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) = logb (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
Andrew Stewart Caldin (Mon,) studied this question.