Claude Fable 5 Disproves 87-Year-Old Jacobian Conjecture Beyond 2D With 3D Counterexample
Updated
Updated · ScienceDaily · Aug 10
Claude Fable 5 Disproves 87-Year-Old Jacobian Conjecture Beyond 2D With 3D Counterexample
3 articles · Updated · ScienceDaily · Aug 10
Summary
A tiny 3D polynomial mapping found with Claude Fable 5 gives a constant Jacobian determinant of -2 yet sends multiple inputs to one output, making it non-reversible.
That formula overturns the Jacobian conjecture in three dimensions and every higher dimension, while the original two-dimensional case remains unsolved.
Levent Alpöge, a mathematician at Anthropic, said the counterexample was short enough to fit in a single X post, helping other mathematicians verify it quickly.
The conjecture was first posed in 1884 and generalized in 1939, surviving many failed proof attempts and computational checks that confirmed the 2D case only up to degree 100.
The result adds to a run of AI-assisted math breakthroughs and suggests large language models may be especially useful for searching vast spaces for unexpected mathematical objects.
If an AI easily disproved a famous conjecture in higher dimensions, what secrets remain hidden in the unsolved two-dimensional case?
How did an AI shatter a century-old math mystery using just a strikingly simple three-dimensional formula?
Could an AI's shocking discovery of a simple counterexample forever change how human experts approach unsolved scientific mysteries?
July 2026: AI and the Disproof of the Jacobian Conjecture—A Turning Point for Mathematics
Overview
On July 20, 2026, Levent Alpöge announced that the Jacobian conjecture was false, thanks to a simple counterexample found with the help of Anthropic’s AI, Claude Fable 5. The AI bypassed human limitations by searching high-dimensional symbolic spaces and pinpointed a brief, three-variable polynomial map that disproved the conjecture. The global math community quickly verified the result, and Terence Tao soon explained its structure, making it understandable. Mathematicians extended the counterexample to higher dimensions, proving the conjecture false for all n ≥ 3, but the two-dimensional case remains unsolved, now the main focus for future research.