Unconditional Analytical Closure of the UFT-F Modularity Constant $C_{\text{UFT-F}}$: Derivation of the Base-24 Correction $1+\frac{1}{240}$ and Proof of Exponential Decoherence Suppression

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:

1. 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.

2. 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).

3. 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 A speculative, now-superseded transcendental ansatz for $R_{\alpha}$ is retained in Appendix A.1 for full transparency.

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