Reduced Integer–Fractional Dynamics of Hydrothermal Memory in Volcanic Gas and Isotope Signals

Volcanic gas and isotope time series are indirect observables of coupled magmatic and hydrothermal dynamics. We formulate a reduced integer–fractional model in which ordinary differential equations describe deep recharge, pressure, gas-phase volatile inventory, and source mixing, whereas Caputo equations describe shallow hydrothermal pressure, thermal excess, gas pathway effectiveness, permeability, and scrubbing. Under explicit local regularity and admissibility assumptions, the mixed-order Volterra problem is locally well-posed and the physically admissible state set is positively invariant. We derive componentwise dissipative estimates and state conditions for global continuation under bounded trajectories and analyze finite-interval consistency with the integer-order limit and local stability of a frozen commensurate hydrothermal linearization. Conservative observation equations link hidden states to gas ratios, fluxes, and isotope ratios. The inverse problem is treated diagnostically; global identifiability is not claimed. Local sensitivity screening, Fisher information concepts, and scalar recovery tests are used only as preliminary local diagnostics of information content under known or misspecified forcing. Synthetic demonstrations and a reference forward solver illustrate how hydrothermal memory and sulfur scrubbing can reshape carbon dioxide/sulfur dioxide (CO2/SO2) anomalies before site-specific calibration.