the two stacks

the same nine shares, side by side. a real pool makes the right stack a copy of the left.

the formula

one decade of a log scale. the law is nothing but these gaps.

( method )

01what is counted

the hook takes the leading digit of the eth amount of every swap through the pool and adds one to that digit's counter. nine counters, one instruction, no weighting by size. a swap of one eth and a swap of nineteen eth both add one to the counter for 1.

02the floor

swaps under a thousandth of an eth are not counted at all. every dust amount begins with the same handful of digits, so without a floor anyone could push the counters toward the law for almost nothing and buy themselves a cheaper rate. the floor is the whole defence against that and it cannot be changed.

03why the first digit

in real quantities the leading digit is not uniform. it is 1 about thirty percent of the time and 9 under five. quantities that span orders of magnitude are spread evenly in log, not in count, so the span from 1 to 2 is wider than the span from 9 to 10 and more numbers land in it.

04what the law is

the share of numbers beginning with digit d is log10 of one plus one over d. that is the whole formula. it has no parameters, nothing was fitted to anything, and it is the same on every pool on every chain.

05the window

the counters hold the last 8192 counted swaps and no more. when the window is full the oldest digit is removed as the newest arrives. a clean early history does not buy cover for what the pool is doing now, and a bad patch does not follow it forever.

06the distance

the squared gap between each counter's share and the law's share, divided by the law's share, summed over nine digits. zero is a perfect match. it is computed on shares rather than raw counts so it does not grow with volume alone.

07the correction

honest flow still scores above zero, because a sample is a sample. the expected amount of that noise is eight divided by the number of swaps in the window, and exactly that much is subtracted. what is left is zero for real flow at any window size.

08the charge

the adjusted distance is measured against 0.4017, which is where a perfectly flat digit spread sits and where invented numbers tend to drift. the surcharge runs from zero to three hundred basis points across that range, is taken in token, and is burned. it is zero until the window holds 1024 swaps.

09what is fixed

the law, the window, the arming count, the ceiling, the floor and the burn were set at deploy. no address can change any of them. there is no owner, no treasury, no keeper, no mint, no pause and no upgrade path.

sourcenewcomb 1881, benford 1938, pinkham 1961, hill 1995

plates

f-x

five ways to see the same flow

what to look for

the plates are five views of one set of nine numbers. none of them contain anything the others do not. they exist because the same deviation is obvious in one view and invisible in another, and which view catches it depends on what produced it.

a flat fill across all nine digits is the signature of sizes chosen by a machine that was not thinking about size. on the first plate it looks like nine bars of nearly equal height inside outlines that are not. on the residual it looks like a strong negative bar on 1 and positive bars on 6 through 9.

a spike on a single digit is one address reusing one size. it barely shows on the first plate and it is unmistakable on the fourth, where a reused size draws a hard horizontal stripe across the scatter of the log fraction.

a lean toward 1 and 5 is a person typing round numbers, which is not manufactured volume and will still cost more. the hook cannot tell the difference between a round number and a fabricated one, because in digits they look the same.

the second digit grid is the one that catches effort. building flow that satisfies the law on the first digit is easy, and doing it on the first and second at once is considerably harder, so a pool that passes the first plate and fails the grid is a pool where somebody tried.

fig. 2 each digit's law share against its measured share. a real pool puts every dot on the line. a dot above the line is a digit appearing too often.

fig. 3 the expected share of every first and second digit pair, with measured shares filling in on top. the hook stores first digits only, so this is computed here from the log.