Shannon Dynamic Entropy and the Structural Evolution of Information

Entropy is not merely a measure of uncertainty—it is a dynamic field that governs the evolution of structure in time. In classical information theory, Shannon entropy assumes a fixed distribution. But in systems with memory, dissipation, or spectral constraints, the probability distribution itself evolves. This demands a new formulation: entropy not just as a function, but as a dynamical object with temporal and structural derivatives. In this work, we propose that information dynamics is governed by two critical struc- tural attractors: • The temporal attractor at K = 1, where Gamma-structured entropy reaches cur- vature stability, marking the onset of memory-free exponential decay. • The spectral attractor at Re(s) = 1/2 , where phase-symmetric structures in zeta-like systems exhibit minimal entropy curvature and maximal information flow. Together, these attractors define a bivariate entropy surface H(K,σ), encoding the coupling of structural scale K and spectral spread σ. The curvature and drift of this surface form the foundation of the Information Dynamical System (IDS): a theoretical framework where entropy flow, attractor dynamics, and structure generation are unified. This view enables us to describe irreversible time, spectral constraints, and complexity emergence from a single entropic principle. Through a series of differential and integral formulations, we show how dynamic entropy evolves, converges to attractors, and shapes the behavior of complex systems in physics, AI, and beyond.