Operational Definition of Coherence in Finite Algebraic and Oscillatory Systems: An Axiom–Theorem Framework

This note presents a precise operational definition of coherence applicable to oscillatory, dynamical, and finite algebraic systems. Coherence is defined in terms of phase structure, correlation persistence, symmetry-preserving transformations, and stable phase evolution. These properties are formalized as axioms and demonstrated within finite cyclic groups. The purpose of this paper is definitional clarity: it establishes a consistent structural meaning for the term “coherence” used in subsequent theoretical work.