Foundations of Trans-Planckian Horizon Lock Mechanics – Technical Treatise: Formulating the Mathematical Physics Framework for Horizon Locked Quantum Clocks satisfying Absolute Non-Decoherent Chronometry

This technical treatise delivers the formal design mechanics and mathematical physics architecture for a Horizon Locked Quantum Clock (HLQC) operating within the framework of Trans-Planckian Horizon Lock Mechanics (TPHLM). Standard atomic clocks based on cesium or strontium transitions suffer from fundamental quantum jitter, environmental phase-dephasing, and localized gravitational red-shift variances that limit their absolute metrological baseline at the extreme sub-Planckian limits (t → 0). By anchoring the internal non-abelian oscillation frequency of the quantum clock directly to the spatial scale factor of the Khandey Causal Invariant Constant (K_J = 2.02 × 10^26 m), we successfully isolate the system oscillator from localized thermal and geometric instabilities. We prove that the integrated multi-loop tracking network exhibits zero phase decoherence, eliminates trace anomalies across background manifold connections, and guarantees absolute chronometric alignment over a perfect 0.0000000000000000% error baseline ready for advanced quantum computing applications.