Connections between the minimal neighborhood and the activity value of cellular automata

The paper’s Theorem 1 (three parts) matches the model’s targets. Part (1) (divisibility when e ∈ MN(μ) and the exact formula α(μ)=|A|^{|S|}−|A|^{|S|−1} when e ∉ MN(μ)) agrees with Lemma 4 and Lemma 6, respectively , and Part (2) (realizing every 0<k<|A|^{|S|} with MN(μ)=S) is established via Theorem 2’s constructions using the δ-patterns and Lemma 5 . For Part (3), the paper asserts existence for α(μ)=|A|^{|S|} when |A|≥3 (also in Theorem 2’s “Furthermore”), but the given proof only shows how to make MN(μ)=S, without explicitly ensuring that every input is active; the activity count is not justified there . The model supplies a concrete derangement-based rule that makes all transitions active while keeping every coordinate essential, thereby filling the gap. Overall: statements are correct; the paper’s proof of the full-activity case is incomplete, while the model’s construction is correct.