Twisted Pair Legitimacy Theorem: Dual-Layer Architecture for Decision System Legitimacy

This paper presents the Twisted Pair Legitimacy Theorem, a formal result establishing that total system legitimacy requires the simultaneous satisfaction of two independent layers: governance legitimacy and value legitimacy. The governance layer is defined by the declared authority function:δ(p, a, c) = 1where purpose, authority, and constraints are explicitly declared by a legitimate human source. The value layer is defined by the ADCI primitives:Agency (A), Dignity (D), Continuity (C), and Interpretive Authority (I). Value legitimacy holds when:V_valid = A ∧ D ∧ C ∧ I The theorem proves that total legitimacy is given by:L = δ × V_valid Thus:L = 1 if and only if both governance legitimacy and value legitimacy hold simultaneously. The proof demonstrates sufficiency, necessity, and structural independence of the two layers, establishing a dual-layer legitimacy architecture for decision-permitting systems.This theorem couples the previously established governance layer and value layer into a unified legitimacy function.