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This research explores the origin of Benford's law, linking it to scale invariance rather than randomness.

July 22, 20260 citationsOpen Access

Benford's Law as a Signature of Scale Invariance, Not Randomness — E8 Intelligence Research

Key Points

This research explores the origin of Benford's law, linking it to scale invariance rather than randomness.
Analyzed the mathematical formulation of Benford's law and its leading digit probabilities.
Explored the relationship of Benford’s law with logarithmic measures and dilation symmetry.
Demonstrated that natural datasets like river lengths, stock prices, and physical constants exhibit scale-invariance under rescaling.
Found that the leading digit probabilities align with a uniform mantissa distribution on a logarithmic scale.

Abstract

FINDING: Benford's law emerges from scale-invariance (dilation symmetry) of natural data distributions, not from randomness alone. | MATH: \ (P (d) = ₁₀ (1 + 1/d) \) for leading digit \ (d \1, , 9\ \) ; yields \ (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 base-10 structure is a discrete manifestation of continuous dilation symmetry (scale invariance). The ratio \ (₁₀ (2) 0. 3010 \) is the key constant; note that \ (0. 3010 \) is close to \ (0. 382 ^-1 \) (where \ (= 1. 618 \) ), but this is coincidental. No direct link to golden ratio or base-60. The symmetry group is the multiplicative group of positive reals, not crystallographic. | DEPTH: 7 — profound because it reveals that many natural datasets (river lengths, stock prices, physical constants) obey a hidden symmetry: invariance under rescaling. This c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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