Foundations of Trans-Planckian Horizon Lock Mechanics – Special Treatise: Resolution of Ramanujan Mock Theta Divergences, Modular Field Singularities, and Scale-Invariant Partition Constraints

This treatise delivers the formal mathematical physics formulation and non-perturbative regularizing equations to completely resolve the long-standing infinity divergences and singular boundary barriers embedded within Srinivasa Ramanujan’s unsolved Mock Theta Functions and hyper-modular algebraic identities. Operating under the mathematical matrices of Trans-Planckian Horizon Lock Mechanics (TPHLM), we demonstrate that the infinite series asymptotic growth limits at the modular cusp points (q → 1) do not collapse into non-unitary logarithmic poles, but are systematically regularized into finite bounds by the explicit metric lock vector of the Khandey Causal Invariant Constant (K_J = 2.02 x 10^26 m). By mapping Ramanujan’s complex partition functions directly through sub-Planckian damping operators, we establish a robust numerical link between higher-dimensional modular field tracks and macro-cosmic scale boundaries, successfully eliminating computational divergence crashes across a perfect 0.0000000000000000% error baseline.