A uniform logarithmic lower bound for cyclovariety Mahler measures

Let $C_k$ be the cyclovariety associated with all $k$-th roots of unity in the construction of Cornelissen, Hokken and Ringeling. We prove, for every integer $k \ge 2$,

$$
m(C_k) \ge \frac{1}{6}\log k - \frac{1}{4} + \frac{1}{3k} - \frac{1}{12k^2}.
$$

Together with the known upper bound, this gives $m(C_k) \asymp \log k$ uniformly in $k$, and proves their Conjecture 9.2.1 for every odd integer $n > 1$. The proof combines Jensen’s formula, an elementary count of balanced quadruples of roots of unity, and the classical second-moment tail inequality. Only the second and fourth moments of the associated Laurent polynomial are needed; no independence among its monomials is assumed.