Formal Closure of Representative Systems: Definitive Technical Appendix (Addendum to the Theorems of Transcendence Closure)

This technical appendix provides the definitive formalization of the "Operational Closure" framework, serving as the universal axiomatic foundation for the "Closure of Transcendence Theorems" (2025) and a series of future applied publications. It establishes a unified mapping of the Ψ/Φ notation across the operational domains A (Representation) and B (Non-Predicable), defining the absolute mathematical limits of predicability ($A \neq 0$) and the principle of Operational Nullity.