Hydrothermal Memory Kernels for Volcanic Gas and Isotope Signals: Volterra–Spectral Structure and Boundary-Aware Synthetic Comparison

Volcanic gas and isotope anomalies can retain delayed effects of shallow hydrothermal storage, transfer and selective chemical removal before they are measured at the surface. A single Caputo order provides a parsimonious memory law, but finite cutoffs, multiple relaxation slopes and discrete exchange domains may require broader descriptions. This theoretical study formulates hydrothermal memory first as a causal Volterra process and distinguishes positive-spectral and Sonine-compatible refinements of that class without assuming that either implies the other. Exact initialized diffusive realizations, positive finite-reservoir approximations, a quadratic storage identity and finite-window kernel-to-observable error bounds are developed. Synthetic experiments compare memory classes through scalar, gas-ratio and isotope responses. The principal statistical benchmark is deliberately restricted to the nested Caputo and exponentially tail-tempered families, thereby testing whether one additional cutoff parameter is supported. In a key negative-control experiment, omission of a slow inherited mode caused AICc to prefer tail tempering in 72.0% of replicates even though the post-observation kernel remained Caputo. This demonstrates that incorrect memory initialization can masquerade as a constitutive cutoff. The study has not yet been validated against real volcanic gas or isotope records, and no site calibration is claimed.