Estimating the Local Jacobian of an Aqueous System from Perturbation-Response Spectroscopy

Near-infrared spectroscopy of aqueous systems currently yields a state: a vector of band intensities describing what is present in the sample. We show that the same measurement procedure, without new instrumentation, also yields the local generator of the system's evolution — a matrix whose spectrum contains the relaxation times of collective modes, the structure of inter-band coupling, the distance to a loss of stability, and the minimum energy required to steer the system to a target state. We define the state vector on the standard twelve-band WAMACS grid, linearize the dynamics about a stationary point, and give three mutually independent estimators of the resulting matrix: from free relaxation after a step perturbation, from the autocovariance of unperturbed fluctuations, and from the static susceptibility measured across stationary points. The three estimators use three different experiments and must agree; their disagreement is a falsification criterion, not a fitting residual. We state explicitly that the estimated object is not the Jacobian of the molecular dynamics but a finite-dimensional Koopman projection onto the chosen observables. Two by-products follow from the same estimate: a dimensionless index of detailed-balance violation, computable from the fluctuation route alone, and a controllability Gramian that turns "minimal intervention" from a qualitative principle into a number with units. The estimators are validated against systems whose generator is known exactly: the fluctuation route recovers all five slow modes to 8–10 % down to 10 dB, the relaxation route carries a projection bias that longer records do not remove, and the non-equilibrium index requires a record of at least (2/η)² correlation times before a current of strength η becomes distinguishable from a finite-sample floor. Six observations are listed that would refute the method. No experimental data are reported; the paper is a method, a preregistered protocol and its synthetic validation. Preprint.