PROVER: Precision-Indexed Risk Order via Euclidean Reachability in Gaussian-Affine Flow Decoding

When does one source representation supply more useful information than another for velocity prediction? Classical comparison settles this when one representation is a random degradation of the other, but leaves many pairs unordered. We prove an exact order for flows that linearly mix a target with Gaussian noise. At each noise precision, a matrix records the posterior error removed by the representation. Matrix dominance at every precision is equivalent to no larger optimal Euclidean velocity risk on every complete flow. The new step is Euclidean reachability: any positive precision and error metric can be realized together on a normalized noise-to-data path. Complete flows therefore test the full Gaussian-affine quadratic prediction order. This comparison principle also orders frozen predictors with fitting error included and determines when a preferred system remains preferable throughout a prescribed path family.