Exact Quantum Maxima of the $n$-Cycle Overlap Inequalities

We extend the three-state overlap analysis to determine the exact quantum maximum over finite-dimensional pure-state realizations of the $n$-cycle overlap inequalities, $S_n^{\max}=n\cos^2(π/(2n))-1$, for arbitrary cycle length $n\ge3$. The bound is saturated by an explicit family of coplanar qubit