FIXED-CARDINALITY JOINT DIFFUSION FOR COMBINATORIAL PROTEIN DESIGN

Fixed-cardinality design seeks a batch of sequences with exactly K mutations relative to a reference, a constraint libraries impose at cloning that post-hoc filtering cannot restore. We introduce FC-JD (Fixed-Cardinality Joint Diffusion), a two-stream diffusion model. Its position stream diffuses mutation masks on the Johnson graph J(N, K), making exact-K a construction invariant. Its simulation-free objective replaces forward rollouts with an analytic posterior over reverse transitions. We answer three questions. (Q1) Can FC-JD guarantee exact-K generation without post-hoc correction? (Q2) How does FC-JD compare with search-based methods (simulated annealing, Bayesian optimization) under the same batch budget? (Q3) When is FC-JD useful, and what are its limitations? In a pre-registered, oracle-scored benchmark (three complete-oracle landscapes, one partial-oracle case study, one regression-only dataset, four arms), FC-JD produced exact-K designs in every round, achieved substantially higher valid rates on partial-oracle landscapes, delivered batches in under two seconds, and descriptively matched or exceeded GP-BO on the extreme tail in every oracle-evaluable cell at K ∈ 5, 7. Surrogate-guided simulated annealing retained the best mean fitness on all four oracle-evaluable landscapes, and pre-registered criteria confirm this division of labor: FC-JD meets the generation criteria (valid, time) while SA96 keeps the best mean. We position FC-JD as a fast, diversity-preserving generator for exact-K design. Additionally, ESM-2-conditioned variants and cross-target transfer are reported as supporting analyses.