Cross-Scale Geometric Threshold for Dynamical Stability of Hierarchical Multi-Star Systems

This research is based on the core hypothesis that physical laws share homologous characteristics across microscopic and macroscopic scales. By establishing an equivalence relation between microscopic particles and macroscopic celestial bodies, this paper argues that the reciprocal of the fine-structure constant α≈1/137 not only underpins microscopic quantum stability but also functions as a universal geometric threshold governing gravitational equilibrium in large-scale stellar systems. With the construction of a spatial geometric framework, the Spatial Geometric Balance Center O_G is defined alongside a dynamic boundary expansion mechanism. Retrospective cross-check against astronomical observational data shows that the scaling ratios of gravitationally stable astrophysical systems converge near the numerical value 137. Using a simplified orbital-plane model, the study establishes quantitative zoning criteria: absolute stability, metastability and orbital instability separated by the 137 threshold, alongside calibrated baseline characteristic frequency 72.5 and fundamental equilibrium position L0. Practical verification confirms the framework accurately accounts for the long-term orbital stability of the 61 Cygni binary system and enables structural prediction for the Sirius system, solidifying 137 as a cross-scale fundamental cosmic constant.