Nonasymptotic diffusion analyses often decompose sampling error into score estimation, continuous reverse-time propagation, discretization, and terminal conversion. We isolate the propagation module on certified scalar-isotropic reverse-SDE windows, with terminal quadratic-Wasserstein reporting as the goal. The propagated object is not $W_2^2$, but an affine-tail transportation cost adapted to the learned drift. Reflection coupling exposes the learned reverse drift through a worst-case pairwise radial profile and reduces stability to a one-dimensional comparison. This reduction separates consistency from stability. Score-modeling and solver residuals quantify error injection and enter as additive forcing; radial load--reserve geometry quantifies error amplification and supplies the Wasserstein stability certificate. The obstruction is a barrier: an increasing concave cost must spend slope to cross adverse radial load before exploiting a contractive tail reserve. Hardy capacity measures this bottleneck, finite load before reserve yields an explicit affine-tail cost, and the main theorem propagates this adapted cost with separate score, solver, geometry, and terminal-reporting inputs. Terminal tails, moments, or bounded support are used only afterward to convert the affine-tail bound into $W_2^2$. The framework recovers uniformly dissipative propagation, converts bounded-amplitude perturbations into finite inverse-radius load, and gives analytic certificates for common-covariance Gaussian-mixture smoothing windows. We also prove that one-sided adverse height, even with eventual reserve, does not determine the radial Hardy scale, and realize this separation by smooth one-dimensional drifts. For fixed learned drifts, we provide deterministic and PAC compact certification templates.

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