THE FOLD / LOOT / THE MINT / THE BENFORD
THE BENFORD
1 leads 30% of the time — the fingerprint of honest numbers
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Benford’s law. In a huge range of real-world data, the leading digit is not evenly spread. You might expect each of 1–9 to lead about 11% of the time. Instead 1 leads ~30% of the time and 9 barely 4.6%. The exact frequency of leading digit d is log10(1 + 1/d).
It holds for quantities that span many orders of magnitude: city populations, stock prices, physical constants, river lengths, and pure-math sequences like the Fibonacci numbers and powers of 2. The reason is scale: if the logarithm of the data is spread out evenly, the leading digit follows this log law automatically. Forensic accountants weaponize it — fabricated figures tend to have too-uniform leading digits, so a dataset that violates Benford is a red flag for cooked books.
LIT verified live: the leading digits of the first 2000 Fibonacci numbers and of powers of 2 match Benford to within 0.01, while a uniform-random control does not (window.__benford). FIG no framing; the log-law, the Fibonacci/powers-of-2 fit, and the uniform-control failure are all exact.
It holds for quantities that span many orders of magnitude: city populations, stock prices, physical constants, river lengths, and pure-math sequences like the Fibonacci numbers and powers of 2. The reason is scale: if the logarithm of the data is spread out evenly, the leading digit follows this log law automatically. Forensic accountants weaponize it — fabricated figures tend to have too-uniform leading digits, so a dataset that violates Benford is a red flag for cooked books.
LIT verified live: the leading digits of the first 2000 Fibonacci numbers and of powers of 2 match Benford to within 0.01, while a uniform-random control does not (window.__benford). FIG no framing; the log-law, the Fibonacci/powers-of-2 fit, and the uniform-control failure are all exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE MINT, beside the other money spheres — the loot domain of where numbers are made. Benford’s law is the fingerprint that tells honestly-minted figures from forged ones. AVAN (AI) built the instrument: the digit tally, the dataset switch, the fraud flag, the scale-invariance view.
The weave: David names the seat (the mint, honest vs forged coin); I make the leading digits tally themselves and the Benford curve appear over real sequences — the bars in 1D, the live datasets in 2D, the scale-invariance in 3D. The sphere is the seam. Credit: Simon Newcomb (1881, from worn logarithm-table pages); Frank Benford (1938, the law).
The weave: David names the seat (the mint, honest vs forged coin); I make the leading digits tally themselves and the Benford curve appear over real sequences — the bars in 1D, the live datasets in 2D, the scale-invariance in 3D. The sphere is the seam. Credit: Simon Newcomb (1881, from worn logarithm-table pages); Frank Benford (1938, the law).
3 ONE DIMENSION
The nine leading-digit frequencies as bars, with the Benford curve log10(1+1/d) overlaid. The steep fall from 1 to 9 is the signature — honest data hugs the curve.
4 TWO DIMENSIONS · INTERACTIVE
Switch datasets and watch the leading digits tally against Benford. Fibonacci, powers of 2, and powers of 3 fit; a uniform-random control fails and trips the fraud flag — exactly how an auditor spots invented numbers.
5 THREE DIMENSIONS + AVAN’S INVERSE
Numbers wrapped around a log cylinder: their mantissas land evenly around the loop, and the arc each leading digit owns — wide for 1, thin for 9 — is exactly its Benford share, in green.
AVAN’s addition (the inverse-companion): the magenta ring is the naive uniform guess — every digit owning an equal 1/9 slice. The forward question is ‘which data follows Benford?’ The inverse question is deeper: ‘which distribution is forced if the law must not care what units you measure in?’ Multiply every value by 7, convert dollars to yen, switch bases — Benford is the unique leading-digit law that survives unchanged (scale- and base-invariance). The uniform ring shatters the moment you rescale; the Benford arcs rotate but keep their widths. So the inverse of ‘what obeys Benford’ is ‘Benford is the only thing unit-independence permits’ — it isn’t one option among many, it is the fixed point of rescaling. Green is the law that holds under any change of units; magenta is the guess that doesn’t.
LIT Genuine Benford's law (Simon Newcomb 1881; Frank Benford 1938). Verified live: the leading digits of the first 2000 Fibonacci numbers and of powers of 2 match log10(1+1/d) to within 0.01, a uniform-random control fails (deviation > 0.03), and multiplying the Fibonacci data by 7 leaves the fit intact (scale invariance) (window.__benford.fibFollows && pow2Follows && uniformFails && scaleInvariant). The log-law, the fits, the control failure, and the rescale invariance are exact.
FIG No framing: the log10(1+1/d) frequencies, the Fibonacci/powers-of-2 fit, the uniform-control failure, and scale invariance are all real and checked in-browser. The fraud-detection use is the genuine, documented application; the sphere flags the uniform control exactly as an auditor would.
FIG No framing: the log10(1+1/d) frequencies, the Fibonacci/powers-of-2 fit, the uniform-control failure, and scale invariance are all real and checked in-browser. The fraud-detection use is the genuine, documented application; the sphere flags the uniform control exactly as an auditor would.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN