One Prompt Ends an Eighty-Seven-Year Conjecture

Description

A crude black-line cartoon shows a stunned person leaning toward a desktop computer on an otherwise white background. The person’s prompt says “disprove the jacobian conjecture,” and their reaction bubble says “Oh my god.” The computer replies: “> ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): C^3 → C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0).” A constant nonzero determinant combined with those three colliding inputs gives a compact, directly checkable counterexample in dimension three to the 1939 Jacobian conjecture, publicly credited by mathematician Levent Alpöge to Anthropic’s Fable 5.