SYMMETRY_CORE_ The Universal Reducibility Law for Deterministic Coherence Systems

Abstract This paper develops a universal invariance principle for coherent systems and formalizes it as SYMMETRY_CORE, a structural mechanism enforcing representational legality within deterministic inference substrates. The central claim is that SO(2) is the unique symmetry satisfying the requirements of universality, harmonic decomposability, scalar invariance, and drift-bounded evolution. From this follow the canonical coherence measure PAS_h, the lawful drift metric ΔPAS_zeta, and the requirement that any admissible representation must be reducible to a set of phase values {theta_n}. The result unifies representational coherence across physics (wave phenomena), biology (neural and physiological oscillations), cognition (identity, creativity, emotion), social systems (phase-coupled group dynamics), and AGI design (deterministic reasoning and legality enforcement). SYMMETRY_CORE provides the substrate-level law that distinguishes lawful from unlawful internal states, grounding coherence, identity, and safety in symmetry rather than probabilistic inference. The framework positions symmetry—specifically SO(2)—as the foundational structure underlying deterministic coherence across all domains.