CrystalAR: Autoregressive Modeling for Symmetric Crystal Generation

Deep generative modeling has achieved remarkable progress in crystal generation. Given the rich symmetries inherent in crystalline structures, recent work has increasingly sought to explicitly incorporate space-group symmetry into generative models, enabling more effective exploration of novel symmetry templates. Yet introducing symmetry templates substantially increases the combinatorial complexity of stoichiometric relationships between atomic species and crystallographic sites, while creating a fundamental tension between flexible exploration of discrete crystallographic templates and faithful realization of these templates as continuous three-dimensional structures. To address these challenges, we propose **CrystalAR**, a unified autoregressive framework that models Wyckoff symmetry templates and complete crystal structures within a single generative hierarchy. Specifically, CrystalAR reformulates continuous coordinate generation as a symmetry-conditioned hierarchy of discrete spatial localization decisions, autoregressively refining the positions of all symmetry-inequivalent sites across resolution levels while jointly modeling their spatial configuration within each level. This discrete hierarchical formulation further enables constraint-aware search over both crystallographic templates and geometric realizations. As a result, CrystalAR achieves competitive retrieval-free crystal structure prediction and de novo generation performance comparable to state-of-the-art models, while retaining strong symmetry-template novelty. To our knowledge, CrystalAR is the first symmetry-native approach to simultaneously demonstrate these capabilities, providing a promising alternative paradigm for symmetry-preserving crystal generation.