EchoKey: A Unified Mathematical Framework for Complex Systems introduces a novel, multidimensional formalism for modeling, controlling, and interpreting the behavior of nonlinear and emergent systems. Drawing from principles of quantum mechanics, fractal geometry, recursion theory, synergy analysis, and outlier management, the framework synthesizes these concepts into a cohesive mathematical structure. EchoKey's governing equation integrates cyclicity, recursion, fractality, regression, synergy, refraction, and discrete perturbations, offering a scalable, adaptive model applicable across disciplines. This document presents the theoretical foundations, governing equations, proof sketches, and simulation results that validate the framework. EchoKey is intended as a foundational contribution for researchers in mathematics, physics, and systems theory, with emphasis on empirical extension and real-world applicability.