Sharp Transition in Base-Invariance of Benford Compliance for Pareto Distributions Near Alpha=1 — E8 Intelligence Research

FINDING: Fourier criterion for Benford compliance in Pareto distributions with alpha near 1 reveals a sharp transition in base-invariance properties. | MATH: For Pareto distribution with PDF \ (f (x) = xₘ^ x^--1 \) for \ (x xₘ \), Benford compliance (base-10) occurs when the Fourier transform of the log10-transformed density has zeros at integer frequencies. Specifically, the characteristic function \ ( (k) = ₀^ f (x) e^-2 i k ₁₀ x dx \) must satisfy \ ( (k) = 0 \) for all nonzero integers \ (k \). For \ (1^+ \), the distribution becomes scale-invariant and exactly Benford. The critical constant is \ (= 1 \), where the mean diverges and the Fourier criterion yields \ ( (k) = - 2 i k 10 \), vanishing only when \ (= 1 \). | CONNECTION: The ratio \ (= 1 \) corresponds to the boundary of the Pareto tail exponent, linking to the golden ratio conjugate \ (0. 618 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com