>The conjecture was first stated for two variables by Ludwig Kraus in 1884[1] and then stated in full generality in 1939 by Ott-Heinrich Keller.[2] It was subsequently widely publicized by Shreeram Abhyankar,[3] as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.
>The Jacobian conjecture was notorious for the large number of published and unpublished proofs that turned out to contain subtle errors.[4][5]
>On July 19, 2026, Anthropic employee and mathematician Levent Alpöge presented an explicit counterexample in three-dimensional space, discovered by Anthropic's large language model Claude Fable 5, which disproves the conjecture for n > 2
If that won't convince you that LLMs do more than parrot existing ideas, you've got your head in the sand.
It is deeply unlikely that a random guy at Anthropic just happened to solve it so they could pass it off as the LLM's work.
It doesn't.
In a nearby comment: https://news.ycombinator.com/item?id=48983413
> In mathematics (including information science, CS) there are all sorts of problems that are essentially searches for a solution, and many have the property that the search is computationally difficult, but verifying the solution is relatively cheap. E.g. finding integers such that a^2 + b^2 = c^2 isn't easy, but given a claim that some proposed <a, b, c> satisfies this equation is easy to check. The LLM is like that: it solves a search problem that can be fairly hard.
Funnily enough, one of the attempted solves in the litterature is exploring the problem space in two-dimensional space; the LLM found one in three-dimensional space.
So far we know very little as to why and how it found the solution.
It may very well have been directed to brute force 3D space, or even "elected" to" by expanding the known-failed 2D approach to 3D as pure mimicry.
> It does so unreliably, but if you can cheaply verify the solution, there is a win there.
This circles back to what the LLM advocates are pushing for: build the harness that keeps the agent in check, guardrails all the way because it's driving like a demolition derby.
You are desperately searching for ways to excuse it, to explain how it can't be what it obviously is.
Oh the irony...
I wrote in the opencode thread that when it came out I put it behind a vm and its own user, and I never allowed it to run outside of it. But I know of people that as soon as they noticed that it worked well like 99% of the time, they let their guard down and give in to YOLO mode. And in orgs I've even seen CEOs treat their agents less like a user/employee/contractor, and try to 'empower' it by giving it ALL the data. Time bomb.
It's like fucking with condoms just the first couple of times. And then simultaneously ditching it and joining the free love movement.
And then during inference the light goes out and the "agent" staggers randomly like a zombie along those preset paths. Stochastic parrot.
So you and your AGENT.md and your skills files and your harnesses will never make your Claude perceive something that is not in its model checkpoint.
ML experts and neurobiologists free to correct me.