What was already known
The full element-order spectrum was known to distinguish a simple group in a related question; it was open whether the weaker totient sum did.
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
An explicit counterexample settles the narrower invariant question.
Limitations
The counterexample concerns the totient-sum test; it does not undermine results based on the full element-order spectrum.
Technical verification
The companion metadata names declaration kourovka_19_25 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_19_25.
- Checker
- Lean 4
- Scope
- A simple group and a non-simple group both have order 6048 and totient sum 23984, disproving the proposed simplicity test in Problem 19.25.
- 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.
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