No-Harm Missing-Friction Correction for Potential-of-Mean-Force-Matched Coarse Graining.

Potential-of-mean-force (PMF)-matched coarse graining (CG) can make a retained coordinate structurally correct but dynamically wrong. Eliminated bath degrees of freedom carry projected memory and friction that control response curves, relaxation times, kinetics, and transport. We ask whether the observable part of this missing friction can be recovered cheaply without harming generated dynamics. Residual-force fits and scalar rescalings alone cannot answer that question because they may expose the signal or improve one summary while worsening another. We distinguish the measured finite-window error between the high-resolution reference (labeled AA throughout) and CG dynamics from the component supported by residual-memory diagnostics, formulate that component as an observable projected-memory object, and approximate its consequence with a finite-history or surrogate candidate. The candidate is deployed only when held-out continuation rollouts improve both response-curve and relaxation-time errors. In two complete experiments, the fitted increment was added to the same detailed-balance PMF transition used as the uncorrected baseline. For C6 and C5 end-to-end distances, selected strength 0.10 retained PMF histogram L1 distances of 0.054 and 0.061, and the one-sided 5% lower bounds of both dynamical error reductions were positive. Separate precorrection diagnostics assigned acetone C-O to a richer state description and ethanol heavy-atom end-to-end to a finite memory kernel, so neither received the low-cost correction. The 13-coordinate breadth panel and diagnosis-informed tests provide breadth and selectivity evidence. Against a C6-calibrated scalar clock, the finite-history candidates improved both errors in 4/4 point-estimate comparisons versus 3/4 and gave 2.8 times the mean joint gain; experiment-specific uncertainty is reported separately. A distinct OpenFF 2.3.0/OpenMM surrogate showed the same point-estimate improvement on four molecules. The resulting reproducible, cost-bounded rule corrects only when structural and rollout evidence support correction.