a rational curve

supply is a function of position. position is the running total of eth that has ever entered the curve. the function is K times one minus S squared over the square of S plus position. K is 33 554 432 tokens. S is 512 eth. the function starts at zero, reaches half of K a little past 212 eth, three quarters of K at exactly 512 eth, and fifteen sixteenths at 1 536 eth. it never reaches K.

why the cube

differentiate supply with respect to position and invert it and you get the marginal price: the cube of S plus position, over twice K times S squared. at position zero that is 7 629 394 531 250 wei per token. at 512 eth it is eight times that. at 1 536 eth it is sixty four times. the price of the next token is a cubic in the depth of the curve, which is a steeper law than the square used by most curves of this shape and a much steeper law than a constant product pool.

integers only

the contract never evaluates a square root, a logarithm, or an exponential. a mint computes supply before and supply after with one full precision multiply and divide each, rounded down, and issues the difference. the site reproduces that arithmetic in bigint and the worked mint plate prints every intermediate. the twelve positions table is the same code evaluated at twelve points.

the wedge

book is the cistern divided by supply. with zero pressure and no burns, book at a position is that position's eth over that position's supply, and it always sits under the marginal price because the marginal price is the price of the last token while book averages every token that came before it. pressure pushes book up further, because surcharge eth enters the cistern without tokens. burns push book up further still, because the fee stays. the gap between the amber line and the blue line on the plate is the region a burn can never reach into.