Köthe's conjecture asserts that the sum of two nil left ideals of a ring is nil. Here an ideal is nil when each of its elements is nilpotent, with the exponent allowed to depend on the element. This is weaker than requiring the ideal itself to be nilpotent. The analogous assertion for right ideals is equivalent.
An equivalent matrix formulation asserts that 
is nil whenever 
is a nil ring. This illustrates the
difficulty: nilpotence of each matrix entry does not by itself force nilpotence of
the matrix. The equivalence and the original conjecture
are discussed by Adamczewski (2026).
Adamczewski (2026) released a proposed counterexample constructed autonomously by GPT-6 Astra. It consists of a nil ideal
over the algebraic closure of the two-element
field together with a nonnilpotent matrix in 
. The Lean proof was checked
against an independently supplied formal statement, using only Lean's standard logical
axioms. As of Sep. 7, 2026, no independent review by a ring
theorist was reported, and the equivalence with the original ideal-sum
formulation was not part of the formalization (VibeMathed 2026).
