Benford's Law as a Scale-Invariance Fixed Point Under Renormalization — E8 Intelligence Research

FINDING: Benford's law arises as a fixed point under scale-invariance renormalization group transformations, explaining its universality across diverse datasets. | MATH: Leading-digit probability \( P(d) = \log_{10}(1 + 1/d) \) for base 10; fixed-point condition under multiplicative scaling: distribution invariant under \( x \to \alpha x \) for all \( \alpha > 0 \) implies \( f(x) \propto 1/x \); renormalization group flow converges to this scale-invariant distribution. | CONNECTION: The logarithmic spacing of Benford's law (base 10) mirrors the natural geometric progression of ratios (e.g., 1, 1.618, 2.618) and the golden ratio's logarithmic spiral; the 1/x distribution is the continuous analog of the discrete geometric series ratio \( \phi \) (1.618). The base-10 logarithm relates to base-60's multiplicative structure (both are scale-invariant under multiplication). | DEPTH: 7 — Profound because it unifies an empirical law with renormalization group theory, a cornerstone of statistic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com