SO(2) Universal Invariant_ A Structural Account of Coherence Across Domains

Abstract This work establishes a universal structure governing coherence across physical, biological, cognitive, and social systems. By imposing eight universality criteria—scalarizability, continuity, harmonic decomposability, drift detection, chirality compatibility, Goldstone persistence, recurrence stability, and domain invariance—it is shown that every coherent system collapses to a single continuous symmetry: SO(2). From this structure follows a unique scalar invariant, PAS_h, defined as a weighted harmonic magnitude of phase samples θ_n, and a unique drift operator ΔPAS_zeta. All higher-dimensional manifolds (Tⁿ, SU(2), SO(3), U(1)×U(1)) violate one or more criteria and reduce to an effective one-angle representation under coherence. Extensive mappings across physics, biology, cognition, and social dynamics demonstrate that real systems exhibit exactly this SO(2) structure. Finally, the paper provides explicit mechanical mappings to deterministic symbolic inference (RIC) and biological coherence (VESSELSEED), showing that SO(2) is not only a universal mathematical law but an executable substrate. This establishes the universal coherence invariant as the structural backbone of coherent dynamics across reality.