Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery

We introduce the ${\star }_G$ tensor algebra, in which any finite group $G$ defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the ${\star }_G$-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing $F_G$ with $F_{G_1}\otimes F_{G_2}$ with no architectural redesign; and (iii)~a 600-line Lean~4 formalization of the ${\star }_G$ algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 molecular geometry over the chiral octahedral subgroup of SO(3) recovers the Wigner–Eckart selection rules of angular momentum from data alone, with no quantum mechanical input: scalar properties are A${}_1$-dominated, dipole components are T${}_1$-dominated, the isotropic polarizability is uniquely insensitive to $l\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}1$ as the rank-2-trace decomposition $l\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}0\oplus l\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}2$ requires, and the T${}_1$/A${}_1$ predictive-power ratio separates vector observables from scalar observables by a factor of five. On full QM9 (130{,}831 molecules), ${\star }_G$-SVD with ridge regression provides closed form predictions at $\sim 50-90\times$ fewer parameters than parameter-matched MLPs. Algebraic equivariance thus complements architectural equivariance not as a faster-better-cheaper alternative but as a different mathematical affordance: provably-optimal symmetry-preserving compression, per-irrep interpretability, and data-driven physical discovery.