Benford's Law as Logarithmic Scale Invariance — E8 Intelligence Research

FINDING: Benford's law arises from scale invariance under multiplication, linking first-digit distribution to logarithmic group structure on positive reals. | MATH: P(d) = log₁₀(1 + 1/d) for d=1..9; key constant log₁₀(2) ≈ 0.3010; scale invariance: P(d) unchanged under multiplication by any constant c>0. | CONNECTION: The ratio log₁₀(2) ≈ 0.3010 is not a classical harmonic ratio, but the underlying multiplicative group ℝ⁺ is isomorphic to additive ℝ via log, revealing a deep symmetry akin to logarithmic spirals (golden ratio φ appears in natural growth contexts). The base-10 logarithm choice is arbitrary; base-φ would yield different constants. | DEPTH: 6 — Benford's law is a profound consequence of scale invariance in multiplicative groups, but its constants (0.3010, 0.1761, etc.) are not directly the golden-ratio-based numbers you listed. The connection to geometric harmony is indirect: the law reflects the uniform distribution of logarithms modulo 1, which is the same structure unde Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com