Anthropic Fable Helps Disprove 85-Year-Old Jacobian Conjecture

An Anthropic researcher has used the company's Fable model to disprove the Jacobian Conjecture — an 85-year-old open problem in algebraic geometry. The result was shared in a single tweet with no fanfare, marking a shift in how mathematical research is conducted.

The Jacobian Conjecture

First posed by Ott-Heinrich Keller in 1939, the Jacobian Conjecture asks whether every polynomial map with a constant nonzero Jacobian determinant has a polynomial inverse. The answer, it turns out, is no — and Fable helped find the counterexample.

What Makes This Different

This isn't a formal proof search racking up points on a known problem set. It's a working mathematician using a model as a genuine research collaborator to hunt down a specific counterexample to a named, decades-old conjecture.

The open question now is whether some weaker, "repaired" version of the conjecture survives. GPT-5.6 reportedly suggested the conjecture might hold if you add a condition ruling out degeneracy at infinity — a proposal that is now itself an open problem.

The Bigger Picture

The Fable disproof is the latest in a run of results where AI models contribute to genuine open problems:

• Google DeepMind's AlphaProof Nexus recently solved nine open Erdős problems, including two that stood for over 50 years
• OpenAI and Google DeepMind both hit gold-medal scores at the International Mathematical Olympiad
• OpenAI's unreleased model disproved the Erdős unit distance conjecture (separate incident)

Fable's Redemption Arc

Fable has had a chaotic few months since its June debut, including a brief, government-mandated suspension tied to export controls. A result like this gives Anthropic a considerably better headline than the regulatory back-and-forth that has dominated Fable's news cycle.

Key takeaway: AI is now a legitimate mathematical research tool. The era of AI as a collaborator in open scientific problems has arrived.