IX_ Rigorous Mathematical Formalization of the Unified Theorem_CBD_RAG_RES

This work presents a rigorous mathematical formalization of the Unified Theorem of Endogenous Governability within the CBD-RAG-RES framework. It establishes a structural bridge between the micro-geometric torsion observable δT(t) and the global divergence metric W2(t), providing a minimal bound linking local dynamics to macroscopic structural change. The accessible variety Ω(t) is formally defined as a measurable topological set representing the real plasticity of adaptive systems. A mimetic contraction theorem shows how increasing internal correlation reduces the space of admissible trajectories. The framework identifies a universal structural threshold S∗≈2.14 corresponding to the onset of irreversibility. This threshold is demonstrated to be invariant under metric scaling and dimensional transformations. A probabilistic extension models regime transitions as stochastic threshold crossings and defines structural risk through hitting-time theory. Empirical observations across AI systems, collective dynamics, and market regimes support the robustness of the theoretical structure. The work contributes a consolidated mathematical block strengthening the analytical foundations of the CBD-RAG-RES corpus.