Budgeted Quotient-Residual Guidance for Frozen Pocket-Conditioned Molecular Diffusion

Pocket-conditioned molecular diffusion updates ambient atom coordinates, but many lead-optimization objectives are expressed on quotient features such as distances, contacts, and anchored substructures. We introduce budgeted quotient-residual guidance (QRG), an inference-time correction that makes these quotient objectives active without retraining the molecular generator. QRG lifts quotient covectors to metric-horizontal ambient directions and delivers them through a trust budget set by the frozen sampler's own step norm: quotient geometry chooses the direction, while sampler motion bounds the scale. We derive the horizontal lift, closed-form sampler-budget update, KL/kinetic interpretation around a frozen reverse step, equivariance conditions, and a product-budget split for budget-capped section and residual controls. Controlled quotient tasks confirm that sampler-relative delivery activates signals that raw local quotient gradients leave dormant. On frozen TargetDiff backbones, official seed-0 CBGBench ligand-generation/editing sweeps show practical quality-runtime gains: Local-QRG improves validity from 0.815 to 0.864 on fragment growing, 0.664 to 0.707 on scaffold hopping, and 0.681 to 0.712 on linker design, while PredNext-QRG improves fragment/scaffold and remains near-neutral on linker. Novelty remains 1.000 and diversity is preserved in the matched multi-seed molecular slice, giving task-dependent improvements without sampler retraining or backbone modification. Overall, QRG provides a lightweight route to quotient-aware inference for frozen molecular samplers with explicit runtime accounting.