Spin-Weighted Spherical Harmonics Enable Complete and Scalable \mathrm{E}(3)-Equivariant Networks

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 Lean4 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 $l\!=\!1$ as the rank-2-trace decomposition $l\!=\!0 \oplus l\!=\!2$ 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 $\sim50-90\times$ 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.

SO(3) equivariant graph neural networks have become the dominant paradigm for atomistic foundation models, achieving high accuracy and data efficiency by building rotational symmetry directly into the architecture. Yet the computational cost of their higher-order tensor operations creates a tough trade-off between model accuracy and inference efficiency. In this paper, we propose a structural pruning method for SO(3) equivariant atomistic foundation models to bridge this accuracy-efficiency gap. The pruning is applied along the channel and order dimensions, with each irreducible representation kept or removed as a complete block, thereby retaining SO(3) equivariance. Starting from a large checkpoint, the pruned model substantially reduces the inference cost while retaining higher accuracy than an independently trained small model. The pruned MACE-MP model outperforms the official from-scratch trained small model on 7 of 9 metrics on the Matbench Discovery leaderboard. In terms of efficiency, compressed MACE-MP and MACE-OFF models contain 1.5$\times$ to 4$\times$ fewer parameters and require 2.5$\times$ to 4$\times$ less pre-training compute than training a small model from scratch. For downstream applications, fine-tuning the pruned model reduces energy and force errors by 70.1% and 34.4% compared to training task-specific models from scratch across eight representative downstream datasets. We demonstrate that the method generalizes to other SO(3) equivariant architectures (SevenNet, eSCN) and can be combined with quantization and knowledge distillation for further gains.

Equivariant graph neural networks provide a powerful modeling language for three-dimensional scientific data, but their reuse is often limited by implementations tied to specific tasks, outputs, and training regimes. We present GEqTrain, a configuration-driven framework that separates dataset semantics, model composition, and training objectives. Raw data are mapped to typed node-, edge-, and graph-level fields, while model stacks, losses, and training workflows are assembled declaratively through Hydra configurations. A shared equivariant backbone and training infrastructure can therefore be retargeted to a new task primarily through configuration. We demonstrate this flexibility on three different problems handled within one software stack: coarse-grained-to-atomistic backmapping of biomolecular systems, prediction of NMR chemical shifts in molecular solids, and equivariant generative modeling. Our aim is not to surpass individually optimized task-specific systems, but to show that a shared representation and training infrastructure can achieve competitive accuracy across qualitatively different tasks at the cost of a configuration change. We further introduce GEqDiff, a generative extension based on equivariant flow matching. GEqDiff treats user-defined equivariant fields as first-class generation targets, jointly transporting Cartesian positions and non-scalar node fields spanning representations up to l=3 within a single equivariant flow. We validate this capability on a controlled synthetic benchmark inspired by protein secondary-structure motifs, showing that fields with heterogeneous transformation properties can be reconstructed jointly and with high fidelity. By reducing the software overhead of moving between predictive and generative, scalar and tensorial settings, GEqTrain aims to make equivariant modeling more reproducible, extensible, and reusable.