Benford's Law as Logarithmic Uniformity from Scaling Invariance — E8 Intelligence Research

FINDING: Benford's law arises from the multiplicative symmetry group of positive reals, specifically the invariance of the leading-digit distribution under scaling (multiplication by a constant). This is equivalent to the distribution being uniform on a logarithmic scale (base 10 or any base b > 1). MATH: - Probability of first digit d (base 10): P(d) = log₁₀(1 + 1/d), for d = 1,2,…,9. - General base b: P(d) = log_b(1 + 1/d). - The underlying measure is the Haar measure on the multiplicative group of positive reals: dμ(x) = dx/x (logarithmic measure). - Scale invariance: For any constant c > 0, the distribution of leading digits of cX is the same as that of X. - Fourier analysis shows that a random variable is Benford iff its logarithm modulo 1 is uniformly distributed on [0,1). This connects to equidistribution theory and Weyl's criterion. CONNECTION: - The logarithmic measure dx/x is invariant under scaling, which is the continuous analogue of the multiplicative symmetr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com