Gigli’s characterization of Alexandrov curvature

What OpenAI claims

Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support RCD ((n-1)κ,n) condition with reference measure ℋⁿ and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced. (Lean)

Papers (2)

1. Gigli’s distributional curvature characterization of Alexandrov spaces

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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 ℋⁿ, it is a full-support RCD((n-1)κ,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.

2. Weak Hessian bounds along every geodesic in RCD spaces

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract On a full-support RCD(K,N) space with 1 < N < ∞, 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.