When Do Local Score Models Extrapolate Across Size? A Diagnostic Theory and Benchmark

Scientific generative modeling often requires size transfer, where models trained on small systems are evaluated on larger ones. While translation-invariant architectures enable this evaluation, we show that architectural locality alone does not guarantee stable size extrapolation. Instead, stable extrapolation depends on how far the Gaussian-smoothed score responds to a perturbation. Even when clean interactions are local, posterior correlations can make the smoothed score respond to distant perturbations. We formalize when this response remains sufficiently local for fixed-receptive-field models to transfer across system sizes, and prove a system-size-independent error bound for generated distributions on fixed local regions. We also introduce Finite-Depth Local Flow (FDLF), a white-box diagnostic benchmark with exact clean scores, densities, and controllable response ranges. Across 2D and 3D FDLF experiments, models transfer stably when the teacher response lies within their receptive fields. A related size-dependent pattern emerges near criticality in the standard 2D Ising model. In a 2D Ising test, the center-score change error of a fixed local model grows with system size near criticality but not away from it.