Geometric Reconstruction in the Hydrogen Atom and a Possible Micro–Macro Scaling Relation

This work investigates a geometric reconstruction model for the hydrogen atom in which the observable orbital motion of the electron is interpreted as a projection of an internal trajectory propagating at the speed of light. The model introduces a mixed geometric scale defined by the geometric mean of the Bohr radius and the classical electron radius. From this scale, a reconstruction number naturally emerges and reproduces the known orbital velocity relation of the hydrogen atom. In this framework, the inverse fine-structure constant appears as a geometric reconstruction quantity associated with the relation between orbital scale and electron width. The analysis also identifies the same dimensionless structure in the classical gravitational compactness parameter, suggesting a possible geometric link between atomic binding and gravitational binding regimes. The work proposes that geometric reconstruction may provide a useful bridge between microscopic atomic structure and macroscopic gravitational geometry.