A digestion of the Jacobian conjecture counterexample

https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/

314 98 1 day ago

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tptacek4• 1 day ago

The introduction to this piece was easy to follow, but as soon as he got into recapitulating it with algebra he lost me (because I'm bad at math). But he includes the GPT5 prompts for his conversation, which are easier to follow:

https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...

minimaxir• 24 hours ago

Also note the timestamp: he started working on this thread a few hours after the tweet.

CamperBob2• 21 hours ago

I like how he goes one-on-one with it like Gandalf fighting Yoda on fifteen planes at once for 80% of the transcript, and then we read "OK, I've activated Pro."

ignoramous2• 19 hours ago

I'm bad at math ... he includes the GPT5 prompts for his conversation, which are easier to follow

You were kidding, right?

brookst• 18 hours ago

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism {X \cong {\bf C}^3} and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian.

I mean, clearly, right?

You and me both, pal.

ChrisArchitect• 24 hours ago

Claude Fable produced a counterexample to the Jacobian Conjecture

https://news.ycombinator.com/item?id=48973869

Human mathematicians are being outcounterexampled

https://news.ycombinator.com/item?id=48983382

hyperhello• 24 hours ago

Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?

sfpotter2• 23 hours ago

No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.

imtringued2• 12 hours ago

It's not immediately intuitive what it means for something to be globally and locally invertible. After all, it is obvious that it is both in the 1D case.

You can get the inverse of the Jacobian at any point, but you cannot describe the inverse of the Jacobian through a polynomial, which is a function. You need a more complex object to describe the inverse, because the global inverse is not a function due to the potential of overlapping values.

mswphd• 22 hours ago

it doesn't overturn much. For example, here is a post from 2004

https://www.math.columbia.edu/~woit/wordpress/?p=105

it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open.

Anyway, in that post it says

It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables *(for more variables, many people believe it is not even true)*

so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.

monster_truck2• 22 hours ago

Not much.

But it does give credible plausibility to the concept that we might be mistaken about the exact boundaries of hardness for adjacent (but not equivalent) polynomial systems. Most (all?) of which have also stood up to a whole lot of undeniably sharp people poking at them for about as long.

impendia2• 21 hours ago

For the Jacobian determinant to be constant is a massive coincidence, in general it is some complicated and messy polynomial. The conjecture was that this coincidence couldn't happen, except for simple special cases.

So Alpoge and Fable found an example of a function that was believed to be too strange to exist.

jibal• 20 hours ago

It overturns the Jacobian conjecture (i.e., speculation) for dim >= 3, which we now know was an overgeneralization. Tao characterizes it as "can be viewed as an assertion that local invertibility implies global invertibility". It was already widely suspected to be false. Assuming that it was true was never warranted, so this really doesn't change anything. The significance is that an AI was able to find a relatively simple counterexample. Its "chain of thought" would be very interesting to see.

kingstnap• 20 hours ago

I'm not a mathematian but I know enough linear algebra and vector calculus to understand the conjecture. This is my interpretation:

Firstly, the determinant of the Jacobian is measuring if at any point the function is crushing space / flattening out.

If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. A small change in X along any line will always produce a non zero change in Y. Not flattening out means that locally you can invert it.

What was conjectured is that this local invertibability property everywhere would mean global invertibility.

Turns out to not be the case.

For a simple case, the falsified conjecture is trivially true in 1D.

Specifically consider f(x) = x^2

This function happens to flatten out right at x=0. At that x coodrinate the function flattens out and folds over on itself. This fold means you can't invert x^2. It's also not locally invertible around x=0.

If a function f(x) has constant derivative evewhere then it would flatten out nowhere and it would be invertible everwhere. It would also be globally invertible.

The Jacobian conjecture was stating that the extension of this property holds in higher dimensions. That if the function had no fold in space then it would be invertible globally.

The counterexample shows that you can create a simple function in 3 variables, where the function demonstratably is invertible evewhere, but is not injective globally (they specifically show 3 points that map to the same output).

What's interesting is this is like if someone showed you a parabola where somehow you got back to the same y coordinate without a kink bending over back to itself.

efavdb• 19 hours ago

Thanks, I was curious for the motivation behind the conjecture, but it wasn’t mentioned in the Wikipedia article.

ignoramous2• 18 hours ago

Sounds like 3 more iterations on K3/Fable, and we'll able to ask it to break computational hardness assumptions like DLP?

QuesnayJr• 10 hours ago

There was no particular reason to think it was true. It's easy to find examples using exponentials or trig functions where it's not true. But it would be neat if it was true, and nobody found an example where it wasn't true in 75 years, so it was tempting...

