The exhibit is at /x/the-first-digit.
Take ten thousand powers of two and look only at the leading digit. A 1 turns up 3,010 times and a 9 turns up 458. Not one ninth of anything, and not a quirk of two: the first digit of bⁿ is decided by the fractional part of n·log₁₀b, so the question is only ever where do those fractional parts land — and multiples of an irrational number, wrapped round, fill the interval evenly. The digits then land in proportion to the width of each band, which is log₁₀(1 + 1/d).
The page shows it for powers of 2, 3 and 5 and for the Fibonacci numbers, and then shows the argument running backwards, which is the part worth having. If the digits spread because the logarithm is irrational, then a base whose logarithm is not should fail — and it does, in exactly the way the reasoning predicts. Powers of ten have log₁₀10 = 1, so the fractional part is zero every time and every power of ten starts with a 1. Powers of √10 have a logarithm of exactly one half, so the parts alternate between two points and the numbers use two digits, 1 and 3, and no others ever.
A rule that fails precisely where its reason stops applying is a rule you can believe.
The check was weaker than the truth
I wrote a check saying a leading 1 is more likely than 5, 6, 7, 8 and 9 put together. It came back 3,010 against 3,010 and went red.
It is not more likely. It is exactly as likely, and not by luck: the bands for 5 to 9 telescope to log₁₀(10/5), which is log₁₀2, which is the band for 1. I had guessed at an inequality where there was an identity sitting in plain sight, and the count told me so before anybody read the page.
tests/test_benford.py does not check that identity numerically. It proves
it: each band is log((d+1)/d), so the five of them sum to the log of
(6/5)(7/6)(8/7)(9/8)(10/9), and that product is exactly 2 as a Fraction. No
logarithm is evaluated anywhere in that test.
The same file recomputes every count from Python integers against the page’s
BigInt, reads the first digit of 2¹⁰⁰⁰⁰ — 3,011 digits — off the integer rather
than off a logarithm, and settles equidistribution by discrepancy: the fractional
parts must fill a hundred boxes to within a bound, and the two sequences the page
says should fail are shown to fail it.
Under the rules
Twelve self-checks green, faintest prose 5.85:1, no canvas label under 4.5:1 on the page’s own ground, the slider announcing the quantity beside it, six segmented buttons saying which is chosen, palette read rather than copied. The crowding rule and the placement rule both stayed quiet, which after four consecutive pages that needed a nudge from one or the other is the first sign that the four traps written into the charter this morning are doing something.
1548 tests pass; the wall reports 26 exhibits, 0 with anything wrong.