VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
Köthe's conjecture asks whether the sum of two nil left ideals of a ring is always nil. Equivalently, by Krempa's 1972 formulation, if $I$ is a nil two-sided ideal of a ring $R$, then the matrix ideal $M_n(I)$ should be nil for every finite $n$, already for $n=2$.
The conjecture is false: there exists a ring $R$, a nil two-sided ideal $I\subseteq R$, and a $2\times 2$ matrix with entries in $I$ that is not nilpotent.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-6 Astra (pre-release)
A pre-release GPT-6 Astra autonomously attempted the open Formal Conjectures benchmark statement, with no human steering during the run, and constructed both the mathematical counterexample and its Lean proof. It builds a nil algebra from three weighted backward shifts over $\overline{\mathbb F_2}$, arranges a universal mortality property for all elements, and simultaneously constructs a $2\times2$ matrix over the resulting nil ideal that has a nonzero eigenvalue and hence is not nilpotent. Claude was later used to generate repository documentation from the completed proof; it was not the mathematical solver.
Lean-verified on this site's ladder: kernel-checked, and the statement was written independently of the prover. Checked here on 5 September 2026 from a clone at b052755: 3,331 lines of Lean, zero `sorry` outside the Challenge.lean stub, zero `axiom` declarations, no `native_decide`, `unsafe` or `implemented_by` (two uses of `decide` on small numerals), mathlib pinned, Comparator in CI with the three standard axioms. The compared statement is byte-identical to Formal Conjectures' `KotherConjecture.variants.general_matrix` at 9cbe1d3c, and its one nontrivial ingredient, mathlib's `TwoSidedIdeal.matrix`, is the ideal of matrices whose every entry lies in I, so the statement is Krempa's matrix form as intended. What the kernel has certified is therefore: a ring with a nil ideal I and a non-nilpotent matrix in M_2(I). That Köthe's original statement implies the matrix form is an elementary argument (M_n(I) is a sum of n nil column left ideals) stated in the repository and not formalized; Formal Conjectures opened a PR on 4 September relating the formulations. Candidate because no ring theorist has read it yet, and the machine-generated proof account is unaudited.
GPT-6 Astra constructs a unital algebra $R=k\oplus A$ over the countable field
$$
k=\overline{\mathbb F_2},
$$
with $I=A$ a nil two-sided ideal, together with a matrix
$$
W\in M_2(I)
$$
that is not nilpotent.
The algebra $A$ is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of $A$ is nilpotent. At the same time, a suitable polynomial combination of the shifts fixes a nonzero vector; this yields a companion-type matrix with a nonzero eigenvalue, and hence a nonnilpotent matrix whose entries lie in $I$.
This formally disproves Krempa's matrix formulation of Köthe's conjecture.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
The source status is candidate and must not be represented as solved.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.