What question did this study set out to answer?

This research aims to demonstrate that Benford's law arises from scale-invariance rather than random chance.

July 22, 20260 citationsOpen Access

Benford's Law Arises from Scale-Invariance, Not Random Chance — E8 Intelligence Research

Key Points

This research aims to demonstrate that Benford's law arises from scale-invariance rather than random chance.
Analyzed the mathematical expression underpinning Benford's law, P(d) = log₁₀(1 + 1/d).
Explored connections to geometric ratios and logarithmic measures.
Discussed implications of dilation symmetry groups in relation to self-similar structures.
Benford's law shows that leading digit probabilities are not uniformly random but scale-invariant.
Demonstrated that the law's invariance relates to geometric ratios corresponding to logarithmic intervals.
Highlighted that variations in base logarithms maintain the law's core properties, regardless of the base used.

Abstract

FINDING: Benford's law emerges from scale-invariance (dilation symmetry) of natural data distributions, not from random chance. | MATH: P (d) = log₁₀ (1 + 1/d) for leading digit d ∈ 1, …, 9; P (1) ≈ 0. 3010, P (9) ≈ 0. 0458. The law is equivalent to the mantissa distribution being uniform on [0, 1) under logarithmic measure. | CONNECTION: The logarithmic spacing 1. 0 → 2. 0 → 3. 0 … → 9. 0 → 10. 0 corresponds to geometric ratios: the interval [1, 2) occupies log₁₀ (2) ≈ 0. 301 of the log cycle, matching the golden ratio's logarithmic footprint (φ ≈ 1. 618 → log₁₀ (φ) ≈ 0. 209, not directly 0. 301). However, the law's invariance under scaling (multiplying all data by a constant) is a dilation symmetry group isomorphic to ℝ⁺, which is the same continuous symmetry underlying self-similar structures in nature (e. g. , fractal geometries, logarithmic spirals). The base-10 logarithm is arbitrary; in base b, P (d) = logb (1 + 1/d). For b = φ² ≈ 2. 618, the digit 1 probability becomes log₂. ₆₁₈ (2) ≈ 0. 724, a high Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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