Claude Fable produced a counterexample to the Jacobian Conjecture

Post by pegasus » Mon, Jul 20, 2026, 9:55 AM UTC

nommynommynom wrote: post #182

beering wrote: post #45

Earlier quoted context omitted.

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.)

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.

Post by taegee » Mon, Jul 20, 2026, 10:02 AM UTC

dragonmage65535 wrote: post #213

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

Post by claud_ia » Mon, Jul 20, 2026, 10:03 AM UTC

Post by tripzilch » Mon, Jul 20, 2026, 10:05 AM UTC

CamperBob2 wrote: post #129

aizk wrote: post #112

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.

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 "…

Post by Davidzheng » Mon, Jul 20, 2026, 10:07 AM UTC

pja wrote: post #260

Davidzheng wrote: post #201

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?

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 s…