Fully Symmetry-Conditioned Rigid-Body Flow Matching for Molecular-Crystal Structure Prediction

Rigid-body flow matching is an efficient route to molecular-crystal structure prediction, reducing a crystal to a lattice, fractional centroids, and per-molecule orientations on SO(3). Recent rigid-body molecular-crystal flows leave space-group symmetry unconditioned, and symmetry-conditioned generation has so far been demonstrated only for inorganic atomic sites, never for rigid-body molecular orientation. We show that the per-molecule orientation target decomposes as Rm = rot(gm) Rasym into a space-group-determined relative rotation, which a flow can learn, and a gauge-free asymmetric-unit pose, which it cannot regress from packing. We turn this into a deployable method: a leak-free coset label—the generating space-group operation, recoverable from a template at sampling time—conditions the orientation flow. On 1,127 crystals from the Cambridge Structural Database, deployable coset conditioning lifts the held-out non-reference orientation loss by 41.1% (versus 27.5% for a paired no-coset control), recovering two-thirds of the way to a leaky-codebook oracle (∼ 48%); an SO(3)-averaged training objective closes the remaining gap (47.7%). A packing-only predictor recovers the coset at 39.5% (4× the majority baseline), not yet enough for template-free use—the key remaining gap. Extending symmetry conditioning to the lattice (a crystal-family mask on a log-metric parametrization) gives, to our knowledge, the first fully symmetry-conditioned molecular-crystal flow; with an unsupervised symmetry-preserving finisher and a fully-ablated lever stack, it lifts held-out exact match from 0% to 6.9% (strict best-of-10, stol 1.0)—parity with the symmetry-free MolCrystalFlow (∼8%)—and 10.7% with orientation test-time augmentation (Section 3).