Gigli’s distributional curvature characterization of Alexandrov spaces

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.