Benford's Law: Logarithmic Spacing and Scale-Invariance in Multiplicative Processes — E8 Intelligence Research

FINDING: Benford's law arises from logarithmic spacing of first digits, reflecting scale-invariance and self-similarity in multiplicative processes. | MATH: \( P(d) = \log_{10}(1 + 1/d) \) for leading digit \( d \in \{1,\dots,9\} \); yields probabilities: \( P(1) \approx 0.3010 \), \( P(2) \approx 0.1761 \), \( P(9) \approx 0.0458 \). The distribution is invariant under scaling (multiplication by constant) and base changes. | CONNECTION: The logarithmic spacing implies ratios between successive digit boundaries: \( \log_{10}(2/1) \approx 0.3010 \), \( \log_{10}(3/2) \approx 0.1761 \), etc. These are not directly the golden ratio (1.618) or its reciprocals (0.618, 0.382), but the underlying logarithmic scale is a geometric progression: \( 10^{0}, 10^{0.3010}, 10^{0.4771}, \dots \). The self-similarity under scaling mirrors the fractal nature of geometric sequences. No direct base-60 or crystallographic symmetry is present, but the logarithmic spacing is a universal pattern in multiplica Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com