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This research investigates how scale invariance and multiplicative symmetry can explain the distribution of first digits in datasets according to Benford's law.

July 22, 20260 citationsOpen Access

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Scale Invariance and Multiplicative Symmetry Explain Benford's Law — E8 Intelligence Research

ACAndrew Stewart Caldin

Key Points

This research investigates how scale invariance and multiplicative symmetry can explain the distribution of first digits in datasets according to Benford's law.
Analysis of first-digit distributions based on Benford's law formula: P(d) = log₁₀(1 + 1/d) for digits 1 to 9.
Exploration of multiplicative symmetry groups and their relation to logarithmic invariance under scaling transformations.
Connection made between Benford's law and logarithmic structures such as the golden ratio and crystallographic symmetries.
The distribution of leading digits follows Benford's law with digit 1 occurring ~30.1% of the time and digit 9 ~4.6%.
Scale invariance leads to the conclusion that P(x) ∝ 1/x on a logarithmic scale, showcasing the unique Haar measure.
Connections found between Benford's law and the logarithmic spiral of the golden ratio, suggesting deeper mathematical relationships.

Abstract

FINDING: Scale invariance in natural data generates Benford's law via multiplicative symmetry groups, linking first-digit distribution to logarithmic invariance under scaling transformations. | MATH: Benford's law: P(d) = log₁₀(1 + 1/d) for d=1..9; leading digit 1 occurs ~30.1%, digit 9 ~4.6%. Underlying invariance: P(kx) = P(x) for any scaling factor k, requiring P(x) ∝ 1/x on logarithmic scale. Multiplicative group ℝ⁺ acts transitively on log-space, making Benford's law the unique Haar measure on the multiplicative group of positive reals. | CONNECTION: The logarithmic spacing of Benford's law mirrors the golden ratio's logarithmic spiral (φ = 1.618) and the base-60 sexagesimal system's natural logarithmic divisions. The 30.1% for digit 1 approximates 1/φ² ≈ 0.382, while digit 9 at 4.6% approximates 1/φ⁶ ≈ 0.056 (φ⁶ ≈ 17.944). The scale-invariant group structure aligns with crystallographic scaling symmetries in quasicrystals (e.g., Penrose tilings with φ scaling). | DEPTH: 8 — Conne Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Scale Invariance and Multiplicative Symmetry Explain Benford's Law — E8 Intelligence Research

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