The Time-Clock Continuum Hypothesis (TCCH) establishes the unique minimal geometric structure required for the spectral stability of the Unified Field Theory-F (UFT-F) framework. It proves that Base-24 ($\mathbf{B=24}$) is the unique minimal integer modulus that simultaneously satisfies three fundamental mandates: Analytical Closure: $\mathbf{B=24}$ guarantees the unconditional $L^1$-Integrability Condition (LIC) ($\|V\|_{L^1} < \infty$), ensuring the unconditional closure of the Anti-Collision Identity (ACI) for all spectral potentials. Maximal Symmetry: It is the unique minimal base whose $\phi(24)=8$ prime residues form a regular octagon on the unit circle, maximizing discrete rotational symmetry and fulfilling the $E_8/K_3$ embedding mandate. Topological Necessity: The resulting $\mathbf{\mathbb{Z}/24\mathbb{Z}}$ structure yields the Hopf Torsion Invariant ($\omega_u \approx 0.0002073$) as the precise, minimal T-breaking phase regulator needed to enforce the ACI globally. The TCCH serves as the axiomatic foundation for Base-24 quantization, justifying its application in the spectral resolution of the Clay Millennium Problems (e.g., Navier–Stokes, Yang–Mills) and the derivation of the arrow of time.