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

notes

f-x

where the law came from

lineage

1881

simon newcomb

a canadian-american astronomer at the naval observatory, working out of a shared book of logarithm tables. the early pages were grey with use and the later ones were clean, and he took that as evidence rather than wear. he published two pages, gave the frequencies for the first and second digit, offered a short argument for why, and moved on. nobody followed it up.

note on the frequency of use of the different digits in natural numbers. american journal of mathematics 4(1), 1881, pp. 39–40read the two pages →

1938

frank benford

a physicist at general electric who found the same pattern fifty-seven years later without knowing newcomb had. he did the part newcomb skipped: 20,229 numbers from twenty unrelated tables, rivers and populations and atomic weights and street addresses and newspaper figures. the distribution held in all of them, and the law took his name.

the law of anomalous numbers. proceedings of the american philosophical society 78(4), 1938, pp. 551–572

his result, drawn from the formula

1961

roger pinkham

showed that if any law of first digits exists at all, and it does not change when every number is multiplied by the same constant, it has to be this one. change dollars to yen or metres to feet and a real dataset keeps its digits. that is why the law survives a change of units, and why invented numbers, which have no reason to be scale-free, fail it.

on the distribution of first significant digits. annals of mathematical statistics 32(4), 1961

1995

theodore hill

gave the proof that closed it. he showed that if numbers are drawn from a mixture of many different distributions, their first digits converge to the law even when none of the sources follow it on their own. that is the situation of any large pool of real transactions, which is what the counters on this site are.

a statistical derivation of the significant-digit law. statistical science 10(4), 1995, pp. 354–363

newcomb found it, benford measured it, pinkham explained why it could not be otherwise, and hill proved it. the hook only counts.