Spectral Stationarity Predicts Neural-Operator Transfer Across Resolutions

Neural operators may accept a new grid without preserving the physical response learned at the training resolution. We show that spectral stationarity predicts this transfer before target-resolution labels are available. At each physical frequency, it separates the stability of the learned response from its representability on the target grid. This yields a label-free estimator from band-limited probes evaluated on the training grid and one additional grid. In controlled one-dimensional studies, the estimator reduces false acceptance of resolution-anchored models from 68.6% to 2.8%. In a fixed-architecture two-dimensional study, eight probes achieve 0.962 concordance in paired model selection, compared with 0.464 for global training error at identical 0.259 coverage. The analysis also distinguishes failures caused by target-grid sampling, discretization-anchored representations, and uninformative probes. Spectral stationarity therefore identifies unsafe grids and usable frequency bands before labels are generated at the deployment resolution.