Asymptotic-Preserving A Posteriori Analysis of Diffusion and Flow-Matching Samplers

Diffusion and flow-matching samplers integrate a learned probability-flow ODE from a large noise scale down to a small terminal floor $\sigma_{\min}$, at which the score is stiff and the flow develops a boundary layer.

We treat $\sigma_{\min}$ as a singular-perturbation parameter and determine which fixed-step samplers are asymptotic-preserving (AP), that is, stable and uniformly accurate as $\sigma_{\min}\to0$, casting the criteria as an a posteriori audit: residual functionals with $\sigma_{\min}$-uniform coefficients, computable on a pretrained checkpoint without ground-truth scores or exact trajectories.

On the terminal layer, Euler in the $\sigma$-clock, the deterministic DDIM update, is the unique layer-exact discretization up to affine reparameterization, with rectified flow its flow-matching counterpart; the $\lambda$-clock is stable only for steps $h\le h_\star=1+W(1/e)$, and the uniform-$\sigma^2$ heat clock stalls a $\sigma_{\min}$-independent distance from the data.

On two solvable models (rank-deficient Gaussian, symmetric two-point mixture), deterministic samplers remain first-order uniformly accurate with no $\log(1/\sigma_{\min})$ factor, even across a symmetric posterior-switching interface whose distributional budget is a universal constant; the logarithm is charged entirely to the Itô term of stochastic samplers, whose path-KL scales as $\Lambda^2/N$ against the ODE's $O(\Lambda^2/N^2)$ budget, with $\Lambda=\log(\sigma_{\max}/\sigma_{\min})$.

On the EDM CIFAR-10 checkpoint, spectra measured once predict held-out residual budgets across step count, schedule, and noise level against pre-specified gates with no per-configuration refitting, and calibrate the Itô coefficient at $M_1=1.00\pm0.01$.

The clock decides stability; the noise, not the geometry, charges the logarithm.