Benford's Law as the Unique Scale-Invariant Measure on Positive Reals — E8 Intelligence Research

FINDING: Benford's law arises from scale invariance of the multiplicative group of positive real numbers; the leading-digit distribution is the unique probability measure invariant under scaling by any constant factor. | MATH: For base \(b\), \(P(d) = \log_b(1 + 1/d)\), \(d \in \{1,2,\dots,b-1\}\). For base 10: \(P(1) = \log_{10}(2) \approx 0.3010\), \(P(9) = \log_{10}(10/9) \approx 0.0458\). The underlying group is \((\mathbb{R}^+, \times)\), with Haar measure \(dx/x\). Scale invariance: \(P(d) = P(d \cdot c)\) for any \(c>0\) modulo cyclic group of powers of \(b\). | CONNECTION: The ratio \(\log_{10}(2) \approx 0.3010\) is not a classical harmonic ratio, but the logarithmic spacing relates to the golden ratio via \(\log_{10}(\phi) \approx 0.2089\) (not a leading digit). However, the multiplicative group structure is isomorphic to the additive group of reals via logarithm, linking to exponential growth patterns in nature (e.g., spiral phyllotaxis, logarithmic spirals with golden ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com