Fisher-constrained flow matching for transferable free energy estimation

Cross-pair amortization of alchemical free-energy estimation requires a transferable within-pair conditioning coordinate. We identify the cumulative Fisher arc length ℓalch, the path length along the alchemical path under the Fisher metric, as the unique (up to affine rescaling) parameterizationinvariant coordinate whose equal increments correspond to equal Kullback–Leibler distances between adjacent intermediates. The canonical coupling parameter λ fails this property because the Fisher information varies across pairs. This information-geometric framing makes a falsifiable prediction: the advantage of ℓalch over raw λ should track the quality of the underlying single topology atom mapping, concentrating where the mapping is well-defined. Substituting ℓalch for λ at training time in an otherwise identical conditional 1flow-matching estimator confirms this: improvements on real-chemistry corpora are concentrated on the cleanly-mapped operating regime, with diagnostic synthetic and Lennard-Jones systems verifying the mechanism in controlled settings. We treat the held-out pair’s MD samples as given; predicting ℓalch from pair features to lift this assumption is left to follow-up work.