Benford's Law Arises from Scale-Invariance, Not Random Chance — E8 Intelligence Research

FINDING: Benford's law emerges from scale-invariance (dilation symmetry) of natural data distributions, not from random chance. | MATH: P(d) = log₁₀(1 + 1/d) for leading digit d ∈ {1,…,9}; P(1) ≈ 0.3010, P(9) ≈ 0.0458. The law is equivalent to the mantissa distribution being uniform on [0,1) under logarithmic measure. | CONNECTION: The logarithmic spacing 1.0 → 2.0 → 3.0 … → 9.0 → 10.0 corresponds to geometric ratios: the interval [1,2) occupies log₁₀(2) ≈ 0.301 of the log cycle, matching the golden ratio's logarithmic footprint (φ ≈ 1.618 → log₁₀(φ) ≈ 0.209, not directly 0.301). However, the law's invariance under scaling (multiplying all data by a constant) is a dilation symmetry group isomorphic to ℝ⁺, which is the same continuous symmetry underlying self-similar structures in nature (e.g., fractal geometries, logarithmic spirals). The base-10 logarithm is arbitrary; in base b, P(d) = log_b(1 + 1/d). For b = φ² ≈ 2.618, the digit 1 probability becomes log2.618(2) ≈ 0.724, a high Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com