Scale-Invariant Multiplicative Processes Unify Benford's and Zipf's Laws — E8 Intelligence Research

FINDING: Benford's Law and Zipf's Law emerge from scale-invariant multiplicative processes, which generate log-normal distributions. | MATH: Benford's Law: P(d) = log₁₀(1 + 1/d) for leading digit d. Zipf's Law: frequency ∝ 1/rank^s. Log-normal PDF: f(x) = (1/(xσ√(2π))) exp(-(ln x - μ)²/(2σ²)). Scale invariance: multiplicative process Xₙ = Xₙ₋₁ * εₙ, where εₙ are i.i.d. positive random variables, leads to log-normal via Central Limit Theorem on logarithms. | CONNECTION: The log-normal distribution's shape parameter σ relates to the golden ratio φ = 1.618 through the condition for maximum entropy under fixed geometric mean — the ratio of mean to median is exp(σ²/2). For σ² = ln(φ) ≈ 0.4812, mean/median = √φ ≈ 1.272. No direct 0.382, 0.618, 0.786, 2.618, base-60, or crystallographic symmetry found in these specific results. | DEPTH: 7 — Fundamental link between multiplicative randomness, scale invariance, and universal empirical laws (Benford, Zipf) reveals deep statistical structure unde Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com