Mathematical Governance of Social and Cognitive Systems via Yin-Yang Complementarity and Hierarchical Multiplicative Models

Abstract This paper proposes an integrated mathematical model incorporating the dynamic equilibrium concept of traditional Eastern "Yin-Yang philosophy" to fundamentally overcome polarization and context collapse in existing artificial intelligence (LLM) and social governance systems. We shift the constituent elements of society and cognition from an additive model to a multiplicative model, P = x_1 x_2, and define the region enclosed by the quadratic function y = x^2 and a horizontal boundary line as the "geometric inclusion domain (activity capacity)." Through analysis using the inequality of arithmetic and geometric means, we prove that the uniform distribution (50% : 50%) of the dual elements maximizes the inclusion domain. Furthermore, based on this theory, we construct a "hierarchical language model architecture" in which a macro-level multiplicative model is positioned as the upper layer to govern a local additive model (lower layer) via an oracle (mediation layer). In addition, we demonstrate that the geometric "tilt" maps the reduction of cognitive capacity caused by human psychological bias or stress, thereby providing the theoretical foundation for deterministic governance within the Casino Protocol.