A Casual Tweet Just Broke an 87-Year-Old Math Problem, With Help From AI

On July 20, 2026, mathematician Levent Alpöge posted a strangely casual message on X: “hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final.” Attached was a compact polynomial map. That was it, no fanfare, no press release. But it was enough to topple a conjecture that had stood since 1939.

The Jacobian conjecture asks a deceptively simple question: if a polynomial map has a Jacobian determinant that’s a nonzero constant, must it always be invertible? First stated for two dimensions by Ludwig Kraus in 1884 and generalized by Ott-Heinrich Keller in 1939, the conjecture was considered important enough that Fields Medalist Stephen Smale included it in his 1998 list of great problems for the next century. Over the decades, several respected mathematicians, including Beniamino Segre and Wolfgang Gröbner, published proofs that later turned out to contain subtle errors.

Alpöge’s counterexample worked in three-dimensional complex space, with a Jacobian determinant of exactly −2. Three distinct points in the domain all mapped to the same output, immediate proof that the function couldn’t be inverted. Because the polynomial was so compact, other mathematicians could verify it almost immediately using tools as simple as Wolfram Alpha.

What makes this different from earlier AI-assisted math results is the nature of the discovery itself. Alpöge worked with Anthropic’s Claude Fable 5, and rather than the AI helping verify or write up a proof a human had already conceived, it played a hands-on role in surfacing the actual counterexample. Within days, other mathematicians built on the result: Gallagher produced an infinite family of examples, Speyer offered a geometric explanation involving tangent lines swept across a plane curve, and Shuhong Gao generalized the construction to arbitrary dimensions, a paper that itself credits Claude Fable 5 with assisting the proofs and writing.

It’s worth noting what wasn’t resolved: the original two-dimensional version of the conjecture, the one that mattered most to algebraic geometers, remains open. And some mathematicians have pointed out that the underlying technique isn’t conceptually new, it builds on methods established by Vitushkin back in 1999. What’s new is where the idea came from, and how fast it spread once it landed.

Still, the story has mathematicians talking less about the specific conjecture and more about what it signals: AI may turn out to be just as useful for stumbling onto unexpected mathematical objects as it is for grinding through formal proofs. Where that leaves human mathematicians is very much an open question of its own.

Original paper: Counterexamples to the Jacobian conjecture in dimensions greater than two 

Source: A Casual Tweet Just Broke an 87-Year-Old Math Problem, With Help From AI