What was already known
Köthe’s 1930 question about sums of nil left ideals was still open. Krempa established the matrix formulation in 1972.
AI involvement
The original construction and Lean proof came from an unsteered GPT-6 Astra benchmark attempt set up by Tom Adamczewski. Böhmler and Marczinzik subsequently checked the AI material used in their paper and explained the construction.
The original nil-algebra counterexample and Lean proof came from an unsteered benchmark attempt; human mathematical extensions are credited separately.
- Supporting source (vibemathed.com) — Verification audit and comparison with Formal Conjectures
Importance
The counterexample resolves a longstanding ring-theory conjecture through Krempa’s classical implication from Köthe’s original statement to the matrix form.
Limitations
Lean checks the matrix statement. Its implication from Köthe’s original wording is a separate classical argument. Böhmler and Marczinzik later explained and strengthened the construction over arbitrary countable fields and answered Rowen’s question; those extensions are their work.
Technical verification
The original matrix-form theorem was checked by Lean/Comparator against the independent Formal Conjectures statement. Böhmler and Marczinzik also report checking the AI material used in their modified mathematical proof. This review did not rerun the kernel or verify their check independently.
Independent checking
Böhmler and Marczinzik report reviewing AI material incorporated into their modified mathematical paper.
- Checker
- Bernhard Böhmler and René Marczinzik
- Scope
- Incorporated AI-origin counterexample, not a line-by-line audit of the original Lean term
- Basis
- Reported by the linked checker
Formal checking
Reported Lean 4 kernel and Comparator check of the negated general_matrix statement; original run not reproduced here.
- Checker
- Lean 4 kernel and Comparator
- Scope
- Krempa matrix-form counterexample
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Böhmler and Marczinzik give a detailed human exposition of the weighted-shift construction and nilness argument. This supports understanding of the central mechanism, while their additional field generality remains separately credited.
Correction and publication history
The original matrix-form counterexample is separate from Böhmler and Marczinzik’s stronger field and Rowen results.
Read this published revision