Strict log-concavity of Garvan's k-rank

Let N_k(m,n) be the number of partitions of n with Garvan k-rank m. We prove that N_k(m,n)^2 > N_k(m-1,n)N_k(m+1,n) for all k ≥ 3, n ≥ k+71, and |m| ≤ n−k−71. Together with the rank and crank inequalities of Mao, Shi and Zhu, this resolves Zhou's log-concavity conjecture. For each fixed k ≥ 3 the inequality fails at m = n−k−70 for all sufficiently large n. The proof uses a partition asymptotic with explicit errors, a residue estimate, and rigorous interval arithmetic to obtain a bound uniform in k for n ≥ 16450. Exact integer computations cover the remaining finite range.