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: \( P(d) = log₁₀(1 + 1/d) \) 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 \( log₁₀(2) ≈ 0.3010 \) is the key constant; note that \( 0.3010 \) is close to \( 0.382 × φ⁻¹ \) (where \( φ = 1.618 \)), 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