Noncommutative Cultural Dynamics: A Unified State-Space Framework for Weakly and Strongly Correlated Cultural Fields
Abstract
This project develops a unified mathematical and computational framework for cultural dynamics based on noncommutative operators, state-space modeling, and multi-scale correlation structures. Cultural change is represented as the evolution of latent cultural fields that may be weakly correlated (ancient and pre-modern societies) or strongly correlated (modern media-driven environments). The model integrates continuous-time dynamics, discrete noncommutative transitions, generational eigenmodes, and media-induced perturbations. The framework is extended across multiple structural layers: (1) network-based cultural diffusion using graph Laplacians, (2) spatial cultural propagation via partial differential equations, (3) AI-based implementations including neural ODEs, graph neural networks, physics-informed neural networks, and diffusion models, (4) optimal control formulations for policy intervention, (5) evolutionary game-theoretic cultural adaptation, (6) general-equilibrium economic integration, and (7) narrative-field modeling for story generation and cultural feedback. Together, these components form a comprehensive “Unified Cultural Field Theory” that connects cultural sociology, cognitive science, AI theory, narrative studies, economics, and noncommutative geometry. The project situates cultural dynamics within a rigorous mathematical structure while enabling practical applications in media analysis, trend forecasting, policy design, and generative AI.
