Geometric Model of Atomic Orbits Based on Hyperbolic Functions and Interdisciplinary Invariant of Material Stability

An innovative relativistic approach to calculating the energy states of atomic orbits based on a system of hyperbolic functions (the Boltian model) is proposed. It is demonstrated that Sommerfeld's fine-structure constant strictly defines the spatial angle for the first energy state, whereas the conventional Bohr quantum formula is merely a special limiting case of the derived general geometric dependence. By employing a modified expression for the half-Lorentz factor, precise agreement of theoretical calculations with fundamental constants (Rydberg constant, Hartree energy) and experimental data for multi-electron systems is achieved. A universal spatial angle of material stability (≈ 33.26^∘ - 34^∘) has been discovered. It acts as a shared invariant for the stability of atomic nuclei, the spatial architecture of DNA, and geomechanical macro-systems, pointing toward a unified geometric nature of the Universe.