RiMES: Riemannian Flow Matching with Equivariance and Stereochemical Regularization for Antibody CDR Co-Design

Antibody complementarity-determining region (CDR) co-design requires compatible sequence and structural predictions under a specified framework and antigen context. We present **RiMES** (Riemannian flow Matching with Equivariance and Stereochemical regularization), an iterative full-atom co-design model that couples geometric message passing to Riemannian flow matching on residue frames. A local-frame velocity head supplies translation corrections between refinement rounds, while a differentiable rotamer-compatibility term and a soft C$\alpha$ overlap penalty connect sequence prediction to local geometry. The flow objective uses mixed prediction- and noise-based starts with detached targets and shared encoder representations. Under known-interface conditioning, we report PLM-free amino-acid recovery of 53.5% on RAbD CDR-H3, 77.1% on joint six-CDR design, and 47.9% on the cluster-date-ordered SAbDab evaluation. Component ablations support the mixed-start configuration and intermediate flow feedback, while exposing trade-offs between recovery and local structural accuracy. We separately evaluate parent-conditioned redesign using protein language model embeddings. Masking the designed CDR before PLM encoding and retraining removes the parent-conditioned recovery advantage, demonstrating the importance of distinguishing the two input regimes. Together, these results support flow-assisted conditional CDR co-design, with all-atom packing, antigen-relative placement, and experimental function remaining important limitations.