Information-Causality Field and the Unified Field Theory Based on 11-Dimensional T^3 Torus Fractal Geometry -- Paper IX: Geometric Decomposition of Point-Line-Plane Phase Operators and the Universal Probabilistic Isomorphic Convergence Theorem

This paper decomposes the Phase-Latching Pipeline (PLP) operator system into a three-fold geometric structure of Point, Line, and Plane, and completely derives Shannon entropy and Landauer's thermodynamic threshold limit of classical Information Theory from it. Furthermore, we derive the fundamental causal anchor equation M^2 = (L · c) / G that binds microscopic quantum action quantities with macroscopic causal structures, and verify its complete consistency with Planck unit system boundary conditions. When applying the speed-of-light c constraint condition and the Causal Phase Parallelism condition between the start and end boundaries to an arbitrary high-dimensional probability density function defined over a D-dimensional space, we mathematically prove that regardless of the initial function shape or dimension D of the system, all state spaces undergo homomorphic contraction into a singular 1-dimensional toroidal phase ring Ω* ≅ Z_N. This paper forms the absolute foundational pillar for the 105 Information-Causality lattice formula system to be unfolded in Volume 10.