Sharp arity bounds for clonoids from mixed-exponent groups

Author: Henry Zweiman. Date: September 9, 2026.

This manuscript gives a proposed complete answer to Question 60 of Fioravanti, Kompatscher and Rossi, and treats the larger family

[ A=\mathbb Z/p^2\mathbb Z\oplus(\mathbb Z/p\mathbb Z)^s. ]

Every clonoid from this group to a module on which p is invertible is generated by its (s+2)-ary functions. The paper proves a uniform identity over Z[1/p] and gives an explicit finite-field-valued function whose generated clonoid requires s+2 variables. The proof is theoretical; the accompanying computations are exact diagnostics.

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• Proof and scope assessment
• Internal review
• Exact checker and results

Run python3 check.py using Python 3.8 or later. The checker uses only the standard library. It includes all 262144 matrices over Z/4 of size three, plus the documented projector, quotient and additional lower-bound checks. It does not replace the arbitrary-prime proof.

The cyclic case s=0 and the finite-field matrix lemma are credited to earlier work. Groups with multiple cyclic p^2 factors and higher exponent are outside the result. The paper is one manuscript, including its upper bound, lower bound and corollary.

Status: internally checked research manuscript, publicly shared for scrutiny. Independent correctness, historical priority and significance assessment remain outstanding. No journal acceptance or external peer review is claimed.