Emergence of Conversion Constants from Vacuum Saturation in the TUO–NS Framework.

In this work, we complete the TUO–NS program by demonstrating how both electromagnetic and gravitational couplings emerge from distinct geometric regimes of a single saturated vacuum mode. Starting from the reconstructed vacuum operator (Article A) and the unique $H^1(\mathbb{R}^3)$-admissible saturation profile identified with the electron (Article B), we show that physical constants are not fundamental inputs but conversion factors and geometric invariants of vacuum dynamics. The fine-structure constant $\alpha$ is identified as a pure dimensionless ratio $\alpha \equiv \mathcal{Q}\star / \mathcal{J}\star$, where $\mathcal{J}_\star \approx 355.287$ is the universal saturation invariant characterizing the nonlinear core of the localized mode. Conversely, gravitation is shown to originate from the exponentially suppressed asymptotic tail of the same mode, $u(x) \sim A_\infty e^{-\kappa x}/x$. In the near-critical regime, this tail yields a controlled Newtonian limit over macroscopic distances. This unified core–tail mechanism explains the hierarchy between electromagnetic and gravitational interactions as a structural consequence of vacuum saturation, eliminating the need for point particles, fine-tuning, or renormalization. Supplementary material provides reproducible numerical diagnostics (Python/CSV) auditing the extraction of $A_\infty$ and $\kappa$.