Abstract

Abstract This paper establishes the mathematical foundation of CODES (Chirality of Dynamic Emergent Systems), introducing a unifying framework for structured emergence across disciplines. We formalize prime-driven resonance equations, a novel class of nonlinear phase-locking dynamics, and a generalized coherence metric to quantify system stability across physical, biological, and cognitive domains. By extending harmonic analysis, prime number theory, and topological invariants, we propose a universal resonance function that governs the transition from stochastic disorder to structured order. This framework: • Resolves fundamental paradoxes in probability theory by demonstrating that randomness is a projection of underlying resonance structures. • Redefines symmetry-breaking as a phase-locked emergence process, replacing traditional group-theoretic formulations. • Introduces a computable coherence model that predicts emergent stability across complex adaptive systems. Finally, we explore implications for cosmology, AI, and quantum gravity, demonstrating that mathematical reality is fundamentally a structured resonance field, not a probabilistic space.