Bansal Fiber Framework: A Stratified Organization of Schrödinger Electronic Structure

Nonrelativistic Schrödinger quantum mechanics successfully describes electronic structure across physics and chemistry, yet it admits multiple, apparently distinct representations of the electron, including localized particles, delocalized waves, continuous charge densities, fractional occupations, and transient carriers in open or nonequilibrium systems. These descriptions coexist within standard theory but lack a unified structural explanation. In this work, I introduce a stratified geometric framework that organizes Schrödinger-based electronic information without modifying the Schrödinger equation or its solutions. The central idea is to treat the electron as a derived, context-dependent projection of the Schrödinger state rather than as a primitive ontological object. Electronic information is decomposed into structurally distinct components corresponding to charge and occupation (R), phase and coherence (C), transient or virtual structure (D), and orientation-dependent internal degrees of freedom (H). This stratification is formalized using a graded algebra and a fibered geometric construction, referred to as the Bansal fiber. An algebra-valued Schrödinger equation is introduced as an organizational extension of the standard formalism. Its scalar projection recovers the ordinary Schrödinger equation exactly, while controlled nilpotent couplings encode short-lived electronic contributions without altering spectral properties or unitarity. A functional-analytic formulation based on self-adjoint operators and Kato-semigroup theory establishes well-posedness, projection consistency, and leading-order stability. Electron number, localization, and fractional occupation emerge naturally through energetic stabilization of charge-carrying projections. This leads to a nonlinear Poisson-projection equation that expresses electrostatic self-consistency in terms of projection admissibility, connecting the framework directly to quantum chemistry, density functional theory, and self-consistent field methods. The resulting theory provides a unified geometric and operator-theoretic language that clarifies the coexistence of multiple electron models within standard Schrödinger quantum mechanics.