What is jacobian?

This repository holds a formally verified mathematical proof, written in the Lean 4 programming language. Lean is a tool mathematicians and computer scientists use to write proofs that a computer can check step by step, so there is no ambiguity about whether the logic actually holds together. The specific proof here answers a long standing open question in mathematics called the Jacobian conjecture. That conjecture asked whether a certain kind of polynomial transformation, one where a mathematical quantity called the Jacobian determinant equals one, is always reversible, or injective, no matter what number system you are working in. This project provides a counterexample: an explicit polynomial transformation of three dimensional space where the Jacobian determinant is one, but the transformation is not reversible, for any field. A field, in this context, is a number system with addition, subtraction, multiplication, and division, such as the real numbers or a finite set of numbers used in certain areas of computer science. The main result is stated as a formal theorem in the code, and the repository also includes a version tied to a specific numeric condition from the original counterexample, plus a version specialized to complex numbers. To verify the proof yourself, the README gives two commands to run: one that downloads a cache of prebuilt files, and one that builds and checks the whole proof. This project is aimed at mathematicians, logicians, and researchers interested in formal verification rather than general software developers. It does not include a wider library or reusable tools, just the counterexample itself and its proof. There is no license or additional documentation beyond what is described here, so anyone wanting to build on this work should reach out to the author directly.