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 \(k≥ 2\),\[m(C_k)≥16log k-14+1{3k}-1{12k^2}.\]Together with the known upper bound, this gives \(m(C_k)log k\) 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.