Certified Quotient Natural Gradients in Wasserstein Space: Retractions, Ravines, and Higher-Order Collisions

Degenerate Wasserstein pullback metrics arise from redundant parameters, unresolved transport potentials, and rank-changing generators. These mechanisms require different remedies. We develop a residual-certified quotient natural-gradient framework that distinguishes them. A matrix-valued Poisson residual bounds the true metric from above and below, certifies equality of nullspaces, and controls the induced distributional step without an absolute eigenvalue threshold in raw coordinates. Compatible horizontal lifts transfer retraction-based descent and local convergence to redundant generators. For genuine rank changes, we construct moment-balanced Ornstein-Uhlenbeck generators with collisions of arbitrarily high order. Their cumulant coordinates have nonzero Wasserstein tangents even when the raw metric and loss vanish to orders $2m-2$ and $2m$. Gauss-Hermite quadrature gives explicit $K$-component examples with $m=2K$. We further reduce the true metric of rotated product generators to one- and two-dimensional Poisson problems, identifying precisely the rotations lost at Gaussian strata. Complete proofs cover inexact metric certificates, mixed-order ravines, chart contraction, angular cone limits, and statistical lower bounds. This is a pure theory study: it does not claim a globally regular chart for arbitrary neural generators or empirical superiority.