AI Model 'Claude Fable' Produces Counterexample Disproving 85-Year-Old Jacobian Conjecture
- Mathematician Levent Alpoge posted that Anthropic's Claude Fable model produced a polynomial counterexample disproving the Jacobian Conjecture, an open problem in algebraic geometry that had stood for 85 years.
- The counterexample is an explicit map from \C^3 to \C^3 with constant Jacobian determinant -2, yet three distinct points, (0,0,-1/4), (1,-3/2,13/2), and (-1,3/2,13/2), all map to the same output (-1/4,0,0), proving the map has no polynomial inverse.
- The claim was verified independently with Wolfram Alpha and later checked by a Hacker News commenter using GPT-5.6, which wrote SymPy code and confirmed no algebraic error, calling for serious independent verification.
- Hacker News commenters noted the counterexample is surprisingly low-degree with small coefficients, raising the question of why brute-force search had not found it earlier despite decades of attempts to prove the conjecture true.
- The announcement triggered active edit wars on the Jacobian Conjecture Wikipedia page, and commenters flagged the episode as a possible case study in AI sycophancy causing even credentialed mathematicians to move quickly on an unverified claim.
Hacker News 의견들
So the map sends three different points, (0,0,-1/4), (1,-3/2,13/2), and (-1,3/2,13/2), all to the same output (-1/4,0,0). That means the map isn't injective even though the Jacobian determinant is a nonzero constant (-2), so it has no polynomial inverse. That's exactly how it disproves the conjecture.
He just tweeted it out. No paper, no formal writeup, just a tweet with a polynomial map and a Wolfram Alpha link.
I fed the same polynomial to GPT 5.6. It wrote SymPy code, checked it with exact symbolic algebra, and said it doesn't see an algebraic catch and that this deserves serious independent checking rather than casual dismissal.
The conjecture held for 85 years and the counterexample got announced in a tweet that expires after seven days.
What's wild is how simple the counterexample is, low degree, small coefficients. Makes you wonder why nobody brute forced this decades ago with a computer.
Some people are still saying stochastic parrot, spicy autocomplete, wake me up when AI does something real. But the goalposts keep moving, even r/betteroffline has shifted from AI is useless to AI is good but the bubble will collapse.
I bet the LLM synthesized this because of all the published proofs of the conjecture that turned out to be wrong, it had a lot of failed attempts to learn from.
That's speculative honestly. Those wrong proofs were trying to prove the conjecture true, not helping find a counterexample. Either way, nobody had found this before despite plenty of people having access to the same failed proofs.
ELI5: the Jacobian determinant is a single number you get from a matrix of partial derivatives of a multivariable function. If that number is a nonzero constant, the conjecture says the function must have a polynomial inverse, meaning no two inputs can map to the same output. This example has a constant determinant of -2 but three different inputs mapping to one output, so no inverse exists.
Nice thing about these cleanup type results is nobody wastes more years trying to prove the conjecture true now.
Eh, that's the MBA mindset. Attacking hard problems even unsuccessfully builds insight and tools you use elsewhere, it's not wasted time.
This is already triggering edit wars on the Jacobian Conjecture Wikipedia page, people arguing over whether the claim is even correctly stated.
I'd wait for actual mathematicians to verify before trusting a tweet. This could be AI psychosis making people believe unverified claims.
Worth noting the original poster has a PhD in math from Cambridge. If this turns out wrong, it's a case study in AI sycophancy making even experts drop their guard.
The lack of properness in the map is just another way of confirming it's a real counterexample, since Jacobian invertible implies etale, and properness would force it to be an isomorphism, which affine space can't support nontrivially.