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Mathematician Levent Alpöge (Harvard University/Anthropic) announced on 20 July 2026 that Anthropic's AI model Claude Fable 5 helped find a counterexample disproving the Jacobian Conjecture, open since 1939.
The result disproves the conjecture for polynomial maps in three or more variables; the two-variable (planar) case remains unresolved.
The counterexample was independently verified using the SymPy computer algebra system and the Lean formal proof assistant, and checked by outside mathematicians including Jared Duker Lichtman of Stanford University.
The episode is being cited as one of the most significant instances yet of AI contributing directly to a solution in pure mathematics.
For a system of polynomial equations in several variables, the Jacobian determinant measures how the map locally stretches or compresses space. The conjecture asked whether a constant, non-zero Jacobian determinant everywhere guarantees the map can always be reversed by another polynomial map (global invertibility) — a natural-seeming claim that turned out to be false in three or more dimensions.
Simple Analogy: Like assuming that if a machine never jams locally at any single point while processing an item, it must always produce a unique, reversible output overall — the 2026 counterexample shows that assumption can fail even when it looks locally safe everywhere.
GS Paper III > Science and Technology, Artificial Intelligence
General Awareness > Science and Technology
A quantity derived from the partial derivatives of a multivariable function that measures local stretching/compression of space by a map
Software (such as Lean) that checks mathematical proofs step-by-step against strict logical rules, providing machine-verifiable certainty