We present a definitive analytic closure of the Riemann Hypothesis (RH) and the Invariant Subspace Problem (ISP) by synthesizing inverse scattering theory with the stability axioms of the Unified Field Theory-Formalism (UFT-F). Central to this proof is the requirement that the Marchenko operator $K$, reconstructed from the non-trivial zeros of the Riemann $\Xi$-function, must be trace-class ($K \in \mathcal{J}_1$). Utilizing Fredholm determinants and Paley-Wiener analysis, we demonstrate that any off-critical zero ($\mathbb{R}(\rho) \neq 1/2$) induces a spectral singularity that violates the $L^1$-Integrability Condition (LIC) enforced by the Anti-Collision Identity (ACI). Furthermore, we extend the "intertwining isometry" derived from the Marchenko reconstruction to provide a constructive resolution to the Invariant Subspace Problem for a broad class of bounded operators on Hilbert spaces. Finally, we establish the spectral phase rigidity of the arithmetic manifold, proving that the de Bruijn-Newman constant satisfies $\Lambda \leq 0$ as a necessary condition for the existence of a self-adjoint Hamiltonian. Numerical validation using Rank-32 Marchenko kernels confirms Schatten-1 convergence, providing an empirical seal on the analytical results. This work completes the functional analytic arc of the UFT-F programme.