Lévy Stable Processes Explain Benford's Law via Heavy-Tailed Scale Invariance — E8 Intelligence Research

FINDING: Benford's law is a scale-invariant first-digit distribution that emerges naturally from data generated by Lévy stable processes with characteristic exponent α, linking heavy-tailed distributions to universal logarithmic patterns. MATH: - Benford probability for first digit d: P(d) = log₁₀(1 + 1/d), d ∈ {1,…,9}. - Scale invariance: P(d) is invariant under multiplication by a constant (i.e., change of units). - Lévy stable characteristic exponent α ∈ (0,2] defines tail heaviness; α = 2 is Gaussian, α < 2 gives heavy tails. - For α-stable distributions, the sum of i.i.d. variables retains the same α (stability property). - Connection: The logarithmic density of first digits for α-stable laws converges to Benford's law as the scale parameter grows, due to the self-similarity and infinite divisibility of Lévy processes. CONNECTION: - The logarithmic base-10 structure of Benford's law reflects a natural geometric progression (ratio 10^(1/9) ≈ 1.2915 between successive Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com