Spectral Representation of Causal Memory Dynamics: A Practical Guide for Implementation

We develop a spectral formulation for causal memory dynamics generated by linear semigroups. The approach applies uniformly to classical systems and open quantum systems governed by non-self-adjoint generators. Observable time dependence is shown to be fully determined by the spectral structure of the generator, including decay rates, oscillatory modes, andequilibration timescales characterized by the spectral gap. We provide complete algorithmicprocedures, test cases, and parameter guidelines for immediate implementation. A fully reproducible numerical implementation is publicly available.