For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure $\mathcal H^n$, it is a full-support $\mathrm{RCD}((n-1)\kappa,n)$ space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.

On a full-support $\mathrm{RCD}(K,N)$ space with $1\lt N\lt \infty$, we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.