Measure-Theoretic Proof of Benford's Law Scale Invariance and Allometric Links — E8 Intelligence Research

FINDING: Scale invariance in logarithmic mantissa distributions (Benford's Law) is proven via measure-theoretic invariance under base scaling, with deep links to allometry and self-similarity. | MATH: Benford's Law: P(d) = log₁₀(1 + 1/d) for leading digit d ∈ {1,…,9}. Scale invariance condition: P(ax) = P(x) for all a > 0, satisfied only if P(x) ∝ 1/x on logarithmic scale. Mantissa distribution uniform in log₁₀: P(mantissa ≤ t) = log₁₀(1 + t). Allometric scaling: y = α x^β, log-log linearity. | CONNECTION: The logarithmic measure (1/x) is the Haar measure on the multiplicative group ℝ⁺, isomorphic to additive ℝ via log. The golden ratio φ = 1.618 appears in allometric exponents (e.g., metabolic scaling β ≈ 3/4, but φ-related ratios occur in biological fractal branching). The mantissa uniformity implies that ratios like 0.618, 1.618 are equally likely as any other, but their frequent occurrence in natural scaling laws (e.g., phyllotaxis, spiral growth) reflects underlying self-similarit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com