AI disproves the Jacobian conjecture! Counterexample posted on twitter, LOL

[00:00:00]Yet another victory for AI against mathematics.

[00:00:05]Now we have a report that Anthropic's Fable has found a counterexample to the Jacobian conjecture and it's posted on Twitter in the on Twitter in the most irreverent format. Here's this grinning guy with a picture in the background, a picture of Alan Turing if you really squint and he says, "Hello there. The Jacobian conjecture is false thanks to my close friend Akhil for asking about it and my other close friend Fable, that's an AI, that's not really his friend, for working during the World Cup final."

[00:00:42]The >> [clears throat] >> function which takes XYZ in the complex fields and sends it to the Sorry, it takes X, Y, Z in complex three space and sends it to another complex three point namely to one plus XY quantity cubed Z plus Y squared times one plus XY all times four plus three XY in the first component and in the second component Y plus three X times the square of one plus XY Z plus three XY squared times four plus three XY.

[00:01:20]And in the third component two X minus three X squared Y minus X cubed Z.

[00:01:25]That function has Jacobian determinant minus two and sends three different points all to the same point namely it sends zero, zero, minus one fourth and one, minus three halves, thirteen halves and minus one, three halves, thirteen halves all to the same point minus one fourth, zero, zero.

[00:01:47]And he posts a link to Wolfram Alpha to verify the calculation of the determinant and so on.

[00:01:55]>> [clears throat] >> And it's more links to Wolfram Alpha.

[00:01:59]Anton replies saying, "Thank you, blessed math fairy."

[00:02:05]Internet poster replies saying, "Investing at zero likes in case this is true."

[00:02:11]Levent replies to that saying, "This is the most hilarious {slash} horrifying problem because it is like the canonical crank graveyard. Thank you for investment."

[00:02:21]Lil Hardy, with a picture of G.H. Hardy as his avatar, says, "Is this the same as in Iitaka-Johns conjecture or or thesis?"

[00:02:32]And links to some thesis.

[00:02:34]And Levent replies saying, "Yo, he is so incredible. I was at his first talk on small gaps. The story was that he relied on a result which had a gap, right?" I'm not sure what the reference there is talking about.

[00:02:48]Yeah.

[00:02:49]Very interesting.

[00:02:50]Let's see what hackers are saying. Oh, this is interesting. Julian Burns Julian Burns says, "Maybe a stupid question, but doesn't this also imply that Dixmier conjecture and Poisson conjecture are false in general?" And he posts a screenshot from Wikipedia which says, "It is well-known that the Dixmier conjecture, that every endomorphism of a file algebra is an automorphism, implies the Jacobian conjecture." Well, then that means the Dixmier conjecture is also falsified.

[00:03:18]Conversely, the Wikipedia article continues, "It is shown by Yoshifumi Tsuchimoto and independently by Alexei Belov-Kanel and Maxim Kontsevich that the Jacobian conjecture for two began variables implies the Dixmier conjecture in began dimensions."

[00:03:36]Oh, well, since the Jacobian conjecture is falsified, the implication here doesn't have any implication at all.

[00:03:44]A self-contained and purely algebraic proof of the last implication is also given by Kossev Adjamagbo and Arno van den Essen who also proved in the same paper that these two conjectures are equivalent to the Poisson conjecture that every endomorphism of the big end complex Poisson algebra is an automorphism.

[00:04:04]Uh which two conjectures?

[00:04:06]The Dixmier conjecture and Jacobian conjecture? It's a little ambiguous because we hear it when it says uh that these two conjectures are equivalent to the Poisson conjecture.

[00:04:20]Now, if either of those if either of these two conjectures includes the Jacobian conjecture or the Dixmier conjecture, then I guess that means that the Poisson conjecture is also falsified. Yes.

[00:04:34]>> [clears throat] >> Dominos are falling. Somebody posts a picture of Ramanujan and under it says, "It was revealed to me in a dream." They cross out the word dream and replace it with chat session.

[00:04:46]Yeah, maybe maybe that goddess who Ramanujan claimed to commune with was actually just an AI that time traveled or something. And Aldo Cortesi posted that picture and said the absolute state of maths. Yeah, if I was employed full-time to prove um random conjectures or something, I'd be worried about my job security. But nobody's actually employed in that capacity, so those fears are a little overblown. It does, however, really emphasize that math is changing. We are at a historic moment.

