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.

翻译：非渐近扩散分析通常将采样误差分解为分数估计、连续逆时传播、离散化和终端转换。我们在经认证的标量各向同性逆时SDE窗口上隔离传播模块，并以终端二次Wasserstein报告为目标。传播对象并非$W_2^2$，而是一种适应于学习漂移的仿射尾部运输成本。反射耦合通过最坏情况下的成对径向剖面揭示学习逆时漂移，并将稳定性简化为一个一维比较。这种简化将一致性与稳定性分离。分数建模和求解器残差量化误差注入，并以加性强迫的形式进入系统；径向载荷-储备几何结构量化误差放大，并提供Wasserstein稳定性证书。障碍是一个屏障：递增凹函数必须在利用收缩尾部储备之前跨越不利径向载荷，消耗斜率。Hardy容量度量这一瓶颈，储备前的有限载荷产生显式仿射尾部成本，而主要定理以分离的分数、求解器、几何和终端报告输入传播这一适应成本。终端尾部、矩或有界支撑仅在后处理中用于将仿射尾部界转换为$W_2^2$。该框架恢复均匀耗散传播，将有界振幅扰动转换为有限逆半径载荷，并为公共协方差高斯混合平滑窗口提供分析认证。我们还证明即使存在最终储备，单侧不利高度也不决定径向Hardy尺度，并通过光滑一维漂移实现这种分离。对于固定学习漂移，我们提供确定性和PAC紧致认证模板。