Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups

Let $\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $π$, and let $A_{ψ_n}$ be the Laplacian with word length $$ ψ_n(k) = \min(k,n-k). $$ For every $n\ge4$, we prove the sharp log-Sobolev inequality $$ \text{Ent}_π(|f|^2) \le 2π(\bar{f}A_{ψ_n} f), \qquad f:\mat