The CM Quantum Hall Theorem: Analytic Derivation of the von Klitzing Constant via Topological Information Locking in Mn(C)

The Quantum Hall Effect (QHE) is a cornerstone of modern condensed matter physics, distinguished by the extraordinary precision and universality of the von Klitzing constant RK. Traditional theories explain this quantization through the topology of wave functions and Chern invariants, yet the underlying logical necessity—why these constants emerge as invariant geometric ratios regardless of material impurity—remains a subject of empirical observation. This paper presents the CM Quantum Hall Theorem, reinterpreting the QHE as the terminal phase of the Cognitional Mechanics (CM) transport hierarchy. Following the CM Friction Theorem (dissipation), the CM Superconductivity Theorem (phase-locking), and the CM Magnetism Theorem (parallelization), this work introduces the concept of "Topological Information Locking." In the CM framework, the Hall plateau represents a state where the manifold's geometry prohibits "unitary leakage" (logical stress), rendering sigma identically zero. We demonstrate that the bulk of a QHE system functions as a closed logical loop, while the boundaries form chiral information buses. This chirality enforces unidirectional flow, making the information current topologically immune to backscattering. Crucially, this theorem derives RK = h/e^2 by identifying the Planck constant (h) as the minimum unitary action packet and the square of the elementary charge (e^2) as the spatial imprint intensity. The resulting ratio defines the fundamental impedance of information transfer in a locked manifold. By mapping the integer quantization to the winding number of information cycles within a dynamic Mn(C) manifold, where n corresponds to the Landau level occupancy, we eliminate the need for empirical fitting. The theorem accurately predicts the standard quantization sequence for GaAs/AlGaAs and the shifted Dirac fermion sequence for Graphene, accounting for its SU(4) symmetry. This work completes the logical bridge between algebraic geometry and topological transport, positioning CM as the unified predictive logic for the fundamental constants of nature. Key Features: Geometric Derivation of RK: Establishes the von Klitzing constant as a universal ratio of action packets to imprint intensity. Topological Integrity: Defines the QHE as the terminal transport phase where non-commutative stress is structurally prohibited (sigma = 0). Dynamic Mn(C) Framework: Generalizes the matrix dimension to account for variable Landau level occupancy and system symmetries. Predictive Universality: Accurately matches experimental quantized resistance for both standard heterostructures and graphene. This paper completes the "Transport Tetrad" alongside the Friction, Superconductivity, and Magnetism theorems, providing a comprehensive information-geometric foundation for condensed matter physics.