It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.

vanderZwan• 23 hours ago

While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{3}-1 = 1329} coefficients, which is much larger than the {\binom{7+3}{3} = 120} degrees of freedom for a generic degree seven polynomial of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.

Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.

j2kun3• 20 hours ago

I was reading another source that claimed this example was inspired by an existing (rational polynomial) example from the literature (created in 1999 by a Russian mathematician Vitushkin).

The seed is almost certainly Vitushkin's old rational "counterexample."

From https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0

This is not quite lore laundering, but it seems to be close.

gus_massa3• 19 hours ago

I still can't understand all the details, but it's very interesting to read that chat. Anyway, instead of close to lore laundering, for me it's "standing on the shoulder of giants".

marvinborner2• 14 hours ago

I hate that Anthropic seemingly tries to make Claude act as if it was conscious or had feelings

It's a strange feeling to admire the cleverness of something I did and can't remember doing.

pred_• 13 hours ago

I guess we won't know if that's what was used (and maybe even provided as part of the prompt given that both Alpöge and Mathew are mathematicians) since they decided against sharing their Fable conversation and instead opted for a memey tweet as their avenue of publication. We really ought to normalize full transparency in how results come about.

Anyway, if I read Tao's post and comment correctly, there's still a gap from the Vitushkin construction to a counterexample, but chances are that was in the training data. In general, it is just a serious problem for their practical applicability that the models are outputting proofs with absolutely terribly reference hygiene.

zzzeek• 23 hours ago

reading through this I eventually realized a situation similar to my experience of it is what my dog sees if I attempt to explain Python programming to him.

tgrowazay• 22 hours ago

Some people downvoting you, but I think it is a valuable illustration of IQ gap.

And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.

left-struck• 20 hours ago

I really don’t think IQ has much to do with it. Understanding this stuff is like a skill you practice. Yes, granted, if you had a low IQ your chances of ever understanding it goes down, if you have a high IQ maybe you can gain the prerequisite understanding faster.

A lot of maths is about both being able to wrap your head around hard problems and gaining the prerequisite knowledge to make it easier to do so.

richard_chase• 20 hours ago

The difference is your dog will never understand the Python code but you could probably understand this post in a matter of days or weeks if you really wanted to. Can we all please stop acting like this Terry guy is so special?

tclancy• 20 hours ago

Goats though, they get it. Bang your head against a monitor until things start working.

bugufu8f832• 20 hours ago

I certainly do not think that the average HN reader "could probably understand this post in a matter of days or weeks". I think the level of background information you need is something like an undergraduate degree in mathematics and at many universities that's probably not enough either.

jrflo• 19 hours ago

I'm not sure if you're taking the piss or genuinely don't know who he is

fragmede• 16 hours ago

I'm not dumb. I'm relatively smart. I am, however, smart enough to see that people like Terry or Ramanujan are actually that special.

pertymcpert2• 13 hours ago

Yeah this Tao guy is just your average scrub. You would be able to tell the difference, right richard_chase?

drivebyhooting• 23 hours ago

Can we audit the CoT and work the AI did to generate such a remarkable cancellation?

castedo3• 22 hours ago

I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing.

I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.

p-e-w• 22 hours ago

a human prompting with deep math expertise

The original tweet implied that the whole thing was done while the author was watching the World Cup final.

I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.

jasonfarnon• 22 hours ago

Not just an assumption, I saw the LLM saying it used sympy.

tptacek4• 22 hours ago

I think you can reasonably assume that frontier models are using SymPy or something like it any time interesting math gets into the picture, and the person driving Fable here is an accomplished mathematician, but I don't think we can reasonably assume either extensive prompting or brute-force compute in any sense other than what it normally takes Fable to, say, whip up a calculator app.

impendia2• 21 hours ago

I have no idea what actually happened behind the scenes, but the human prompter, Levent Alpöge, indeed has deep math expertise. Princeton PhD, Harvard postdoc, and some excellent research (prior to this) to his name.

https://alpo.ge/

vatsachak• 19 hours ago

Yeah more or less. I proved a SOTA result using Gemini 3.1 Pro a year ago and it was a lot of back and forth.

We're definitely still in the computer chess phase.

netvarun• 22 hours ago

https://news.ycombinator.com/item?id=48986943

jmward01• 22 hours ago

Finding a different way of thinking about a problem often leads to a breakthrough. This is what an ecosystem in nature shows us, that diversity matters in finding hard solutions. I think the great thing here is we are getting a chance to find whole new ways of thinking about problems that were hard. I suspect many old problems will fall because of it and, hopefully, some really new interesting ones will replace them.

gerdesj• 21 hours ago

"problems that were hard"

They are still hard problems - As we say in the UK: "one swallow does not a summer make".

