The Mathematical Foundations of Generative Game Theory: Higher-Order Self-Reference and Dimensional Emergence via ROA and STCT

This study establishes the mathematical foundations of ‘Generative Game Theory’ by unifying Rough Operator Algebra (ROA) and the Seonggil Theory of Composite Torsion (STCT). Classical game theory, epitomized by Nash Equilibrium, presupposes a fixed strategy space and payoff matrix, thereby confining itself to describing static stability within a closed commutative topology. We reformulate the cognitive and response processes of agents as non-commutative operators on a Hilbert space H, expanding higher-order self-reference (meta-competition) through the Baker-Campbell-Hausdorff (BCH) series. We rigorouslyprove that the cognitive friction generated in this infinite regress manifests geometrically as the STCT Torsion Tensor. When the norm of this accumulated torsion exceeds the topological threshold, the system departs from local Nash stability, triggering a spontaneous symmetry breaking governed by the Seonggil Field Equation (SFE), which inevitably leads to the algorithmic emergence of novel strategy dimensions.