun123 2mo ago
          Because SAS triangle identity proofs are relatable and I'm trying to make a point...?
• jdw64 2mo ago
  I've come to understand that while an LLM is a parrot, it's a parrot that's smarter than I am.
  • sobellian 2mo 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 2mo ago
      "If I have seen further it is by standing on the shoulders of Giants." -Isaac Newton
    • deleted 2mo ago
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  • thinking_cactus 2mo 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 2mo 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 2mo ago
        Same here
  • meowface 2mo 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.
• jdw64 2mo ago
  But I'm curious—can Fable handle cases where n=2 as well?
  • oliculipolicula 2mo 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 :) https://xcancel.com/BrunsJulian1541/status/2079073462560133469#m https://xcancel.com/BrunsJulian1541/status/20790734625601334... https://arxiv.org/abs/2410.06959 https://arxiv.org/abs/2410.06959
    • krackers 2mo ago
      >Wasn't it proven true in general 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)
      • oliculipolicula 2mo ago
        Paper linked above? Edit my bad. I was more familiar with Dixmier and doing the translation I assumed A1 -> n=2. A2 is open ofc. Maybe to fall in 2031
  • QuesnayJr 2mo ago
    This seems like the next step, but the smallest counterexample in n=2 is degree greater than 100. (This is a paper of Moh. Wikipedia has details.)
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• brcmthrowaway 2mo ago
  For the people saying goalposts have moved etc, whats your endgame?
  • deleted 2mo ago
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  • perching_aix 2mo 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?
• randompartytime 2mo ago
  [dead]
• laszlojamf 2mo ago
  Apparently ChatGPT-verified: https://xcancel.com/__alpoge__/status/2079045382940573896#m https://xcancel.com/__alpoge__/status/2079045382940573896#m
  • laszlojamf 2mo ago
    boo on the downvote. The response is pretty funny IMO
• aizk 2mo ago
  I want to see the Collatz Conjecture next!
• aizk 2mo 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.
  • SubiculumCode 2mo 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
  • minimaxir 2mo ago
    Who knew DOES NOT COMPUTE would be an actual thing?
    • left-struck 2mo ago
      The unexpected part is it does compute!
      • deleted 2mo ago
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    • TMWNN 2mo ago
      Captain Kirk defeated multiple evil computers this way
  • CamperBob2 2mo 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."
    • 2mo ago
• SubiculumCode 2mo ago
  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.
  • trentor 2mo ago
    Post hoc ergo propter hoc.
    • SubiculumCode 2mo ago
      Are you sure that this applies?
      • trentor 2mo ago
        Yes, why shouldn't it?
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  • charlieyu1 2mo ago
    I mean of course an employee has cheap tokens
  • Jweb_Guru 2mo ago
    "Anthropic and OpenAI" it's basically been all ChatGPT outside of this one.
  • bcstyle 2mo 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 2mo ago
  Interesting announcement on social media rather than publishing somewhere like arXiv.
  • teravor 2mo ago
    a short counterexample makes it likely the best medium for such an announcement. would have been inappropriate for a positive proof.
  • se4u 2mo 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.
• senderista 2mo ago
  Now that is some spicy autocomplete.
  • baq 2mo ago
    "the Riemann Hypothesis is " <tab>
    • gilgoomesh 2mo ago
      ...not going anywhere.
  • TMWNN 2mo ago
    The day is coming when some *BIG* problem is solved by AI just because someone jokingly asks about it.
• pumuck 2mo ago
  Would be really interesting to see the full conversation and understand how Fable came up with the counterexample in the first place.
• tacomonstrous 2mo 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.
  • QuesnayJr 2mo 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 2mo 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???
• rao-v 2mo 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 2mo ago
    So IIUC no one really worked on that in „professional” or rather university world since then?
  • Someone 2mo ago
    > and yet the counter example is something a grad student in 1997 could have found w a ~3 day computer search Is that true? Even restricting this to f(x,y,z) and coefficients and powers to 1 ≤ x ≤ 10, there are a lot of polynomials to check, and checking requires checking the Jacobian determinant and, if it’s a non zero constant, finding two points for which the polynomial produces the same value. Or is there a way to generate all polynomials with a non-zero Jacobian determinant, and does that speed up things? (My intuition say it wouldn’t, because I guess those with zero determinants are rare)
    • rao-v 2mo ago
      I mean you’d probably just generate random low descriptive length f(x,y,z), check for a const determinant then poke for invertibility. Be fun to ask Fable to write a search program to find more counter examples using only early grad theory to guide the search.
      • Someone 2mo ago
        > I mean you’d probably just generate random low descriptive length f(x,y,z), check for a const determinant then poke for invertibility. That’s true, but for “a ~3 day computer search”, it seems you would have had to be extremely lucky to find a counterexample. Might have had more chance playing the lottery.
  • evenhash 2mo ago
    The search space for a naive brute force of three polynomials of degree <= 7 with integer coefficients <= 12 is roughly 10^500. I think it would take a little longer than that.
• k1t 2mo 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 2mo 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.
    • elbear 2mo ago
      This helps because the ELI5 was too general. To me at least.
  • deleted 2mo ago
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  • QuesnayJr 2mo 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.
  • bananaflag 2mo ago
    Exactly, yes! And the conjecture was for a specific class of processes. And Fable found an example of one concrete* process in that class and three concrete inputs (two were enough of course) giving the same output. *Concrete here means given by a finite string of characters
• Athanase000 2mo ago
  Is there a site or github repository that aggregates all these non-trivial AI-solved results?
• js8 2mo 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?
  • krackers 2mo 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 2mo 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.
  • yorwba 2mo 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 2mo ago
      Ah you're right, thanks.
• sam_lowry_ 2mo ago
  Looking at this poor tweet and only thinking that Telegram has proper LaTeX rendering.
  • amai 2mo ago
    You can have that also on Mastodon: https://mathstodon.xyz/about https://mathstodon.xyz/about
• koito17 2mo ago
  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 2mo ago
    Would this counterexample not be included in their search space?
    • koito17 2mo 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 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 2mo ago
      based on the approach described they were probably in dimension 2, this counterexample is 3 dimensional
      • inigyou 2mo ago
        16 variables
      • deleted 2mo ago
        [deleted]
  • 2mo ago
• dnhkng 2mo ago
  Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
• dnhkng 2mo ago
  Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
• vatsachak 2mo 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.
• Davidzheng 2mo 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 2mo 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 2mo 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 2mo 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.
        • auggierose 2mo ago
          Or we just don't use LEAN but something better.
          • rowanG077 2mo ago
            Does anything truly better exist? I'm not a mathematician but I did use Rocq and Lean during university. And I found lean to be better.
• huflungdung 2mo ago
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• ibarrajo 2mo 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. Here’s my first piece if anybody is interested in number theory: https://arxiv.org/abs/2607.09793 https://arxiv.org/abs/2607.09793
  • deleted 2mo ago
    [deleted]
  • danabramov 2mo ago
    How much do you try to understand while doing it? I.e. how many levels of abstraction down in your own understanding do you go vs vibing at the surface level?
  • igravious 2mo ago
    i'm doing the same thing but for type theory
    • potsandpans 2mo ago
      This is very interesting to me. Care to share your process?
      • igravious 2mo ago
        one agent to write the C code, one agent to write the Agda code, and an agent to bridge the two and make sure that the C code does what the Agda code says it does. https://gitlab.com/igravious/lettuce.git https://gitlab.com/igravious/lettuce.git
        • potsandpans 2mo ago
• mdurana 2mo ago
  Somebody said this on the xcancel thread but this also means that the Dixmier conjecture for the third Weyl algebra is disproven
• tekacs 2mo 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.
• dragonmage65535 2mo 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) https://xcancel.com/__alpoge__/status/2079091571316912542 https://xcancel.com/__alpoge__/status/2079091571316912542
  • taegee 2mo ago
    https://www.wolframalpha.com/input?i=Det%5BD%5B%7B-%281%2Bxy%29%5E2+z+-+y%5E3%281%2Bxy%29%2C+2x%281%2Bxy%29z+%2B+%281%2Bxy%29%5E3+w+%2B+y%5E2%287%2B12xy%2B4x%5E2y%5E2%29%2C+2x%5E2z+%2B+3x%281%2Bxy%29%5E2w+%2B+2y%281%2B10xy%2B6x%5E2y%5E2%29%2C+2x+-+4x%5E2y+-+x%5E3w%7D%2C+%7B%7Bw%2Cx%2Cy%2Cz%7D%7D%5D%5D https://www.wolframalpha.com/input?i=Det%5BD%5B%7B-%281%2Bxy...
• deleted 2mo ago
  [deleted]
• YeGoblynQueenne 2mo 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.
