A Mathematician at Anthropic Posted 216 Characters on X and Killed an 87-Year-Old Conjecture

The post read, in part, "hello there the jacobian conjecture is false thanx." Attached was a formula 216 characters long. With that, a question mathematicians had been unable to settle since 1939 was settled — and it was wrong.

The author was Levent Alpöge, a mathematician working at Anthropic, and the tool was the company's Claude Fable 5 model. The Jacobian conjecture belongs to algebraic geometry and concerns polynomial maps: functions built out of ordinary polynomials that take a set of coordinates and produce a new set. Attached to any such map is a quantity called the Jacobian determinant, which measures how the map stretches or compresses space at each point. The conjecture held that whenever that determinant is a non-zero constant everywhere, the map must be reversible — there must exist a second polynomial map that undoes it.

It sounded like the kind of claim that should either be provable in a few pages or falsified by an afternoon of doodling. It was neither. The two-dimensional version was stated by Ludwig Kraus in 1884. Ott-Heinrich Keller generalized it to higher dimensions in 1939, and it survived from there, accumulating a long trail of published proofs that were later withdrawn. Fields Medalist Stephen Smale placed it on his 1998 list of the most important unsolved problems in mathematics.

What Alpöge found is a polynomial map in three dimensions whose Jacobian determinant is the constant −2, and which sends more than one input point to the same output. A map that collapses distinct points cannot be reversed — there is no way to know which point you started from. That single object disproves the conjecture in three dimensions, and because such counterexamples extend upward, in every dimension above two as well. The original two-dimensional case remains open.

The interesting part, to mathematicians reading it, is where the difficulty had been hiding. The counterexample requires no elaborate construction and no long proof; verifying it is a matter of computing a determinant and checking two inputs. The hard part was locating it in a search space of possible polynomial mappings vast enough that eight decades of human intuition about where to look had produced nothing. That is a different kind of problem from the one AI systems have been used on so far.

Most attention on machine assistance in mathematics has focused on proof — on getting models to construct or formally verify chains of logical reasoning. This result points somewhere else: at using models to hunt for mathematical objects with prescribed properties, in spaces too large and too unstructured for a human to sweep by hand. As Melissa Lee, a senior lecturer in mathematics at Monash University, noted in an account of the episode published this week, the discovery suggests AI may prove as valuable for finding unexpected objects as for building arguments about them.