The parity gap in crystal tensor prediction

Abstract

Crystal symmetry dictates whether a physical response tensor must vanish, establishing a direct test for machine learning predictions independent of property calculations. We derive the parity gap, a group-theoretic metric quantifying the piezoelectric tensor freedom permitted by a crystal's proper rotation subgroup $SO(3)$ but eliminated by inversion symmetry in $O(3)$. Across state-of-the-art equivariant neural network architectures, unconstrained $SO(3)$ models systematically predict forbidden non-zero responses matching the parity gap of each centrosymmetric crystal class, while polar distortion paths dynamically map output responses to the loss of inversion symmetry. Regression controls confirm that enforcing full $O(3)$ parity incurs no consistent accuracy cost across predictive tasks. Crucially, while training interventions using explicit zero labels reduce violation magnitudes, they leave residual forbidden outputs. Exact physical compliance instead requires structural enforcement through $O(3)$ representation design or explicit output antisymmetrization. The parity gap thus provides a unified framework to distinguish empirical error reduction from exact structural compliance with physical law.