Benford's Law as a Signature of Scale Invariance, Not Randomness — E8 Intelligence Research

FINDING: Benford's law emerges from scale-invariance (dilation symmetry) of natural data distributions, not from randomness alone. | MATH: \ (P (d) = ₁₀ (1 + 1/d) \) for leading digit \ (d \1, , 9\ \) ; yields \ (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 base-10 structure is a discrete manifestation of continuous dilation symmetry (scale invariance). The ratio \ (₁₀ (2) 0. 3010 \) is the key constant; note that \ (0. 3010 \) is close to \ (0. 382 ^-1 \) (where \ (= 1. 618 \) ), but this is coincidental. No direct link to golden ratio or base-60. The symmetry group is the multiplicative group of positive reals, not crystallographic. | DEPTH: 7 — profound because it reveals that many natural datasets (river lengths, stock prices, physical constants) obey a hidden symmetry: invariance under rescaling. This c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com