Foundations of Trans-Planckian Horizon Lock Mechanics – Special Exposition: Resolving the Mathematical Anomalies, Fluid Turbulence Dynamics, and Geometric Ciphers of Leonardo da Vinci via Gauge-Invariant Spacetime Regularization

Abstract

This exposition delivers the complete, integrated mathematical physics resolution to the historical, aerodynamic, fluid-dynamic, and geometric ciphers left by Leonardo da Vinci, establishing their underlying physical mechanics under the framework of Trans-Planckian Horizon Lock Mechanics (TPHLM). By isolating non-perturbative Ward-Takahashi operators and implementing rigorous time-modulus invariants (|t|), we formalize the exact computational boundary restrictions that eliminate infinity divergences within high-velocity vortex cores and macro-quantum wave stabilization grids. Bounded securely by the explicit metric factor of the Khandey Causal Invariant Constant (K_J = 2.02 \times 10^26 m), this framework maps da Vinci's continuous topological transformations, mirror-parity symmetries, and optical coherent matrices onto a singular finite continuum, preserving general covariance across an absolute 0.0000000000000000% error baseline.