Interaction Degree Predicts When Nonlinear Capacity Improves Neural Operators

Neural-operator capacity is usually chosen from errors observed after training, although the target function itself determines whether nonlinear interactions are needed. We show that interaction degree, measured from reference-normalized response profiles, predicts when added nonlinear capacity will improve accuracy. Whitening removes invertible affine coordinate effects, and homogeneity carries the profile across input radii. Complete homogeneous regression with independently sampled signed rays recovers synthetic polynomial support, agrees across viscous Burgers regimes, and keeps a matched heat operator at degree one. Across 27 Burgers regimes, interaction degree correlates 0.986-0.998 with the gain from a denser polynomial approximation. Across Burgers and Allen-Cahn, the profile identifies 10 of 12 beneficial settings near the training radius; under larger shifts, the mean gain reverses. Interaction degree thus directs nonlinear modules to regimes where they help and warns when distribution shift makes the same capacity harmful.