Continuum-Equivalent Neural Operators Can Learn Different Interactions

Neural operators should recover the same physical interaction as input resolution increases, yet vanishing weight decay can erase interactions that remain fully expressible in the continuum. Guarantees of continuum expressivity and consistent representation cannot predict this selection: a fixed interaction may require diverging coefficients, and standard parameter penalties charge those coefficients rather than the finite interaction they encode. This mismatch makes architecture and regularization decisive for whether learned continuum models preserve the underlying physics. We establish a realization-cost principle that characterizes consistent recovery through target approximations with vanishing penalized parameter cost. Counterterm-coupled regularization implements this principle in finite Wick coordinates, preserving the intended interaction objective and restoring consistency. We demonstrate the mechanism using cubic responses of two-dimensional Gaussian free fields with a fixed continuum target and smoothing observation, where continuum-equivalent architectures erase or recover the same target under identical vanishing weight decay. For standard ReLU channel networks, the principle yields the sharp threshold ${\lambda }_{\epsilon }{C}_{\epsilon }^{3/L}\to 0$, linking penalty strength ${\lambda }_{\epsilon }$, field variance $C_{\epsilon }$ at input cutoff $\epsilon$, and depth $L$. Exact population analysis, matched architectural controls, and independently verified polynomial-network training demonstrate the predicted separation and its correction at moderate resolutions.