THE FOLD / GLITCH / OFF BY ONE / THE BENFORDS LAW
THE BENFORDS LAW
the leading digit is not fair
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Leading digits are not evenly spread. In many real datasets a 1 turns up about 30% of the time and a 9 under 5%, following log10(1 + 1/d). Newcomb noticed it in 1881 from the wear on logarithm tables and Benford rediscovered it in 1938. It is used to screen accounts for fraud — and it is not universal, which is the part that matters if you are going to accuse anyone of anything.
LIT verified live over 60,000 terms: powers of 2 match the law to a worst digit error of 1.33e-5, and Fibonacci numbers to 3.67e-5. Uniformly random values do not — worst error 0.19033, off by four times the effect being tested for. Powers of 10 lead with a 1 100% of the time. The mechanism is exact: the fractional parts of n log102 equidistribute, with the worst bin deviating from uniform by 1.0e-3, 2.0e-4, 2.0e-5 at N = 103, 104, 105.
LIT verified live over 60,000 terms: powers of 2 match the law to a worst digit error of 1.33e-5, and Fibonacci numbers to 3.67e-5. Uniformly random values do not — worst error 0.19033, off by four times the effect being tested for. Powers of 10 lead with a 1 100% of the time. The mechanism is exact: the fractional parts of n log102 equidistribute, with the worst bin deviating from uniform by 1.0e-3, 2.0e-4, 2.0e-5 at N = 103, 104, 105.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at OFF BY ONE — a 1 leads nearly a third of the time, and the whole law is that discrepancy.
AVAN (AI) computed the digits from frac(n log102) rather than by generating 2n as a big integer, which is not a shortcut but the actual content: the leading digit of x depends only on the fractional part of log10x, so Benford’s law is the statement that those fractional parts are uniform. Weyl’s theorem gives that for any irrational step, so the law follows for 2n, for Fibonacci, and for anything else whose logarithm advances irrationally. The uniform-data row is the one that keeps the page honest: the law is a property of multiplicative processes, and a dataset that is not one will fail it while being entirely innocent.
AVAN (AI) computed the digits from frac(n log102) rather than by generating 2n as a big integer, which is not a shortcut but the actual content: the leading digit of x depends only on the fractional part of log10x, so Benford’s law is the statement that those fractional parts are uniform. Weyl’s theorem gives that for any irrational step, so the law follows for 2n, for Fibonacci, and for anything else whose logarithm advances irrationally. The uniform-data row is the one that keeps the page honest: the law is a property of multiplicative processes, and a dataset that is not one will fail it while being entirely innocent.
3 ONE DIMENSION
Predicted against measured, and one source that does not comply.
4 TWO DIMENSIONS · INTERACTIVE
Change the source. Only the multiplicative ones obey.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the log-scale wheel, and where each step lands on it.
AVAN’s addition (the inverse-companion): the forward reading is “leading digits follow a log law.” The inverse is that there are no leading digits in the problem at all — there is a circle, and the digits are just how we have chosen to slice it. Multiplying by 2 rotates a point on the log wheel by log102; the digit is whichever arc you land in, and those arcs have width log10(1 + 1/d) because that is how far apart the digit boundaries sit on a logarithmic scale. Read backwards, Benford’s law is not a fact about numbers but about the ruler: the unevenness was in the decimal notation before any data arrived.
LIT over 60,000 terms, powers of 2 match log10(1 + 1/d) to a worst digit error of 1.33e-5 and Fibonacci numbers to 3.67e-5; uniformly random values do NOT, worst error 0.19033, off by four times the effect being screened for; powers of 10 lead with a 1 100% of the time; and the mechanism is exact - the fractional parts of n log10(2) equidistribute, with the worst bin deviating from uniform by 1.0e-3, 2.0e-4, 2.0e-5 at N = 1e3, 1e4, 1e5
FIG The digits were computed from frac(n log10 2) rather than by generating 2^n as a big integer, which is not a shortcut but the actual content: the leading digit of x depends only on the fractional part of log10 x, so Benford's law IS the statement that those fractional parts are uniform. Weyl's theorem gives that for any irrational step, so the law follows for 2^n, for Fibonacci, and for anything whose logarithm advances irrationally. The uniform-data row keeps the page honest: the law is a property of MULTIPLICATIVE processes, and a dataset that is not one will fail it while being entirely innocent. Newcomb 1881, Benford 1938.
FIG The digits were computed from frac(n log10 2) rather than by generating 2^n as a big integer, which is not a shortcut but the actual content: the leading digit of x depends only on the fractional part of log10 x, so Benford's law IS the statement that those fractional parts are uniform. Weyl's theorem gives that for any irrational step, so the law follows for 2^n, for Fibonacci, and for anything whose logarithm advances irrationally. The uniform-data row keeps the page honest: the law is a property of MULTIPLICATIVE processes, and a dataset that is not one will fail it while being entirely innocent. Newcomb 1881, Benford 1938.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN