The complete thermal model: injection, decay, the band, the clock, and the scorch.
The entire thermal state is two numbers: the temperature at the last touch, and the time of the last touch. Everything else is derived. This page reproduces every derivation the contract performs, with the constants it performs them with.
| QUANTITY | VALUE |
|---|---|
| T_last | 21.000 °C |
| t_last | |
| T now | 21.000 °C |
| time since touch | |
| clock running | no |
Heat in is linear in the fraction of supply moved and blind to everything else: dT = 42.0 × amount / 1,000,000,000. The per-block cap of 1.5 °C truncates, never queues; excess warmth is simply lost.
| AMOUNT MOVED | dT (°C) | |
|---|---|---|
| 0.01% | 0.0042 | |
| 0.05% | 0.0210 | |
| 0.10% | 0.0420 | |
| 0.50% | 0.2100 | |
| 1.00% | 0.4200 | |
| 2.00% | 0.8400 | |
| 3.57% | 1.4994 | |
| 5.00% | 1.5000 | cap |
Cooling follows T(t) = 21.0 + (T0 − 21.0)·e^(−λt), half-life six hours, λ = ln 2 / 21600 s. The table gives the survival of excess warmth over ambient.
| ELAPSED | REMAINING FRACTION OF EXCESS | TEMPERATURE AFTER (°C) |
|---|---|---|
| 30 min | 0.9439 | 36.86 |
| 1 h | 0.8909 | 35.97 |
| 2 h | 0.7937 | 34.33 |
| 3 h | 0.7071 | 32.88 |
| 6 h | 0.5000 | 29.40 |
| 9 h | 0.3536 | 26.94 |
| 12 h | 0.2500 | 25.20 |
| 18 h | 0.1250 | 23.10 |
| 24 h | 0.0625 | 22.05 |
| 36 h | 0.0156 | 21.26 |
From the set point, the egg falls out of band in under one hour of silence.
The floor at 36.0 °C is where development stops; the ceiling at 39.5 °C is where it turns to damage. The viable band is 3.5 °C wide. From the set point, silence crosses the floor in 59 minutes; sustained heavy circulation crosses the ceiling in a single hot hour. The band is drawn to scale in figure T2.
The viable band is 3.5 °C wide. Above the ceiling, warmth is damage.
Degree-hours accumulate as the exact integral of (T − 36.0) while positive. Between touches the contract computes the crossing time t* = (1/λ)·ln((T0 − 21.0)/(15.0)) and integrates the closed form up to min(Δt, t*). The full derivation, including the antiderivative and the stopped-clock case, is appendix A2.
| CONSTANT HELD TEMPERATURE (°C) | DH GAINED PER HOUR | DAYS TO 900 DH |
|---|---|---|
| 36.5 | 0.5 | 75.0 |
| 37.0 | 1.0 | 37.5 |
| 37.5 | 1.5 | 25.0 |
| 37.8 | 1.8 | 20.8 |
| 38.5 | 2.5 | 15.0 |
| 39.4 | 3.4 | 11.0 |
Scorch is the same integral taken above 39.5 °C, multiplied by 2.5 and added to the requirement forever. The requirement only rises. The current requirement is the base 900 plus the scorch surcharge:
| QUANTITY | DEGREE-HOURS |
|---|---|
| BASE | 900.000 |
| SCORCH SURCHARGE | 0.000 |
| CURRENT REQUIREMENT | 900.000 |
Adhesion converts neglect into permanent slowness: rate factor = 10000 / (10000 + 25a) basis points, where a is the adhesion count. The table gives the factor at selected counts.
| a | FACTOR | EFFECTIVE DH/HOUR AT SET POINT |
|---|---|---|
| 0 | 1.0000 | 1.800 |
| 4 | 0.9901 | 1.782 |
| 8 | 0.9804 | 1.765 |
| 20 | 0.9524 | 1.714 |
| 40 | 0.9091 | 1.636 |
| 80 | 0.8333 | 1.500 |
| 200 | 0.6667 | 1.200 |
The whole machine, as a circuit: the market is a current source, the shell is a capacitor, the six-hour half-life is a resistor to ambient, and the clock is a meter that only counts while the node voltage sits inside the band. Figure T1 is exact, not decorative: every element is labeled with the constant that governs it.
The market is a current source, the shell a capacitor, the six-hour half-life a resistor to ambient, and the clock a meter that counts only inside the band.