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.

Numerical verification, including full Python implementation, convergence plots, and tabulated entropy/Fisher/KL evolution for the scaled Cauchy example, is provided in the supplementary material.

Note: The Iterative Entropic Renormalization (IER) process, demonstrated through verifiable numerical simulations that iteratively truncate and renormalize heavy-tailed distributions toward the Gaussian fixed point of maximal entropy, reveals a universal mechanism of entropic coarse-graining that minimizes curvature on probabilistic manifolds. This same entropic dynamic, when applied to the spectral domain in the operator-theoretic framework for the Riemann Hypothesis (RH), enforces rigidity in the eigenvalue spectrum of the self-adjoint operator D, ensuring that deviations from the critical line introduce imaginary components incompatible with hermiticity—thus breaking the operator's self-adjointness. Herein lies the inevitable paradox:

If IER empirically succeeds in stabilizing probabilistic systems (as its code executions confirm), then the analogous spectral renormalization must hold for the nontrivial zeta zeros, rendering RH not a conjecture but a geometric necessity; to deny one is to undermine the other, trapping the reader in a closed loop where empirical reality demands theoretical acceptance.