This foundational paper achieves the unconditional analytical closure of the core Modularity Constant $C_{\text{UFT-F}}$ ($\approx 0.003119$) within the Unified Field Theory-F (UFT-F) formal system. This constant is the necessary and sufficient condition for spectral stability, which governs all applications of the UFT-F framework, including its resolutions of the Millennium Prize Problems. The proof establishes a rigorous chain of analytical necessity: The Anti-Collision Identity (ACI) is proven to be equivalent to the $L^1$-Integrability Condition (LIC) for the spectral potentials $V(x)$ via the Gelfand-Levitan-Marchenko (GLM) inverse scattering transform. The LIC is shown to be the condition for the physical existence and stability of the system, guaranteeing the self-adjointness of the associated Schrödinger operator and exponential localization of the eigenfunction (via the Kato-Rellich, Birman-Schwinger, and Agmon theories). The necessary geometric renormalization constant $R_{\alpha}$ required to satisfy the $L^1$ normalization is analytically derived to be exactly the Base-24 modular correction $\boldsymbol{1+\frac{1}{240}}$. This closure results in the mathematically proven exponential suppression of environment-induced decoherence in the weak coupling limit (Davies limit), providing a fundamental, self-consistent mechanism for quantum stability. A speculative, now-superseded transcendental ansatz for $R_{\alpha}$ is retained in Appendix A.1 for full transparency. list of related DOI's below Zenodo Record URL Zenodo Record ID (Identifier) Full DOI (Digital Object Identifier) https://zenodo.org/records/17566371 17566371 10.5281/zenodo.17566371 https://zenodo.org/records/17583962 17583962 10.5281/zenodo.17583962 https://zenodo.org/records/17592910 17592910 10.5281/zenodo.17592910 https://zenodo.org/records/17622862 17622862 10.5281/zenodo.17622862 https://zenodo.org/records/17624288 17624288 10.5281/zenodo.17624288 https://zenodo.org/records/17716751 17716751 10.5281/zenodo.17716751