Mathematical Derivation and Standardized Empirical Verification System for Atomic Orbits

This paper presents a self‑consistent mathematical derivation and standardized empirical verification system for atomic orbitals, built upon first‑principles geometric postulates of the vacuum quantum medium. A fourth‑power scaling relation is rigorously derived from four‑dimensional geometric projection, together with reference constants defined using the proton Compton wavelength. Spherical‑wave geometric corrections and a layered orbital recurrence relation are constructed to describe three‑dimensional atomic orbital hierarchies, yielding exclusive falsifiable predictions for high‑lying Rydberg states. A classical dynamical picture for electron motion inside confining geometry is proposed, and a piecewise radial probability‑density function is given with closed‑form integral verification. A feasible, reproducible spectroscopic experimental protocol is provided to test core theoretical predictions. This framework achieves mathematical closure and offers alternative interpretations for atomic‑scale phenomena, compatible with established experimental observation workflows.