f-x — the influx

fig. 1 — the nine counters against the law. the outlines are drawn from the formula. the fills are reads from the pool.

( method )

01

WHAT 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.

02

THE 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.

03

WHY 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.

04

WHAT THE LAW IS

the share of numbers beginning with digit d is log10(1 + 1/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.

05

THE 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.

06

THE 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.

07

THE CORRECTION

honest flow still scores above zero, because a sample is a sample. the expected amount of that noise is 8 / n, and exactly that much is subtracted. what is left is zero for real flow at any window size.

08

THE 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.

09

WHAT 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.

source — newcomb 1881, benford 1938, pinkham 1961, hill 1995

how far each digit sits from where the law puts it. above the line is too many, below is too few.

window8192 counted swaps, rolling

arming count1024 — no surcharge below this

floor0.001 ETH — anything smaller is not counted

ceiling300 bps

reference0.4017 — a perfectly flat digit spread

in the window nowno reads yet

the window is a queue, not an average. the 8193rd swap removes the 1st. nothing about the pool's past is preserved beyond it, in either direction.

the counter does not care how large a swap is, only what it begins with. this is deliberate: weighting by size would let one large trade set the reading, and the law is a statement about how many numbers begin with a digit — not how much they are worth.

1.4 ETHcounts one toward digit 1

19.8 ETHcounts one toward digit 1

0.0143 ETHcounts one toward digit 1

0.0004 ETHbelow the floor — not counted

9.9 ETHcounts one toward digit 9

PLATE I — THE ORIGINAL NOTE

note on the frequency of use of the different digits in natural numbers

simon newcomb, american journal of mathematics, vol. 4, 1881, pp. 39–40

PLATE I — THE CLAIM

"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

PLATE II — THE REDISCOVERY

the law of anomalous numbers

frank benford, proceedings of the american philosophical society, vol. 78, 1938, pp. 551–572. benford tabulated 20,229 figures from rivers, populations, atomic weights and street addresses, and found the same curve in all of them.

PLATE III — WHY IT HOLDS

on the distribution of first significant digits

roger pinkham, annals of mathematical statistics, vol. 32, 1961. pinkham showed the law is the only digit distribution that survives a change of units — the reason it does not care whether you measure in eth or in anything else.

PLATE IV — THE PROOF

a statistical derivation of the significant-digit law

theodore hill, statistical science, vol. 10, 1995. hill proved that if distributions are selected at random and samples taken at random from each, the leading digits converge to the law. this is why mixed, unplanned flow tends toward it — and why flow that was planned tends not to.

poolnot deployed

counted swaps—

distance—

adjusted—

surcharge—

the tape fills from the hook once the pool is live. nothing on this page is estimated or filled in — a counter with no reads shows a dash, not a number.

the instrument, run on numbers you choose. this is the same arithmetic the hook uses — it touches no chain and reads no pool. it is here so the claim can be checked rather than believed.

THE INSTRUMENThonest flow

swaps 2048

DISTANCE

—

raw

NOISE

—

8 / n, subtracted

ADJUSTED

—

what is charged on

SURCHARGE

—

basis points, burned

honest flow lands near zero at every window size — that is the correction working, not a coincidence. flat invented numbers sit near 0.4017 and take the full ceiling. round numbers sit between the two, which is the point: people inventing amounts reach for round ones, and roundness shows up in the first digit before it shows up anywhere else.

a law about digits not prices

real numbers are uneven

made up numbers are too even

the first pages were dirtier

thirty point one percent

eight thousand swaps is all it keeps

agreement proves nothing

disagreement proves something

( the token )

ticker$f-x

namethe influx

chainrobinhood chain

launched onpons

pairETH

contractnot deployed

ownernone — nothing above can be changed after deploy

buy $f-x on pons x ↗

the surcharge is a fee on trading, taken in token and burned. it is not a yield, a dividend or a payment to anyone — nothing is distributed and no address receives it. agreement with the law is not evidence that a pool is honest, and disagreement is not proof of fraud; the instrument measures one narrow property of one stream of numbers and nothing else. $f-x is a memecoin and can lose all of its value. nothing here is financial advice.