Generative crystallographic phasing through invariant relationships
Crystal structure determination requires the phases of scattered waves -- yet diffraction measures only their intensities. Direct methods exploit phase invariants but become less reliable as diffraction information diminishes. Learned phase prediction has lowered the resolution barrier, yet remains primarily confined to centrosymmetric crystals with binary phases. We introduce PhiGen, a generative reformulation of traditional direct methods that learns origin-independent phase relationships for binary and continuous phasing. Across 210 space groups, including groups absent from training, it recovered high-quality maps for 99.0% of centrosymmetric structures and invariant-consistent phases for 92.8% of non-centrosymmetric structures. From simulated 3 Å zeolite powder data, the network recovered framework maps for 84.2% of held-out structures, versus 1.0% for Superflip. For experimental ZSM-25 and TNU-9, generated phases seeded high-resolution phase extension. These results suggest a route to structure determination from low-resolution, incomplete, and overlapped diffraction data.