arXiv:2610.14164v1 Announce Type: cross Abstract: Deterministic flows can contract Wasserstein distances between probability laws without being contractive in point-state space. We develop a differential contraction theory for pushforward flows on forward-invariant subsets of the 2-Wasserstein space that quotients out the gauge freedom of the continuity equation. Weighted Helmholtz-Hodge decomposition and the Wasserstein contingent cone define the weighted horizontal tangent cone (whtc). Requiring the contraction quadratic form to be nonpositive only on the whtc yields the Intrinsic Wasserstein Contraction Condition (IWCC), which generalizes the classical contraction LMI and recovers it on Dirac measures. An almost-everywhere Euler approximation based on the superposition principle and Banach-valued Lebesgue differentiation theorem proves that IWCC is sufficient for intrinsic Wasserstein contraction. Law contraction without point-state contraction is illustrated by a rotationally symmetric shear example. Finally, a new comparison lemma yields an incremental Wasserstein ISS theorem; for Brockett's integrator, we prove exponential stabilization of a class of axisymmetric laws to the uniform law on a circle via a smooth feedback.