• dragonmage65535 2mo ago
  https://xcancel.com/i/article/2079135211196121363 https://xcancel.com/i/article/2079135211196121363 can any serious mathematicians verify https://xcancel.com/i/article/2079135211196121363 https://xcancel.com/i/article/2079135211196121363
• civilchaos 2mo ago
  Because they've been proven equivalent, so too fall the Poisson Conjecture and the Dixmier Conjecture
  • pedrocr 2mo ago
    That gave me something to play with with these models and this came out as a claimed disproof of the Dixmier Conjecture: https://gist.githubusercontent.com/pedrocr/51157b9b2eed8152e8866d62304f2720/raw/a2f51ecb5d3d028089d64fed920869433a0c4ab6/gistfile1.txt https://gist.githubusercontent.com/pedrocr/51157b9b2eed8152e... It seems mathematics has at least gotten a powerful new tool to automate the work to cascade results after breakthroughs are made.
• claud_ia 2mo ago
  [flagged]
• amai 2mo ago
  See also https://en.wikipedia.org/wiki/Jacobian_conjecture https://en.wikipedia.org/wiki/Jacobian_conjecture
• z7 2mo ago
  > The incredible Yitan Zhang (https://newyorker.com/magazine/2015/02/02/pursuit-beauty 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://x.com/aminkarbasi/status/2079129649830137989 https://en.wikipedia.org/wiki/Yitang_Zhang https://en.wikipedia.org/wiki/Yitang_Zhang
  • Redoubts 2mo ago
    Zhang, Yitang’s life at Purdue (Jan 1985-1991) T.T.Moh - https://www.math.purdue.edu/~ttm/ZhangYt.pdf https://www.math.purdue.edu/~ttm/ZhangYt.pdf kind of a wild document to exist...
    • cma 2mo ago
      Interestingly: "For logic proof, it had been thought that AI could handle all logic problems in the near future, hence logic problems of solving conjectures might not be so interesting in the future."
    • TimorousBestie 2mo ago
      The document says more about the kind of advisor Moh was than the kind of mathematician Zhang is.
    • WhitneyLand 2mo ago
      The world doesn’t need advisors like Moh that are bitter enough to gossip about a family funeral.
      • kjshsh123 2mo ago
        He's not dead, but it is bad to gossip regardless.
        • 2mo ago
• vslira 2mo ago
  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.
• unexpectedtrap 2mo ago
  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.
  • mathgeek 2mo ago
    > I struggle to understand so much hostility to this approach. It's going to be really rough for a lot of folks as machines get better and better at domains that the human brain was exclusively useful for. I tend to view the hostility as a mix of both "unless we have proof this could all be hokum" and "we're going to lose a lot of what we consider makes humans amazing". Compassion is going to be very, very important in the next few years. We're going to hurt, both internally and externally.
    • deleted 2mo ago
      [deleted]
  • jameshart 2mo ago
    Without seeing the Fable trace that produced this, claiming it amounts to ‘advanced brute force’ seems like a leap.
  • grumbelbart2 2mo ago
    The good thing about math is that you can formally verify the results, making hallucinations much less scary. This is similar to games, where large learning models had some of their first big wins (go). Games, like math, have strict rules that can be used to prune invalid outputs. It is already in the making, the next big step in math will be the complete formalization of all existing, relevant theorems and proofs via LLMs.
• Kye 2mo ago
  Discussion on the talk page for the Wikipedia entry: https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#Apparently_disproved https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#Appar...
• narrator 2mo ago
  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.
• exabrial 2mo ago
  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.
  • aurareturn 2mo ago
    I’ve been staggeringly productive with Fable. Opus 4.8 fails a lot more for me. Fable often just “knows” what I want with vague instructions. It also is able to autonomously perform work that lasts an hour long from my experience. I haven’t tested further. Without Fable included in subscriptions, I would have moved my entire team over to Codex 5.6.
• tristanj 2mo ago
  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!
  • ev0lv 2mo ago
    I'm not very excited. Access to the best AI is not a party I was invited to. And the people who are at that party, well, they don't exactly reflect on my best interests.
    • tptacek 2mo ago
      Is mathematics an science or an art? To the extent it's art, it's expressive and rewards the human experience that inspires and that creates it. If it's art, it's drastically less valuable to advance it through automation. But if mathematics is a science, then our entire goal is to increase humanity's understanding of the field. Whether by automation or genius inspiration or as a reward for decades of grinding it out incrementally, it's all the same: knowing more is the point.
