EPOCH: Explicit Projection on Concentric Shells for Multi-Order Molecular Geometry

For many molecular properties, accurate 3D prediction depends on geometric information across multiple angular scales, from coarse global shape to fine directional detail around local environments. Modern equivariant networks expose this structure through spherical harmonics up to a chosen maximum order $l_{\max }$, but tensor-product cost grows as $\mathcal{O}(l_{\max }^6)$ per interaction, pushing practical models to a low-order operating point even when chemistry suggests finer angular content matters. We argue that raising $l_{\max }$ is not the only access path to angular structure: finite directional sampling can empirically preserve a useful mid-frequency angular band beyond the practical low-order operating point, at cost linear in the number of probes rather than polynomial in equivariant order. We instantiate this view as EPOCH (Explicit Projection on Concentric Shells), a shared geometric descriptor obtained by sampling a shellwise zonal-kernel response field, precomputed once from coordinates and inserted into scalar, angular, coordinate-equivariant, and tensor-equivariant backbones through lightweight backbone-aware fusion. On QM9, comparisons across SchNet, DimeNet++, EGNN, NequIP, and the strictly local equivariant Allegro show that this shared descriptor helps within heterogeneous backbone families, with the clearest gains concentrated on angular-sensitive electronic targets. Parameter-matched HOMO controls show that preserving response-to-direction correspondence improves over capacity-matched variants, while also clarifying that the effect is target-dependent. Order ablations on a tensor-equivariant backbone further show that EPOCH continues to help as native equivariant order increases, and that augmenting a low-order backbone with EPOCH can often match or remain competitive with a higher-order baseline at a fraction of its parameter and compute cost. Cross-dataset, cross-task evidence on MD17 further shows that the same design improves force prediction. Together, these results indicate that explicit spherical sampling can provide a portable, target-dependent complement to tensor-product equivariance rather than a replacement for it.