Levent Alpoge, a mathematician at Anthropic, announced on July 20, 2026, via his X account that he discovered a counterexample to the Jacobian conjecture, a mathematical problem that has remained unsolved since 1939. The conjecture posits that for a polynomial mapping with multiple variables, if the Jacobian determinant is a non-zero constant, then the mapping must have an inverse.

The polynomial mapping involving three complex variables proposed by Levent Alpoge maintains a constant Jacobian determinant of -2 for any input. However, because three distinct inputs correspond to the same output, a unique inverse mapping cannot exist, thereby refuting the conjecture for cases with three or more variables. This counterexample can be further extended to four or more variables by adding components that output unnecessary variables unchanged, demonstrating that the general conjecture does not hold. The case for two variables remains unsolved.

Currently, test code to automatically verify this counterexample is available on GitHub for the mathematics software SageMath, and formalization efforts using the theorem prover Lean are underway on a GitHub repository managed by Google DeepMind. The findings represent a report by Levent Alpoge in a personal capacity rather than an official research publication by Anthropic. Specific details regarding the workflow, including interaction logs with Claude or the number of attempts, have not been disclosed.

Levent Alpoge, a mathematician at Anthropic, announced on July 20, 2026, via his X account that he discovered a counterexample to the Jacobian conjecture, a mathematical problem that has remained unsolved since 1939. The conjecture posits that for a polynomial mapping with multiple variables, if the Jacobian determinant is a non-zero constant, then the mapping must have an inverse.

The polynomial mapping involving three complex variables proposed by Levent Alpoge maintains a constant Jacobian determinant of -2 for any input. However, because three distinct inputs correspond to the same output, a unique inverse mapping cannot exist, thereby refuting the conjecture for cases with three or more variables. This counterexample can be further extended to four or more variables by adding components that output unnecessary variables unchanged, demonstrating that the general conjecture does not hold. The case for two variables remains unsolved.

Currently, test code to automatically verify this counterexample is available on GitHub for the mathematics software SageMath, and formalization efforts using the theorem prover Lean are underway on a GitHub repository managed by Google DeepMind. The findings represent a report by Levent Alpoge in a personal capacity rather than an official research publication by Anthropic. Specific details regarding the workflow, including interaction logs with Claude or the number of attempts, have not been disclosed.