A Relativistic Dirac-SUSY Model for the Riemann ξ(s) Function: Analyzing Spectral Mapping and Metric Invariance

We present a novel, mathematically rigorous relativistic quantum mechanical toy model that connects the Dirac equation with analytic number theory. By constructing an energy-dependent matrix Hamiltonian under a supersymmetric (SUSY) algebraic framework on the positive half-line R+, we bypass the divergence and integration inconsistencies found in standard linear models such as the Berry-Keating and Bender frameworks. Through an exact application of the integral Mellin transform, theprincipal spinor component maps bijectively onto the Riemann ξ(s) function without heuristic structural regulators. Finally, we analyze the CPT metric charge operator Cˆ and outline the explicit mathematical boundaries regarding operator linearity, external spectral constraints, and metric non-locality, positioning this framework as a structured phenomenological foundation for future research in quantum number theory