[00:05:21]It is never going to be the same.

[00:05:26]More [clears throat] memes posted. Let's see what hackers are saying about this over at Hacker News.

[00:05:31]The Hacker News headline is Claude Fable produced a counterexample to the Jacobian conjecture.

[00:05:38]At the top level, AIZK says, "This is a rare instance where feeding this groundbreaking information into an AI gives them, the AI, psychosis. I fed this to Claude Court and watched it verify the result in seven different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.

[00:05:57]>> [clears throat] >> Um yeah.

[00:06:00]One freedom says, "This is also just an annoying cloud personality trait."

[00:06:05]Minimaxer 59 says, "Who knew does not compute would be an actual thing?" Yeah, so what what I what I ZK is saying is that he took the counterexample plot and fed it into an AI and said, "Hey, is this really a counterexample of the Jacobian conjecture?" And the AI uh being programmed to emulate human personality and so forth is appropriately quite astonished. Like, "What the heck? It seems to really be a counterexample. I can't believe it."

[00:06:37]>> [clears throat] >> Monokassa says, "I've heard about mathematicians going through some 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.

[00:06:56]Or in the case of themselves, there's a certain blindness that makes it a little more difficult to critically evaluate your own leaps.

[00:07:02]In either case, the way it manifests is by going over it many times in many ways, each time more certain that you missed something until you just kind of break.

[00:07:10]Only then do you publicly start suggesting that there might be something to this new leap.

[00:07:15]Speculum code says, "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."

[00:07:26]Uh it's kind of like P versus NP. The thing is, the solution is indeed quite simple to verify once you have it, but it's quite a complicated solution in the sense that it's totally random and not simple at all.

[00:07:41]You know, if we if we look at the solution, it's this super complicated I mean I mean complicated is relative.

[00:07:50]It's it's not infinitely complicated, but you certainly wouldn't randomly guess this on your own this 1 + XY quantity cubed * Z + etc. I mean who on Earth would guess that?

[00:08:04]Uh yeah, anyway.

[00:08:07]Uh here's a public share.

[00:08:10]Uh of Kelsey having the same result. He says it did the multiple verification sequence before expanding to internet search where it found this thread.

[00:08:19]Uh the AI says, "Okay, so I checked this properly before saying anything because casual chat message disproving an 85-year-old conjecture has an extremely bad base rate. Here's what I got." This is the AI speaking.

[00:08:32]And and the people on Hacker News are reporting that the AI gets flabbergasted.

[00:08:38]The math checks out, all of it. I verified symbolically by hand and then with SymPy that the Jacobian determinant is exactly minus two. The cancellation is beautiful incidentally. The map has hidden structure writing U = 1 + XY and big G = U squared Z + Y squared times 4 + 3 XY. The first two components are big F1 = U big G and big F2 = Y + 3 X big G.

[00:08:59]Oh, that's a nice simplification which gives the polynomial identity U big F2 minus 3 X big F1 = UY and that's what makes the determinant telescope down to a constant. All three point evaluations are correct. I then went further and computed a Grobner basis >> [laughter] >> for the fiber over the triple minus 1/4 {comma} 0 {comma} 0.

[00:09:18]The Grobner basis is spanned by 4 Z + 1 minus 27 X squared {comma} 2 Y + 3 X {comma} X times X minus 1 quantity X + 1. So, the fiber is exactly your three points and for a generic target the eliminant is an irreducible cubic. The map is an honest three to one a tall cover of its image. There's even a cute symmetry. The map is equivariant under sending XYZ to -X -Y Z on the source and big X big Y big Z to big X big - big Y - big Z on the target, which is why two of the fiber points come as a plus or minus pair over a point with big Y equals big Z equals zero. It's consistent with the Zariski's main theorem two, degree three, not by rational, etc. etc. etc. >> [clears throat] >> The less good news for your prior D claim. You found this the same way I suspect a lot of people found it this morning. Levent Alpoge posted exactly this map a few hours ago. Hello there, the Jacobian conjecture is false thanks to my close friend Akil for asking about it and my other close friend Fabel for working during the World Cup final.

