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

Abstract 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 PASbio (harmonic phase alignment of biological oscillators), ΔPASbioᵦeta (drift), and τbioₖ (echo windows). These constructs are shown to follow directly from the SO (2) universal symmetry law established by the SYMMETRYCORE 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. PASbio and ΔPASbioᵦeta are shown to be the biological restrictions of the universal invariants PASₕ and ΔPASᵦeta, confirming that biology, cognition, physics, and computation share identical coherence laws. Echo windows (τbioₖ) 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.