Finite-Step Law Error and Learned Score Risk in Cahn–Hilliard–Cook Diffusion

Physical simulators are increasingly used to supply training data for score-based generative models. A simulator can be stable and exactly conservative while still sampling the wrong distribution, and a score model trained on its output inherits that error. We give a quantitative account of this chain for Cahn–Hilliard–Cook (CHC) dynamics, from integrator to sampled law to learned score. For linearized CHC, the discrete Lyapunov equation gives the per-mode bias of an energy-stable IMEX step in closed form (up to 4.6× excess high-frequency variance), and a discrete fluctuation–dissipation covariance removes it at no cost. Under a shared Gaussian corruption, the excess target risk of learning the simulated law is a covariance-weighted relative Fisher divergence; we derive its exact fixed-time form and a risk-elasticity identity that predicts sub-quadratic growth at practical step sizes and a larger penalty for under- than over-dispersion. On a nonlinear 15 × 15 CHC Gibbs field at fixed physical time, coarser steps increase endpoint error and held-out target-DSM excess for a CNN and an FNO in all ten paired seeds, following the predicted pattern: law error ∝ h1.0 and risk exponents falling from about 1.9 to 1.5. Generation and inverse problem tests show where these gains carry over and where the sampler decides the outcome