[00:10:22]Verbatim down to complex three space to complex three space is currently on the Hacker News front page, where someone already added it to the Wikipedia article citing the tweet, etc. etc. So, yeah, um Claude is able to search the internet and notice that this has already been found.

[00:10:44]>> [clears throat] >> Minimaxer replies to Kelsey Frog saying, "So, the conjecture that survived Keller, Abhyankar, Morse degree 100 verification, and five plus published proofs or published wrong proofs appears to have died via tweet during the World Cup final."

[00:10:59]And RVZ says, "And you guessed it, before Grand Theft Auto VI."

[00:11:04]>> [clears throat] >> Moth Colossus says, "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."

[00:11:15]Laughing my ass off.

[00:11:17]Camper Bob says, "Interestingly, even Claude 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.

[00:11:34]GLM 5.2 whiffed. It insisted the counterexample wasn't valid. Vibe Thinker 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 minus two. Oh, this is hilarious. So, they're talking about some less powerful AIs which did not uh get things right. Quinn 3.6 27B hallucinates that the counterexample was already counterexampled in 2004 by Shasticov, a stereotypical Russian mathematician.

[00:12:10]Perhaps influenced by the famous story of the Poincaré conjecture.

[00:12:15]Uh GLM 5.2, which is actually a fairly sophisticated AI, pretty close to l- l- like only one generation from cutting edge, insisted that the counterexample wasn't valid. This d- proves what we were talking about yesterday that there's a danger of AI reinforcing orthodoxy. The orthodoxy is that the Jacobian conjecture is open. Therefore, if somebody says they've found a simple counterexample, they must be wrong.

[00:12:45]That's the orthodoxy being enforced.

[00:12:48]Vibe Thinker 3B, I've never heard of that AI before, recognized that the counterexample was valid, but kept trying to convince itself that it wasn't over and over since it's an unsolved problem. This is hilarious.

[00:13:00]Ahem. [clears throat] At the top level, K1T says, "Since I actually don't know math, maybe my explain like I'm five 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."

[00:13:18]Dan Burke replies saying, "There is a precondition, constant nonzero Jacobian, to the inverse existing, and the inverse is claimed to be of a specific kind, polynomial. The counterexample satisfies the precondition and by mapping two different inputs to the same output makes any inverse impossible, including polynomial ones.

[00:13:36]But maybe that is already explained like comment seven.

[00:13:42]Uh at the top level Rao V says, "This is so unreasonable. As Al Pouget himself notes, this is classic crank graveyard territory and yet the counterexample is something a grad student in 1997 could have found with a 3-day computer search.

[00:13:55]Wild." This crank graveyard mentioned that people keep talking about it it's referring to the fact that lots of people tried to publish proofs of this conjecture and the proofs have subtle flaws in them. Well, maybe now we'll see a we'll see those proof attempts stopping since a simple counterexample has been found. At the top level Pumukl says, "Would be really interesting to see the full conversation and understanding how Fabel came up with the counterexample in the first place."

[00:14:25]Like site zero says, "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."

[00:14:34]It would be great if an AI could settle the Collatz conjecture next. God knows how many man-years have been burned on that by unsuspecting victims.

[00:14:42]Yes, you know, many people think of the Riemann hypothesis as being the ultimate conjecture, but in my humble opinion, I think the Collatz conjecture is actually much harder than the Riemann hypothesis.

[00:14:52]The Collatz conjecture is also known as the 3n + 1 conjecture.

[00:14:56]And yeah, if the Collatz conjecture falls to AI, oh my goodness, in that case, strap yourself in because we're at the threshold of the stargate and we're about to wake up in a weird hotel at the end of 2001 Space Odyssey.

[00:15:11]Free Bib replies to like site zero saying, "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'll be much harder to prove than your average problem. Yes, indeed.

[00:15:25]Bering replies saying, "I think parents 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."

[00:15:46]>> [clears throat] >> K Maitreya says, "Maybe not." And links to a Wikipedia diff. Let's see what that's about.

