Hierarchical Phase Transitions in Bayesian Flow Networks Enable Training-Free Molecular Design.

Bayesian flow networks (BFNs) jointly model atomic coordinates and atom types for structure-based drug design, but their generative dynamics remain poorly understood. We show that the two channels resolve on different terms: atom types pass through a sharp commitment transition at tc1, whereas coordinate precision improves smoothly with no critical point of its own, reaching sub-Ångstrom resolution only at a stated threshold tc2. Extreme-value theory predicts the type channel's posterior half-maximum in closed form, and the predicted 1/β1 scaling holds across seven β1 values (R2 = 0.996) and over 13 points spanning three BFN architectures and two domains, though measured times run 11%-20% below the predicted absolute values. Exploiting that structure, phase-aware iterative refinement aligns three rounds to these boundaries and trains no auxiliary regressor, reward function, or property predictor. On 100 CrossDocked2020 test pockets at K = 200, it reaches a mean Vina Dock of -8.72 kcal/mol. Gradient-guided methods that need such training stay ahead by a small margin: over the 100-pocket intersection, the paired gap to MolJO (-8.98) is 0.25 kcal/mol (95% CI [-0.13, +0.60], paired Wilcoxon p = 0.022); equivalence testing at a preregistered ±0.5 kcal/mol margin does not establish equivalence within it. CByG (-9.16) leads by 0.44 kcal/mol on unpaired means. Phase-aware iteration improves the Round-1 to Round-3 mean by 1.34 kcal/mol (paired Wilcoxon p = 4.80 × 10-18, Cohen's dz = -1.59) and shifts the median by -1.44 kcal/mol, a shift Best-of-K resampling cannot produce. Paired on the pocket, an audited rerun passes every PoseBusters check 4.7 percentage points more often than MolJO.