A Complete Log-Concavity Classification for Higher-Order Stirling Cycle Rows

Let C_r(n,k)=[n+(r-1)k,k]>=r, where the bracket counts permutations with exactly k cycles, every cycle having length at least r. We prove that every row C_r(n,.) is log-concave for every n if and only if 1<=r<=5, settling the previously open r=3,4,5 layers. We also prove log-concavity of the factorial-normalized rows for every r>=2. The r=5 proof combines exact symbolic certificates, an exhaustive finite prefix of 5,754,989 inequalities, and an effective fifth-order Fourier--Edgeworth estimate. Status: Public Beta; internally verified candidate proof; external mathematical review pending.