The Eigenvalue Spectrum of Atomic Structure: Physical Origins of Chemical Bonds and the Periodic Table

The statistical interpretation of the wave function in quantum mechanics has achieved great mathematical success, but its popular presentation has led to a widespread intuitive misunderstanding — electrons are imagined as diffuse probability clouds surrounding the nucleus, rather than as stable entities confined to specific geometric boundaries. Starting from two geometric axioms, this paper establishes a geometric dynamics framework (GDUT) based on the evolution of three-dimensional spatial curvature, reducing the electron to a stable soliton solution of the modified Ricci flow in a spherically symmetric potential well. The derivation shows that electron shell radii obey an integer eigenvalue spectrum of n^2, the number of electrons in a given shell is constrained by spherical surface density, and a filled shell corresponds to a geometrically rigid state. This framework provides a unified geometric explanation for atomic structure and chemical bonds that requires no externally imposed quantum assumptions, and its predictions are in excellent numerical agreement with experimental data.