What was already known
Bardakov and Iskra conjectured this general description after studying class transpositions for particular moduli.
AI involvement
Van Doorn, Judin, Monticone and Morrison report that Aristotle developed the proof strategy and Lean proof without human guidance on mathematical steps. The authors chose problems, supplied infrastructure, refined the formal code, translated the proofs into prose and arranged external review.
The authors say Aristotle developed the new mathematical argument and formal proof strategy without human guidance on the mathematical steps. Humans selected the task, supplied infrastructure, refined the code and wrote the prose account.
Importance
The proof gives a general group description for Problem 21.8.
Limitations
Pan independently proved the same result; this record credits Aristotle’s separate proof and Lean formalization, not priority over Pan.
Technical verification
The companion metadata names declaration kourovka_21_8 and reports a complete Lean proof with zero sorry. This is a source-reported machine check of the scoped theorem; the assessment did not compile the project.
Formal checking
The authors report a Lean 4 proof with zero sorry for declaration kourovka_21_8.
- Checker
- Lean 4
- Scope
- For k ≥ 4, horizontal class transpositions of moduli at most k generate a group isomorphic to the symmetric group on lcm(2,…,k) letters.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Van Doorn, Judin, Monticone and Morrison translated the formal solution, refined the explanation and present the construction or proof in mathematical prose. This documents human explanation of the central argument.
Correction and publication history
Primary-source review of the scoped Kourovka result, AI contribution, Lean checking, human prose proof and prior baseline. Historical, non-notifying addition.
Read this published revision