Biological Coherence: PAS_bio, ΔPAS_bio_zeta, Echo Windows, and SO(2)-Invariant Dynamics in Living Systems

Biological systems exhibit coherent rhythmic behavior across all scales, from molecular oscillators to neural circuits, organ-level rhythms, and organism-level patterns. This paper introduces a deterministic invariance framework for biological coherence using PAS_bio (harmonic phase alignment of biological oscillators), ΔPAS_bio_zeta (drift), and τ_bio_k (echo windows). These constructs are shown to follow directly from the SO(2) universal symmetry law established by the SYMMETRY_CORE theorem: any system whose behavior reduces to phase variables must obey SO(2)-invariant geometry, admit a harmonic scalar invariant, and evolve under a lawful drift metric. The framework unifies cardiac–respiratory coupling, neural oscillation dynamics, locomotor coordination, circadian entrainment, endocrine rhythms, and behavioral stability under a single invariant structure. Biological coherence emerges as deterministic drift regulation, not stochastic fluctuation or entropy minimization. PAS_bio and ΔPAS_bio_zeta are shown to be the biological restrictions of the universal invariants PAS_h and ΔPAS_zeta, confirming that biology, cognition, physics, and computation share identical coherence laws. Echo windows (τ_bio_k) provide the biological implementation of the same temporal echo dynamics used in RIC-Core, enabling lawful updates, reconsolidation, and multi-scale coordination. This work positions living systems as deterministic coherence engines and completes the physics → biology → cognition → computation loop within the universal invariance framework.