As you well know: birds are not renowned for their arithmetic skills, nor eating encourages the weather!

aayushdutt• 22 hours ago

After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?

clarionbell• 14 hours ago

Not really. I've found that they often believe that they understand the code. They obviously don't. But they do feel like they do.

Sharlin• 14 hours ago

Clearly we use too many natural-language words in programming. Should switch to APL so that the commoners have absolutely no idea what's going on.

lostmsu2• 7 hours ago

This is what I used to feel reading ML papers. Until I didn't anymore. Unstructured learning works, albeit, perhaps, slower.

sashank_1509• 16 hours ago

I don’t understand math but it was amusing seeing Terrence Tao’s chat with chatGPT. Everything Tao said was constantly followed by praise: “That’s exactly the right way to think about it.”,

“Yes, you are exactly right.”

“You have gotten to the core issue.”

And non stop praise. Seems like sycophancy is still an issue lol.

welferkj• 16 hours ago

If there's anyone that deserves praise every time he has something to say about mathematics, it's him.

moralestapia2• 16 hours ago

Tired of this meme, honestly.

square_usual• 7 hours ago

Huh? I went through the first ten messages in his thread and there's no praise there at all.

dumpstate• 15 hours ago

What’s a chance the counterexample was in the training?

TeriyakiBomb• 15 hours ago

Extremely high. Or at least several partial solutions that can be smooshed together.

LLMs really do still just reassemble things in their training data. There’s just a lot of it now, people anthropomorphise and struggle visualising large things. Some people say it’s truly reasoning but hit a topic that is under represented in the data of any LLM and it’ll transport you very quickly back a couple of years and ruin the illusion quickly.

3form• 13 hours ago

The problem being that I don't think there's a definite proof that any of human thinking is more than a sum of high-granularity partial solutions that can be put together.

It could be that with enough tokens, big enough context window, and ability to dig out the relevant partials, many such thought processes could be simulated.

pertymcpert2• 13 hours ago

I think you’re just speculating.

zeusdclxvi• 15 hours ago

I'd wager close to 0%

BurdensomeCount• 14 hours ago

Close to impossible. This is a famous enough problem that anyone who understands what they are doing generally would pretty immediately recognise the significance of the counterexample if shown it.

imtringued2• 12 hours ago

The best part is that we can't know the answer to that.

The necessary precursors to the counter example where definitively in the training set, otherwise the LLM wouldn't know how math works, but at the same time, we can't tell whether there were mathematicians who got 90% of the way, then gave up and the LLM just did the last 10%.

emp173442• 10 hours ago

Anthropic could audit the model to find the answer. It’s telling that they won’t do this.

gus_massa3• 8 hours ago

From a comment by j2kun https://news.ycombinator.com/item?id=49000833 , someone asked Fable and there was an almost counterexample in 2d but it uses division too. [Instead of f=x^2+7xy they have something like f=x^2+7x/y so it's not a polynomial.] As far as I know, nobody know what trick to make to avoid that division. It looks like the new trick was to use a third variable to avoid the division. Note that the implementation of the trick is not straightforward. The almost counterexample was sitting around for almost 30 years, and nobody knew how to fix it.

From another old comment, someone else was trying to find a counterexample with 16 variables using a computer to make thousands of attempts and failed. So it's far from obvious that the trick to add a variable solves the problems.

jeremyscanvic2• 14 hours ago

Also, from the fundamental theorem of algebra, once the Jacobian polynomial {\mathrm{det} DF} is non-zero, it must be constant.

I wouldn't have guessed this is true. I'm wondering what the proof looks like!

amluto• 14 hours ago

I’m fairly confident that the blog post is trying to say something like this:

Given a polynomial function from C^n to C^n, the following statements are equivalent: (a) det DF is nonzero everywhere. (b) det DF = c for some constant c != 0

The backward direction (b implies a) is trivial. The forward direction can be proven by observing that det DF is itself a polynomial function from C^n to C. If n were 1, then this would follow directly from the fundamental theorem of algebra: a non constant polynomial has degree at least 1 and hence has at least one zero. Extending this logic to higher dimension is not especially difficult.

I do find the way it’s stated in the article to be confusing.

jeremyscanvic2• 12 hours ago

Ah right this makes a lot of a sense - thanks!

A digestion of the Jacobian conjecture counterexample

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