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- Hacker News
- 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!by rao-v
- So IIUC no one really worked on that in „professional” or rather university world since then?by misiek08
- > 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)
by Someone - 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.by evenhash
- by kmaitreys
- I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online.
Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.
by rvz - The author has a PhD in math from Cambridge. If it turns out to be a false claim it is an interesting case study on AI's sycophancy causing even experts to drop their guard and make mistakes.by slopinthebag
- This is nonsensical: Properness of the map is equivalent to its being an isomorphism (quick proof: Jacobian invertible implies that the map is etale, and properness would imply that it is finite etale, but affine space doesn't admit non-trivial finite etale covers), so the lack of properness is just another way of verifying that this is indeed a counterexample.
- 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-DaR...
UPD2: There are edit wars happening now: https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu... https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture#c-Sea...
by Tiberium - Since I actually don't know math, maybe my ELI5 understanding can be helpful (or corrected).
The conjecture says that you can always reverse (a process) to determine the original inputs.
But this proof shows multiple inputs creating the same output - which obviously cannot be reversed to determine the input - thus falsifying the conjecture.
by k1t - 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
by bananaflag - It was known to be false in general. But there are many questions that are false in general, but true when restricted to polynomials.by QuesnayJr
- [...] 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.
by danbruc - 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?by Davidzheng
- 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
by ibarrajo - Some speculation in this Claude chat: https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0
linked from here: https://x.com/b_shrir/status/2079094004885668003?s=20
Very short version: there’s an existing false counterexample in the literature which holds almost everywhere except at a pole. It looks like Fable used this polynomial as a base & extended it in a way that eliminated the pole whilst preserving the structure.
by pja - 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.
by tekacs - The poster works at Anthropic, so they likely have internal access to the next generation of Fable. Their internal model is probably an absolute beast at mathematics, and the upcoming benchmark results will likely set a new record for maths performance.
I suspect this is what happened, because the poster is coy about sharing the actual prompt / reasoning trace used to reach this result. That would be covered by an NDA until the model is properly released.
Exciting times!
by tristanj - 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.by ev0lv
- 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.
- Using LLMs to generate piles of code and/or proofs of dubious quality is very questionable thing, and I understand these non-stop debates about it.
But in this case, as using plain brute force is already quite a common thing in searching for counterexamples, using LLMs as a sort of more advanced brute force seems to be just the right thing to do, so I struggle to understand so much hostility to this approach.
- The 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.
by grumbelbart2 - Without seeing the Fable trace that produced this, claiming it amounts to ‘advanced brute force’ seems like a leap.by jameshart
- > 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.
by mathgeek - I’ve been math-vibe coding a few months now.
It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing.
Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them with a ton of scrutiny. And verification has taken more time than it did to make the discoveries.
Here’s my first piece if anybody is interested in number theory: https://arxiv.org/abs/2607.09793
by ibarrajo - i'm doing the same thing but for type theoryby igravious
- 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?by danabramov
- I'm a phyicist by training and have published in the past but I left academia and so haven't for a number of years now.
I tried out using Claude to do some physics problem solving - mix of maths and simulation - and ended up with it getting in quite a mess. It's incredible at setting things up, suggesting approaches you might not have considered. I found it much much worse at interpreting things.
by physicsguy - 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.
by laichzeit0 - I don't think this is how this works. The next step is to determine for what kinds of polynomials the jacobian conjecture is true and for what kinds of polynomials it's false.by IsTom