Core–Tail Origin of Electromagnetic and Gravitational Couplings in the TUO–NS Framework.

In this work, we show that electromagnetic and gravitational interactions emerge from two distinct geometric regimes of a single saturated vacuum mode within the TUO–NS framework. Starting from a reconstructed vacuum operator and its unique $H^1(\mathbb{R}^3)$-admissible localized solution identified with the electron, we demonstrate that the nonlinear saturated core governs atomic structure and yields electromagnetic coupling as a first-order geometric invariant. The fine-structure constant $\alpha$ is identified as a pure dimensionless ratio $\alpha = \mathcal{Q}_\star / \mathcal{J}_\star$, where $\mathcal{J}_\star \approx 355.287$ is the universal saturation invariant. In contrast, gravitation originates from the exponentially suppressed asymptotic tail of the same mode and appears as a second-order elastic response of the vacuum. By computing the interaction energy between two saturation centers from the overlap of their asymptotic tails, we show that the resulting Yukawa interaction reduces smoothly to a Newtonian $1/r$ potential in the near-critical regime of the vacuum operator, where the screening length becomes macroscopically large. Within this controlled macroscopic limit, the gravitational constant $G$ is identified as a derived quantity governed by the square of the asymptotic tail amplitude, rather than introduced as a fundamental parameter. This unified core–tail mechanism explains the large hierarchy between electromagnetic and gravitational couplings as a structural consequence of vacuum saturation, without invoking point particles, independent interaction fields, or fine tuning. The numerical foundations of this work, including the extraction of the asymptotic parameters and the $H^1$ stability audits, are supported by the reproducible scripts and datasets archived in the previous installments of the TUO–NS program (Articles A, B and C).