  • sailingparrot 2mo ago
    Quite a jump in conclusion you are making here. Sol is able to find the same counter example independently [1], so no reason to conclude in the existence of a benchmark destroying math beast Fable 6. [1]: https://x.com/aaron_lou/status/2079218392452530249 https://x.com/aaron_lou/status/2079218392452530249
    • tristanj 2mo ago
      My Fable 6 theory is admittedly speculative, but your “[public GPT-5.6] Sol is able to find the same counterexample” is also a jump in conclusions. Aaron specifically says he used “an internal version of Codex”. When asked whether that meant a different model or harness, he dodged the question, and only said the harness should be the standard commercial GPT-ultra harness [0]. He (intentionally) avoids identifying which model was used, so your claim is similarly unresolved. Given that Aaron works at OpenAI and has access to internal models, that GPT-5.7 is expected to launch in a few weeks and is rumored to be 10T+ parameters, it's very plausible that Aaron used that model in his analysis. Furthermore, public GPT-5.6 pro failed six times to find a disproof to the Jacobian Conjecture [1], even with hints, which is evidence against the claim that the public GPT-5.6 Sol can solve this. Regarding the existence of Fable 6: an internal upgraded version of Fable or Mythos almost certainly exists, given that Anthropic has been testing Mythos internally since April and previously released new models roughly every ~6 weeks. [0] https://x.com/eliebakouch/status/2079237073001730510 https://x.com/eliebakouch/status/2079237073001730510 [1] https://x.com/Tomodovodoo/status/2079172223055863895 https://x.com/Tomodovodoo/status/2079172223055863895
• kkylin 2mo ago
  Related & of interest: https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/ https://xenaproject.wordpress.com/2026/07/20/human-mathemati...
• owenversteeg 2mo ago
  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.
• typon 2mo ago
  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.
• linzhangrun 2mo ago
  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?
• DrAlexHarrison 2mo ago
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• Fredkin 2mo ago
  Lots of other point collisions too: {{{-1, 3/2, 13/2}, {1, -3/2, 13/2}, {0, 0, -1/4}}, {{0, 7/8, -(3479/1152)}, {48/7, 0, 49/1152}, {-(48/7), 7/16, 539/2304}}, {{-1, 7/8, -(40523/648)}, {160/7, 0, 49/5184}}, {{-2, 7/8, 2989/1296}, {32/21, 0, -(49/192)}}, {{1/63 (-120 - 16 Sqrt[3]), 7/8, 0}, {4/7 (-1 + 3 Sqrt[3]), 0, 49/288 (5 + 3 Sqrt[3])}} }; Next steps would be a simpler counterexample or family of polynomials where the conjecture breaks.
• lemma1729 2mo ago
  I am a number theorist and a graduate math student at Bonn. It is eery to see that all of a sudden everyone cares about pure math. Anyhow, I think such incidents like this one are only to happen more often in the coming days, and while many mathematicians are concerned about their role in the community; I believe since we are still very early stages of AI-driven discovery, we still need the appropriate infrastructure for human-AI research, such as with versioning for proofs, a bigger library than mathlib and a central reasoning space so traces aren't lost- which happens to be the most valuable training data for models, and without which, these discoveries do not help much at advancing the field, and remain as blackbox.
  • treszkai 2mo ago
    > without which, these discoveries do not help much at advancing the field, At least the fact that frontier models are not optimized primarily on formal math reasoning is in itself is good for not putting mathematicians out of their jobs, isn't it? How do these results (and the future painted by them) affect your profession; do you expect a similar route as in software, where junior developers are unemployable, LLM-assisted development is the norm, and that great developers stand out partly through better communication with their managers?
    • lemma1729 2mo ago
      I think in short term, it will get better, as more funding flows from industry to academia, you will see a spur in well-compensated PhD positions. However, in the long run, while I doubt that mathematicians will go extinct, they might have to move on to industries like in trading, chip-making, where they oversee AI models write code and also proof that the code works and is consistent with other parts of the software. However, until we have infrastructure for human-AI math research, as Tao himself said that the "roads" for human-AI research is yet to be built, humans would simply be working FOR AI models, and not working WITH AI, which will truly scale discoveries at a massive scale.
• up2isomorphism 2mo ago
  The results are great. They will still need to post the detailed chat session to show how AI is helping though.
• trevoruptain 2mo ago
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