A Recursive Differential Mapping for the Generation of Normalized Hermite-Gaussian Modes

This paper introduces a unified recursive framework for the generation of Hermite-Gaussianeigenfunctions, identifying the ground-state Gaussian as a fundamental DC reference signal. Whilethe differential properties of the Hermite manifold are well-established in standard mathematicalliterature, this study reinterprets the resulting differential mapping as a topological genera-tor of state evolution. By analyzing the recursive chain of state transitions (ψn → ψn+1), wedemonstrate that the linear wavefunction serves as a physical topological archive of causalhistory, where differentiation represents forward action and integration signifies memory restora-tion. This framework reveals a structural basis for stationary causality and provides a mechanicalnecessity for the Aufbau principle, establishing the ψn function not merely as a mathematical toolfor probability density, but as a physical recorder of phase-space events