Asymptotic Preservation and Uniform Accuracy of Diffusion and Flow-Matching Samplers

Diffusion and Gaussian-interpolant flow-matching samplers approach data through a terminal noise floor \varepsilon, a singular limit for manifold-supported or rank-deficient data. We study two properties of a complete sampler specification, comprising its update rule, time grid, and terminal rule. Asymptotic preservation (AP) means a stable and consistent zero-noise discretization with a step count bounded independently of \varepsilon. Uniform accuracy (UA) of order p means that, at numerical resolution h, the endpoint W_2 error is O(h^p) with a floor-independent constant. Bounded log-noise stepping fails AP because its step count diverges. Stopping a stable base solver at a positive switching scale a and appending one map fitted to the analytic normal mode restores AP. On smooth compact boundaryless manifolds, the standard map has exact-input error O(a^2-\varepsilon^2) and sharp zero-floor error Θ(a^2). A base solver with a floor-uniform order-p estimate on the resolved interval retains that order when a=O(h^{p/2}), provided the terminal transfer factor remains bounded. Along exact trajectories, the posterior-mean identity D(x(σ),σ)=x(σ)-σx'(σ) cancels the linear terminal defect and enables higher-order fitted maps. A three-evaluation Hermite construction is uniformly third order for exact switching-scale input over 0\le\varepsilon\le a, and a seven-evaluation construction is fourth order at zero. We classify representative diffusion and flow-matching specifications by AP and UA. On EDM and Rectified Flow checkpoints, a paired decomposition separates base-integration from terminal-completion error and predicts held-out same-seed endpoint errors.