Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules

Abstract

Symmetry determines which physical responses of molecules and crystals are allowed and underlies many models used throughout the physical sciences. Yet several widely used machine-learned models learn symmetry only approximately from data, raising the question of whether approximate symmetry is sufficient for scientific prediction. Using the Orb-v3 force field, we show that it is often not: the relaxed space group of a crystal depends on the orientation of the input for $47\%$ of a $150$-crystal subsample, and relaxation lowers the space group for half of all $399$ crystals studied. To characterize, quantify, and correct such failures, we introduce a tensor algebra, $\star_G$, defined by convolution over a finite symmetry group and applicable to tensors with noncommuting entries. We prove that the symmetry group determines the algebra's block-diagonal transform and use this structure to analyze model errors on symmetric targets. The resulting projections decompose the average squared error into orthogonal components arising from selection-rule violations, inconsistencies across symmetry-related inputs, and residual error. The removable components reveal how much error arises from broken symmetry, provide reference-free lower bounds on prediction error, and can be eliminated without sacrificing accuracy. For operators, projection followed by blockwise truncation yields the optimal symmetric low-rank approximation. Applying these projections to force predictions reduces both rates of space-group change to at most $2\%$. Exact selection rules, reference-free error bounds, and symmetry-preserving corrections are therefore available for any trained model without retraining.