Earlier today, an Anthropic employee posted on X. It was a short enough post, and said it was written during the World Cup final. But the implications in the post were big — it contained a counterexample that disproved the 85-year-old Jacobian Conjecture, and were generated with the help of Fable.
In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse. The conjecture was first stated for two variables by Ludwig Kraus in 1884 and then stated in full generality in 1939 by Ott-Heinrich Keller.
The Math community on X was quick to react to the news. Daniel Litt, who is an Assistant professor (of mathematics) at the University of Toronto, said the proof made him very bullish on the near-term impact of AI on math. “My take on Jacobian conjecture: (1) Very bullish for near-term impact of AI on math. Probably lots more of these coming soon. (2) If you think of the main goal of math research as “solving well-known open problems” (and IMO many people do think this, and it’s a defensible view): it’s a huge deal (3) from the POV of doing good science: of interest because it resolves a natural Q, but plausibly not generative beyond that. Personally I am most excited about (1), but (perhaps idiosyncratically) I don’t care much about this conjecture,” he posted on X.
Mathematician Tim Gowers — who has previously been vocal about how AI is impacting math — called the result “pretty amazing”. “Assuming this is correct, it is for me the first example of an LLM solving a problem not in my area that was nevertheless big enough that I had very definitely heard of it. Again it’s a counterexample, so not in “end of mathematics” territory, but still pretty amazing,” he said.
Bartosz Naskręcki, who is the Vice Dean at Adam Mickiewicz University in Poznań, said that this likely wasn’t the case of Fable solving it autonomously, and would’ve required real insight for someone to create a prompt. “I’m waiting for the full report from (Levent Alpoge). As easy as it looks on X, this wasn’t a simple prompt for Fable. Finding counterexamples like this to the Jacobian Conjecture is like searching for a needle in a haystack, it requires real insight. I’m really interested to see the whole story.
Stanford math professor Jared Duker Lichtman said it was a “remarkable result”, calling the Jacobian Conjecture one of the central open problems in algebraic geometry.
Oxford University’s Math Professor Vidit Nanda hinted that the result could set off a wave of similar discoveries. “Let us celebrate the present and beware the future,” he posted on X.
Przemek Chojecki, who has a PhD in Mathematics, said that the counterexample would create a lot of new mathematics. “he counterexample to the Jacobian Conjecture has many interesting repercussions and is bound to create a lot of new mathemathics. Simply translating the counterexample for JC to one of the other conjectures on the list, and then exploring how it generalizes and how to provide necessary conditions for refined conjectures, is going to inspire dozens of papers to be written. With or without AI,” he posted.
The Fable disproof is the latest entry in a run of results that suggest frontier AI models have moved past competition math and puzzle-solving into work that touches actual open problems. Google DeepMind’s AlphaProof Nexus recently solved nine open Erdős problems, including two that had stood for over 50 years, generating fully machine-verified Lean proofs for a few hundred dollars apiece. A year earlier, OpenAI and Google DeepMind both hit gold-medal scores at the International Mathematical Olympiad, working entirely in natural language rather than relying on specialized proof-formatting pipelines.
What sets the Jacobian result apart is that it 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, then publishing it in a single tweet with no fanfare. And judging by the reaction of professional mathematicians, it could be yet another proof that AI could end up transforming math — and other scientific fields — in the years to come.