# Curved optimal-transport regularity contract

Family 360: Weak MTW curvature gives convexity and regular optimal transport. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Transport researchers can record when weak MTW curvature and bounded positive density classes support bi-Hölder transport on a fixed manifold.

## Applicability and commercial boundary

The constants are uniform over density pairs on one fixed manifold, not arbitrary varying metrics. Finite-family scope does not cover the broader arbitrary-potential injectivity claim. Existence/regularity gives no efficient numerical map or explicit error rate.

## 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

Extend opportunity 015 with manifold, density and curvature fields after extracting effective constants.

## Evidence scope

The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scope notes were also inspected; family 354’s explicit main profile formula was read. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Global Support and Convex Injectivity Domains under Weak MTW](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf).