- Sources: Levent Alpoge (@alpoge), Jacobian conjecture (Wikipedia), HN discussion
- Summary: Mathematician Levent Alpoge posted on 2026-07-19 a concrete counterexample to the Jacobian Conjecture, an open problem since 1939 and number 16 on Stephen Smale's 1998 list. The counterexample is a polynomial map from C^3 to C^3 with a constant nonzero Jacobian determinant of -2 that is not injective: three distinct points map to the same output, which contradicts the conjecture's claim that such a map must be invertible. Alpoge credited Anthropic's Claude for the work, discussed on Hacker News as the Fable 5 model. The map is short enough to check by hand or with a computer algebra system.
- Comments: HN commenters report verifying the map symbolically in Sage and SymPy, confirming the Jacobian determinant is the constant -2 and that the three listed points collide, so the counterexample holds up on inspection. Several note that a machine-checkable counterexample is a stronger claim than the unverified AI proof announcements tracked earlier this month.
- Why it matters: A verifiable counterexample to an 85-year-old conjecture, produced with a language model and checkable in seconds, is a concrete datapoint for AI-assisted mathematics that stands apart from the prose proof claims that could not be independently confirmed.
- Follow-up: Watch for a formal writeup or paper, independent expert confirmation that the map is a genuine counterexample, and clarification of the model's role versus the mathematician's.
send feedback on this story