Orbit weight sharing via group-theoretic orbit decomposition for crystal property prediction

Orbit-conditioned transforms offer a simple way to introduce discrete site information into graph-free crystal-property models. We establish a valid cube-centred action on a 4 × 4 × 4 voxel grid, verify all 48 bijections and 2304 compositions, and obtain four orbits of sizes 8, 24, 24 and 8. An exact element–label histogram factorization identifies the information available to this architecture and its limitations. Controlled experiments on 2741 archived crystals use a common composition-family split, five initialization seeds and validation-selected checkpoints. Formation-energy MAE is 0.2000 eV/atom for Cube4, 0.1892 for a shared-transform model, 0.1826 for a deterministic geometric partition, 0.1962 for a parameter-matched crystallographic stabilizer-order encoding and 0.1415 for a wide shared model. A separate protocol diagnostic increases the shared-model MAE from 0.1917 to 0.9655 eV/atom when a common elemental projection is replaced by independent per-crystal projections. This identifies a substantial source of the archived error scale without claiming exact historical reproduction. Band-gap and space-group-holdout checks do not establish a Cube4 advantage. The study provides an auditable orbit-conditioned parameterization and shows why mathematical validity, representation content and capacity-matched comparisons must be established separately before attributing predictive gains to symmetry.