[00:15:57]Um somebody modified Wikipedia saying, "Removed false claim of assertion. The map is not proper. For map Y = -1/X, Z = 5/X squared big F of XYZ = 0, 2/X, 0. So, a sequence is giving to infinity in the domain has images converging to 0 0 0. By Hadamard's criterion, this failure of properness allows a map non-zero constant Jacobian to fail global injectivity. So, no contradiction with the Jacobian conjecture is established."

[00:16:29]Let's look at the um Wikipedia page and see what it says right now. Well, let's see what people replied here.

[00:16:40]>> [clears throat] >> Tiberium says, "I think the editor themselves misunderstood the conjecture."

[00:16:47]Uh the edit got reverted and there's this on the talk page now.

[00:16:52]If we follow the link, it says, "Hadamard gives the sufficient implication proper local diffeomorphism implies global diffeomorphism. Failure of properness merely makes Hadamard inapplicable. Properness is not a hypothesis of the Jacobian conjecture."

[00:17:05]"Since the displayed polynomial map has constant Jacobian determinant minus two and maps three distinct points to the same point, it directly contradicts the conjecture. Its properness is required for Its non-properness is required for such a counterexample. Okay, so Hadamard gave a sufficient condition uh but the counterexample fails that sufficient condition, but that's fine because Hadamard did not give a sufficient and necessary condition. And that's the Wikipedia editor failed to understand that.

[00:17:33]And apparently there are edit wars happening right now. He links to something uh oh boy, a Wikipedia diff log. Yeah, I'm not going to read through all that.

[00:17:45]>> [clears throat] >> TPCSATex says Wikipedia's going to Wikipedia. Unless there's a material debate over the Jacobian conjecture itself, there's really no open question here.

[00:17:53]This isn't a complicated proof. It's a straightforwardly checkable certificate of a solution. Q star says, "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."

[00:18:07]Vertue says, "I agree the conjecture never says anything about a proper map."

[00:18:11]Whimsicalum says, "Wikipedia, as with other left-leaning sites, struggles with impartiality regarding AI's. The counterexample is straightforward." Oh, that's an interesting aspect I never thought about. Certain political groups have an anti-AI slant and so will motive be motivated to rage against these sort of breakthroughs.

[00:18:32]Let's [clears throat] go back to the top level.

[00:18:36]At the top level, JS8 says, "I find it interesting that the counterexample uses the complex field as field. The complex field is twisted and weird. Maybe the Jacobian conjecture still holds for reals." Crackers replies saying "Why do you say the complex field is twisted and weird? I've admittedly never done a proper complex analysis course, but I got the impression that the complex differentiability was a very strong condition that results in holomorphic functions behaving nicely in ways that real functions do not."

[00:19:01]Yeah, I agree. The complex field only seems twisted and weird if you're approaching it from like the eyes of a newbie, but really it's quite well-behaved. At the top level Subiculum says for all the hubbub as far as I know all the math breakthroughs via AI that I've heard about have come from Anthropic and Open AI, not Chinese models. I could have missed those announcements, but one might think that between the close to frontier performance plus cheap tokens that they'd be leading the way on these things. That's a good point. Bstyles BCstyles says as with the case with AI and in fact a lot of things in life sometimes you just need to wait. It'll happen rather soon.

[00:19:38]Jared Gurus says Anthropic and Open AI has basically been all chat GPT this one.

[00:19:44]Charlie You says I mean of course an employee has cheap tokens.

[00:19:48]Trencher says post hoc ergo propter hoc.

[00:19:50]Subiculum said are you sure this applies? Trencher says yes why shouldn't it?

[00:19:55]Okay what does this mean? I don't speak Latin.

[00:19:59]Oh yeah, let's skip it. At the top level Well actually I need to get ready to move out of this Airbnb so I'm going to stop there but the dominoes are falling. The double cover conjecture was not a one-off. We are seeing mathematics in a true state of revolution right now. This is super exciting. I am simply thrilled to be alive at this moment in time. This is absolutely the most amazing time you could possibly imagine in mathematics. Wow. I wish that I could bring back people like Euler and Eratosthenes and see what their reactions would be to this whole state of affairs. Simply amazing. I'm Dr. Samuel Allen Alexander mathematician and philosopher. Please like, comment, and subscribe and I'll see you in the next video.