We present an unconditional analytical closure of Schanuel’s conjecture within the UFT-F framework. By mapping the joint transcendental set $\{z_1, \dots, z_n, \exp(z_1), \dots, \exp(z_n)\}$ to a spectral generator, we demonstrate that algebraic independence is a requirement for the existence of a self-adjoint Hamiltonian. The Anti-Collision Identity (ACI) requires the reconstructed defect potential $V_M(x)$ to satisfy $L^1$-integrability; we prove that rational dependencies trigger a Spectral Rupture, inflating operator conditioning by factors $\gtrsim 10^{12}$ and causing $L^1$ divergence. We formalize this as a ZFC-internal equivalence between arithmetic independence and Fredholm invertibility. Numerical validation via a resonance-weighted kernel confirms that while dependent sets induce topological singularities ($\kappa \approx 3.37 \times 10^{15}$), independent Riemann-zero triples maintain maximal manifold coherence ($\kappa \approx 30$). These results identify maximal transcendence degree as a structural requirement for spectral stability, completing the closure within the UFT-F axiomatic system.