04. When the Law Does Not Apply

The limits of scale invariance

Benford’s law is frequently misunderstood as a universal guarantee that applies to any numerical dataset provided the numbers span a sufficient range of magnitudes. Theodore P. Hill (2020) argues in his arXiv paper that this common shorthand regarding orders of magnitude is actually a widespread error in statistical analysis. To test this, the 2026 AICE units experiment examined 132 of the world's longest rivers, which span only 0.84 orders of magnitude in their measurements. The resulting mantissa KS distance of 0.3764 in this 2026 study demonstrates that narrow-range data often fails to conform to the expected Benford distribution. Conformity is not an inherent property of data scale but rather a conditional outcome that requires specific underlying mathematical distributions to function as predicted. The assumption that a wide span of numbers automatically ensures Benford-like behavior is a heuristic that frequently misleads those who rely on it for fraud detection.

The problem of assigned numbers

Assigned numbers such as phone numbers, ZIP codes, or specific identification numbers are inherently structured by human design rather than natural processes or growth. Because these identifiers are typically formatted to satisfy bureaucratic requirements, they almost never exhibit the logarithmic distribution characteristic of Benford's law. The 2026 AICE units experiment further illustrated this by measuring the mean height of individuals across 195 countries, finding a span of only 0.06 orders of magnitude. This specific dataset produced a mantissa KS distance of 0.7357, showing how imposed minimums or narrow physical constraints lead to total nonconformity in the distribution. When the leading digits shift drastically based on the chosen unit—such as 100% leading-1s in metres versus 99.0% leading-5s in feet—the failure of the law is absolute. Human-assigned data remains fundamentally incompatible with the law because the digits are chosen for convenience or classification rather than emerging from a multiplicative process.

The dispute over the rule

The popular rule requiring several orders of magnitude for compliance is contested by Theodore P. Hill (2020) who labels it a mega-error. This in-house dispute targets the shorthand currently employed by organizations such as Wolfram Research, the Election Integrity Partnership, and Reuters Fact Check. The reliance on this flawed heuristic risks misapplying the law in contexts where it lacks the necessary mathematical foundation to provide valid evidence.

Example: Expecting a list of ZIP codes to obey Benford’s law is like expecting a list of names to obey the laws of planetary motion.

Verdict. The law does not apply to assigned, bounded, or narrow-range data.

Overturned if: A rigorous mathematical proof demonstrates that narrow-range datasets with high KS distances still reliably predict digit frequencies in a meaningful way.

Sources. 2020 · Theodore P. Hill, "A Widespread Error in the Use of Benford's Law to Detect Election and Other Fraud" — arXiv:2011.13015 · 1995 · Theodore P. Hill, "A Statistical Derivation of the Significant-Digit Law," Statistical Science 10(4) — Project Euclid · 2026 · AICE, "The units experiment" — source/experiment.py