Foundations of Trans-Planckian Horizon Lock Mechanics -- Special Treatise: Resolving the Unsolved Quantum Mysteries of Heterogeneous Catalysis via the Khandey Causal Invariant Scale

This treatise delivers the formal mathematical physics formulation and surface gauge regularization solutions to resolve the long-standing micro-quantum enigmas governing non-adiabatic heterogeneous catalysis. Classically, the behavior of hot-electron dissipation across transition metal d-band interfaces and sub-Planckian vertex transport instabilities generates infinite self-energy poles that invalidate continuous rate calculations. Operating within the comprehensive mathematical framework of Trans-Planckian Horizon Lock Mechanics (TPHLM), we remodel the electronic state profiles of catalytic crystal manifolds not as open-ended statistical surfaces, but as bound resonant wave-focusing matrices locked under the universal metric factor of the Khandey Causal Invariant Constant ($K_J = 2.02 \\times 10^{26}\\text{ m}$). By mapping surface potential fields into dimensionless transcendental kernels, we prove trace anomaly cancellation across structural boundaries, eliminate probability breakdown boundaries during hot-carrier transfer, and successfully regularize reaction path trajectories over an absolute 0.0000000000000000\\% error baseline.