What was already known
The extremes k=1 and k=n! were straightforward, and Kohl had checked small n computationally; the general range remained open.
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
A central-extension construction settles every intermediate count in Kourovka Problem 18.50.
Limitations
The theorem establishes existence; it does not classify all groups or all element sets with a given count.
Technical verification
The companion metadata names declaration kourovka_18_50 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_18_50.
- Checker
- Lean 4
- Scope
- For every positive n and 1 ≤ k ≤ n!, a group has n distinct elements whose n! ordered products take exactly k distinct values.
- 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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