A Discrete Nested Spherical-Shell Atomic Model of Dark Matter Fluid

Contemporary quantum mechanics relies on phenomenological postulates—such as probability amplitudes and quantization rules—yet lacks a microscopic dynamical underpinning. Concurrently, the absence of a microscopic origin of gravity creates a fundamental disconnect between general relativity and quantum theory. To bridge this divide, we propose a topological bound-state model rooted in a single foundational assumption: a global, inviscid, compressible dark-matter fluid governed by classical hydrodynamics. By imposing experimental constraints—including atomic spectra, the inverse-square law of gravity, and lepton mass hierarchies—we derive a unique double-square orthogonal shell topology via analytic volume differencing. This yields a five-tier nested spherical shell model that inherently satisfies radial square-normalization constraints.The model identifies a universal geometric factor (3π/2), which serves as a characteristic signature of the underlying topological defect fusion rules. This factor dictates the quantization condition (or N^2=(3π/2)^2⋅α^(-2) ), establishing a precise correspondence between the topological shell number Nand the fine-structure constant α. Within this framework, fundamental interactions emerge as hydrodynamic coupling effects induced by asymmetric nested topologies, while the Cosmic Microwave Background is interpreted as a macroscopic byproduct of gravitational focusing within dark-matter sink fields. Crucially, the governing equation for the vortical dark-matter field is shown to be mathematically isomorphic to the stationary Schrödinger equation. This reveals the fluid-topological origin of the wave function and posits the Schrödinger equation as an effective description of the dark-matter fluid under topological boundary constraints. Our results offer a self-consistent theoretical framework capable of unifying macroscopic and microscopic observations, providing a quantitative pathway toward addressing longstanding foundational challenges in physics.