Sharp log-Sobolev inequalities on finite cyclic groups

Let ${\mathbb{Z}}_n$ be the cyclic group equipped with the uniform probability measure $\pi$, and let $A_{{\psi }_n}$ be the Laplacian with word length $${\psi }_n(k)=\min (k,n-k)\mathrm{.}$$ We prove the sharp log-Sobolev inequality $${\text{Ent}}_{\pi }(f^2)\le 2\pi (f{A}_{{\psi }_n}f),f:{\mathbb{Z}}_n\to [0,\mathrm{\infty }),$$ for every $n\ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their cubic-majorant reduction, which turns the problem into a 3rd moment estimate; the new point is a blockwise 3rd moment estimate adapted to the word-length multiplier. The same 3rd moment argument also recovers the log-Sobolev inequality for Poisson-semigroup on the circle, first proved by Weissler~\cite{Weissler1980}. The same sharp inequalities were also obtained recently by Yao~\cite{Yao2026} by a different method.