SEAL-BG: Scaling Boltzmann Generation with Equivariant Perception and Local-Coordinate Densities

Estimating protein thermodynamics, including conformational populations and free-energy differences, requires sampling molecular equilibrium ensembles, but correlated trajectories and slow conformational transitions make this costly. Boltzmann generators (BGs) offer independent molecular proposals with evaluable densities, enabling energy-based statistical correction. Autoregressive Boltzmann Generators (ArBG) demonstrate this approach using causal Transformers over Cartesian coordinate tokens, with single-system benchmarks extending to Chignolin (10 amino acids). Scaling ArBG to larger proteins motivates presentations that exploit molecular symmetries while distinguishing local geometry from global conformational variation. We introduce SEAL-BG , an atom-autoregressive BG that decouples geometric perception from probabilistic generation. A causal $\mathrm{SE}(3)$-equivariant geometric encoder encodes the Cartesian coordinates of previously generated atoms, while normalized local-coordinate conditionals model bond lengths, bond angles, and periodic torsions. This separation combines equivariant Cartesian conditioning with structured local-coordinate densities, while analytic local Jacobians preserve an evaluable Cartesian proposal density. We evaluate on four released peptide benchmarks and scale to Trp-cage (20 amino acids, 284 atoms) and Homeodomain (54 amino acids, 947 atoms) using separate within-system ensembles. SEAL-BG achieves competitive energy and torsion distances on the peptide benchmarks and improves raw total-energy overlap relative to ArBG from 0.8651 to 0.8926 on Trp-cage and from 0.4176 to 0.7704 on Homeodomain. Across the evaluated training budgets, SEAL-BG uses geometric-mean factors of $15.2\times$ fewer training floating-point operations (FLOPs) and $18.1\times$ fewer A100 GPU-hours than ArBG. These results support separating Cartesian perception from local-coordinate generation as a route toward scalable Boltzmann proposals.