Köthe's conjecture asserts that in any ring the sum of two nil left ideals is nil. It has been open for over 90 years. A related question of Rowen asks whether K(D) equals the upper nil radical n(D) whenever the powers of K(D) intersect to zero.

An AI model (GPT-6 Astra) was given the Lean formalization of the matrix form of Köthe's conjecture from the Formal Conjectures repository and asked to prove or disprove it. It found an initial counterexample with a formal proof, available at github.com/tadamcz/koethe. The authors then developed a modified construction over any countable field. It uses weighted shift matrices in the ring of finite-bandwidth upper triangular matrices, a multihomogeneous common-zero lemma, and matrix pencils to establish the nil property.

For every countable field F there is a nil F-algebra N generated by two elements b and c for which a specific 2x2 matrix W over N is not nilpotent. This disproves Köthe's conjecture. Applied to the unitization of M_2(N), the construction also gives a negative answer to Rowen's question.