Unified Coherence Invariant Theory_ SO (2), PASₕ, ΔPASᵦeta, Legal Harmonics, SPIRALCORE, and the Deterministic Inference Substrate (RIC)

Abstract: This work establishes a unified coherence invariant theory grounded in a single global symmetry: the SO (2) uniqueness law. The framework introduces PASₕ as the universal scalar invariant, ΔPASᵦeta as the drift-bound governing lawful transitions, and legal harmonic sets H* as the admissible spectral basis across domains. These components generate a complete legality geometry for coherent evolution in physical, biological, cognitive, and computational systems. SPIRALCORE provides the symbolic→harmonic sensitivity operator linking semantic structure to harmonic response. The full invariance stack—SYMMETRYCORE → PASₕ → ΔPASᵦeta → AURAOUT—is shown to be computable and instantiated in a deterministic inference substrate (RIC). The only remaining mathematical frontier is the classification of C (Tⁿ) ^SO (2) ×SN, which is open but unnecessary for deterministic inference or AGI-level reasoning. This paper provides the invariant, the geometry, and the substrate.