Structural Emergence of Atomic Scales from Vacuum Saturation in the TUO–NS Framework.

In this article (B) of the TUO–NS research program, we demonstrate how atomic physical scales emerge directly from the reconstructed vacuum operator obtained in Article A, without invoking empirical constants. A saturation postulate is introduced, stating that the electron corresponds to the minimal stationary configuration admissible in the Sobolev space $H^1(\mathbb{R}^3)$. From this single requirement, the electron rest mass, the Bohr radius, and the hydrogenoid energy spectrum follow as structural consequences of the vacuum spectral gap $\omega_0$. The dimensionless geometric saturation constant $K_M$ is determined numerically by minimizing the universal functional $\mathcal{J}[u,\beta]$ over localized admissible modes, yielding the universal minimum $\mathcal{J}_\star = 355.287146357$. The corresponding saturation profile $u_\star(x)$ is archived as a machine-readable CSV file and verified line-by-line through an independent trapezoidal integration. This work establishes a fully reproducible quantitative bridge between vacuum dynamics and atomic structure, preparing the path toward the derivation of the fundamental constants $c$, $\hbar$, $G$, and $e$ from vacuum saturation itself.