Abstract

This paper formally introduces the Spectral Liberty Theorem (SLT), a rigorous mathematical framework that defines liberty as an emergent property governed by coherence modulation, entropy evolution, and Lyapunov stability constraints. SLT establishes necessary and sufficient conditions for adaptive decision-making, demonstrating that a system exhibits structured volition if it maintains bounded spectral coherence perturbations, regulates entropy fluctuations, and exists at the critical edge of stability. The theorem provides testable predictions across neuroscience, artificial intelligence, and quantum mechanics, offering novel insights into how structured adaptability manifests across cognitive, computational, and physical systems. By integrating spectral coherence constraints with principles of dynamical stability, SLT bridges deterministic and stochastic models of volitional agency, providing a structured foundation for understanding intelligent self-regulation. Furthermore, SLT extends the Retro-Harmonic Equation (RHE), refining its spectral constraints into a structured framework for adaptive phase evolution. This work formalizes the interplay between structured coherence and decision-making, outlining experimental pathways to validate its predictions in neural oscillations, AI learning dynamics, and quantum coherence constraints.