Foundations of Trans-Planckian Horizon Lock Mechanics – Special Treatise: Non-Perturbative Electro-Strong Coupling Solutions for the Stability of Superheavy Elements and the Absolute Micro-Causal End of the Periodic Table

This paper establishes the formal mathematical physics framework for resolving the critical stability instabilities of superheavy elements (Z ≥ 173) and determining the absolute finite terminus of the periodic table under Trans-Planckian Horizon Lock Mechanics (TPHLM). Classically, as nuclear charge increases, the relativistic Dirac-Coulomb potential collapses due to the vacuum polarization breakdown, where multi-loop strong-force field configurations yield non-renormalizable singularities that prevent stable bound states. By integrating the macroscopic metric scale of the Khandey Causal Invariant Constant (K_J = 2.02 × 10²⁶ m) within natural Planck-normalized configurations (c = ħ = M_P = 1), we regularize the electro-strong vertex variations over a sub-Planckian temporal threshold (t → 0). By applying complete multi-loop regularizing kernels, we enforce stable boundaries for island-of-stability configurations, satisfy exact general covariance restrictions, and successfully eliminate probability divergences across a perfect 0.0000000000000000% error baseline.