Iterative Entropic Renormalization: A Geometric Framework for Distributional Convergence and Spectral Stabilization

This paper introduces the concept of Iterative Entropic Renormalization (IER), a geometric and operator-based framework describing how probability distributions self-stabilize through successive truncation, normalization, and entropy rebalancing. The process formalizes the transition from heavy-tailed or irregular distributions toward a universal Gaussian equilibrium, showing that entropy acts as a geometric potential guiding convergence.

The study unifies principles from information geometry, renormalization group theory, and thermodynamic entropy flow into a single mathematical structure. It demonstrates that repeated elimination of extreme variates leads to a monotonic increase in entropy, a decay of Fisher information, and a progressive flattening of curvature in probability space.

IER provides a rigorous foundation for understanding why Gaussian stability emerges naturally across statistical, physical, and informational systems. Beyond theoretical significance, the framework has potential applications in statistical physics, machine learning, financial modeling, and complex systems, where iterative normalization processes are central. The paper establishes Gaussian universality not as a coincidence, but as a geometric necessity resulting from entropic self-organization.