# Metric-measure curvature definition audit

Family 356: Gigli’s characterization of Alexandrov curvature. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometry formalization researchers can track the equivalence of Alexandrov and combined RCD/distributional sectional-curvature conditions.

## Applicability and commercial boundary

Full support, exact Hausdorff reference measure and unreduced RCD parameters are part of the statement; finite data cannot certify these global test-class hypotheses.

## Initial business decision

Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product.

## Next verification action

Create an exact definition crosswalk for a formal geometry library.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and formal-scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Gigli’s distributional curvature characterization of Alexandrov spaces](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf).