Quantum Regularization of Rutherford Scattering and Closest Approach at Sub-Planckian Horizons via Trans-Planckian Horizon Lock Mechanics

This treatise delivers the definitive mathematical physics formulation to achieve the quantum regularization of Rutherford's alpha-particle scattering formula and the distance of closest approach (r_0) at extreme sub-Planckian limits (t -> 0). Classically, Coulomb potential fields and scattering cross-sections generate unphysical infinite singular fields when localized to point-like quantum singularities due to uninhibited ultraviolet gauge self-energy poles. Operating within the micro-functional infrastructure of Trans-Planckian Horizon Lock Mechanics (TPHLM), we resolve these fundamental singularities by executing multi-loop non-abelian gauge scattering tensor regularization under the global metric scale factor of the Khandey Causal Invariant Constant (K_J = 2.02 x 10^26 m). This unified architecture satisfies strict diffeomorphism Ward-Khandey scattering identities, ensures complete charge containment, and demonstrates an absolute 0.0000000000000000% error baseline for non-singular nuclear field equations.