Derivations, geometry, biology, and the words.
Newton's law is dT/dt = −λ(T − T_amb). Separating and integrating from the last touch gives the closed form the contract stores nothing extra to use:
The contract computes the exponential with a 64-bit fixed-point series accurate to nine decimals, the same routine reproduced by the site's client so the live needle and the chain can never disagree by more than dust.
Degree-hours are the exact area between the cooling curve and the floor. With E0 = T0 − 21.0 and the floor excess F = 15.0, the crossing time and the integral are:
The same expression with the ceiling excess 18.5 in place of 15.0, taken only while positive, is the scorch integral. Two integrals, one antiderivative, no sampling error.
Scored pip weight is floor(sqrt(burn)). Concavity is the whole argument: sqrt(a) + sqrt(b) is at least sqrt(a + b), with equality only when one of them is zero, so splitting a burn across two pips never scores less and usually scores more. A whale splitting across many addresses gains nothing over the crowd it is imitating, because the zip's sector order still forces the burns to land in twenty-four different draws of keccak256(caller) mod 24. The mechanism cannot tell a crowd from a patient whale, and does not need to: either way the shell comes apart in many hands' worth of pieces.
The circumference is divided into 24 sectors of 15 degrees, numbered clockwise from the top of the blunt end. A caller's sector is keccak256(address) mod 24 and never changes. The zip pointer starts at the first sector ever scored and advances only clockwise, one sector at a time, so closing the circle requires callers whose addresses cover all twenty-four residues in an order none of them chose. The expected number of distinct random addresses needed to cover all residues is 24 × H(24), about 90.6, though coordinated holders who check their residues first can close it with exactly 24.
A caller's residue is fixed by their address; the circle closes only when all twenty-four are covered.
Everything mechanical here is stolen from the hen. 37.8 °C is the true set point of gallus incubation; 900 degree-hours above a 36-degree floor is five hundred set-point hours, the twenty-one days a clutch actually takes. Eggs are turned because the embryo will adhere to the shell membrane if left still, and commercial incubators turn on ninety-minute cycles, which is where turn()'s gap comes from.
The yolk is built in concentric rings laid down daily, visibly dark and light under section, which is what the ring archive draws. The air cell at the blunt end grows as moisture leaves the pores, and candlers grade eggs AA, A, and B by exactly that measure, without ever opening the shell.
A hatching chick pips internally into the air cell first, then externally, then zips: a clockwise rotation of chips around the blunt end's circumference before the cap gives. It is the only part of the process that cannot be done slowly, and the only part that cannot be done alone twice.
| TERM | DEFINITION |
|---|---|
| ambient | 21.0 °C, the temperature of a dead market. |
| band | 36.0 to 39.5 °C, where the clock runs. |
| clock | the degree-hour accumulator. |
| crossing | a stage boundary passed. permanent. |
| degree-hour | one degree above the floor held for one hour. |
| drag | the permanent slowdown adhesion buys. |
| epoch | 86,400 seconds. one ring. |
| grade | AA, A, or B. the air cell's verdict. |
| mis-pip | a burn in the wrong sector. supply pays, shell does not. |
| pair | T_last and t_last. the whole thermal state. |
| pip | a scored chip at the shell. |
| plate | the drawn cross-section built from the rings. |
| roll | the list of everyone who turned the egg. |
| scorch | degree-hours above the ceiling. raises the requirement 2.5 to 1. |
| settle | the instruction that moves time. runs first, always. |
| set point | 37.8 °C. the hen. |
| tone | dark or light. did the day feed the egg. |
| zip | the clockwise closing of all 24 sectors. |