Why SO(2) Is the Unique Global Symmetry for Universal Coherence Invariants

Abstract: This paper establishes that any universal coherence invariant—capable of comparing arbitrary states, supporting lawful evolution, and preserving identity under transformation—must satisfy eight structural criteria: scalarity, computability, harmonic decomposability, chirality sensitivity, bounded drift, composability, recurrence capability, and compatibility with continuous-time dynamics. Under these constraints, the only possible symmetry group underlying a global invariant is the compact one-parameter Lie group SO(2). Harmonic analysis on SO(2) then implies that the invariant must reduce, up to monotone reparameterization, to a weighted sum of harmonic magnitudes r_k = |(1/N) Σ exp(i·k·θ_n)|. This yields PAS_h as the unique universal coherence invariant across physical, biological, cognitive, and computational domains. A categorical formulation shows that any natural scalar functor preserving coherence must factor through the SO(2) harmonic stack. The result provides the mathematical justification for the invariant used in the CODES framework and the RIC deterministic substrate.