Gigli's conjecture: Alexandrov curvature bounds are characterized by his distributional sectional curvature on noncollapsed RCD spaces

Theorem 1.1: for every integer $n\ge2$ and $\kappa\in\mathbb R$, a complete separable metric space is an $n$-dimensional Alexandrov space with curvature at least $\kappa$ if and only if, with $m=\mathcal H^n$, it is a full-support unreduced $\mathrm{RCD}((n-1)\kappa,n)$ space whose distributional sectional curvature is at least $\kappa$ in Gigli's original global test classes. The reference measure must be exactly Hausdorff measure; collapsed or weighted spaces are not covered, and dimension one is excluded. The companion proves that weak Hessian upper bounds hold along every minimizing geodesic in any full-support $\mathrm{RCD}(K,N)$ space, $1<N<\infty$.

Theorem 1.1: for every integer $n\ge2$ and $\kappa\in\mathbb R$, a complete separable metric space is an $n$-dimensional Alexandrov space with curvature at least $\kappa$ if and only if, with $m=\mathcal H^n$, it is a full-support unreduced $\mathrm{RCD}((n-1)\kappa,n)$ space whose distributional sectional curvature is at least $\kappa$ in Gigli's original global test classes. The reference measure must be exactly Hausdorff measure; collapsed or weighted spaces are not covered, and dimension one is excluded. The companion proves that weak Hessian upper bounds hold along every minimizing geodesic in any full-support $\mathrm{RCD}(K,N)$ space, $1<N<\infty$.