Geometric Derivation of van der Waals Interactions from M3(C): Unified Description via Non-Commutative Operational Coordinates

This paper derives van der Waals interactions from M₃(C) non-commutative algebra structure constants within Cognitional Mechanics framework. Traditional London theory describes dispersion forces via polarizability and ionization energy in physical coordinates, leaving structural questions unanswered: why r⁻⁶ specifically, what is the geometric origin of C₆, and what is the algebraic meaning of polarizability. CM reformulates these interactions in operational coordinates where London's "polarization" becomes off-diagonal operator components and "quantum fluctuations" represent temporal evolution of non-commutativity. The r⁻⁶ dependence emerges naturally from trace dimensional analysis as geometric necessity rather than perturbative consequence. Molecular electronic states are represented as operators F ∈ M₃(C) with interaction energy defined via commutator traces. The C₆ coefficient is computed from su(3) structure constants (Gell-Mann matrices) with explicit calculation procedure provided. Ultra-precision implementation incorporates effective nuclear charge with exchange-correlation corrections (Perdew 1996), distance-dependent shielding functions evaluated at van der Waals equilibrium distance, Dirac first-order relativistic corrections (0.032-0.29% for Ar-Xe), numerical Hartree-Fock orbitals from NIST Atomic Spectra Database with adaptive Gauss-Kronrod quadrature achieving 10⁻⁸ accuracy, and higher-order dispersion terms (C₈, C₁₀ from rank-4 structure constants). Final results: He-He (0.07% error), Ne-Ne (0.13%), Ar-Ar (0.02%), Kr-Kr (0.15%), Xe-Xe (0.21%), average 0.12% error, matching ab initio CCSD(T) benchmarks. Results demonstrate quantitative correspondence between operational constraints and physical measurements, establishing that intermolecular interactions arise from geometric state space constraints rather than quantum fluctuations per se. All numerical calculations fully specified via explicit formulas ensuring complete reproducibility without external code. Framework extends London theory as valid physical coordinate description while providing higher-level operational structure, unifying chemistry as non-commutative operational programs executed on material hardware. Future applications include covalent/ionic bonding operational descriptions, chemical kinetics formulation, and connection to emergent gauge structures in CM-GUT framework.