FRANK BENFORD
“the law of probability of the occurrence of numbers is such that all mantissae of their logarithms are equally probable.”simon newcomb, american journal of mathematics, vol. 4, 1881, pp. 39–40
this is the whole law, stated by the first person to notice it. a mantissa is the fractional part of a logarithm. if those are spread evenly, first digits are not, and the rest follows.
in 1881 simon newcomb noticed that the first pages of shared logarithm tables were worn and the last pages were clean. people looked up numbers beginning with 1 far more often than numbers beginning with 9. he wrote down the frequencies and nobody used them.
frank benford checked the same thing in 1938 against twenty thousand figures taken from rivers, street addresses, atomic weights, death rates and the front pages of newspapers. the distribution held across all of them. it holds for anything that spans several orders of magnitude and is not bounded or assigned.
it does not hold for numbers somebody chose. invented figures cluster on round values and repeat sizes, and their leading digits flatten toward uniform. this is why auditors and tax authorities run the test and why it survives as evidence.
the hook runs it on itself. the counters are its own swap flow and the charge is set by how far that flow sits from the law. nothing waits for a caller and there is nothing to claim.
the outlines on the plate never move. they are log10 of one plus one over the digit, computed in the page, and they would be identical on a pool that had never opened. only the fills are chain.
a pool doing ordinary business fills the outlines from the inside and the two shapes converge as the window fills. it will not be exact. eight thousand swaps is a sample, and a sample wanders, which is what the dashed band on the residual tab is for.
the shapes that matter are the ones a sample does not produce. a flat fill across all nine digits is a machine choosing sizes without thinking about them. a spike on one digit is a machine reusing one size. a fill that leans on 1 and 5 is a person typing round numbers.
none of this is an accusation and the hook does not make one. it prices what it measures and burns what it takes. the charge is a statement about the shape of the flow and nothing else.
this is not a fraud detector and nothing here is an accusation. a pool can fail the test for dull reasons, and a small number of very large trades will move the counters more than they should. the hook does not know why the shape moved and does not claim to.
it is not a fee that anyone set. the ceiling was fixed once and the position between zero and that ceiling is decided by the counters. there is no address that can raise it, lower it, waive it for one wallet, or turn it off for a weekend.
it is not a filter. nothing is blocked, nothing is reverted and no swap is refused. flow that looks manufactured is served at a worse price than flow that does not, and that is the entire consequence.
it is not a claim on anything. what the surcharge takes is burned. there is no treasury holding it, no pool distributing it, no wallet receiving it and no function that pays it out. the supply goes down and nothing else happens.
it is not clever. the law is a hundred and forty years old, the statistic is standard, and the whole hook is nine counters and a comparison. the only unusual thing about it is that it is being run by the thing it measures, continuously, in public, with no way to switch it off.