Dough, mould and time

The yeast rule is three smooth curves multiplied together. A warmth curve climbs exponentially to a best temperature and falls away above it. Salt and heavy sugar pull water out of the yeast, as osmotic stress does, while a little sugar feeds it. And the gas made grows with the dose of yeast, with diminishing returns for big doses.

The dough keeps only part of that gas, and how much depends on the gluten: well-kneaded dough keeps more and can hold more. Over one piece of time the gas share has a closed form, the exact solution of a foam filling towards what its gluten can hold. That's what lets a long step stay accurate without many small ones.

As it makes gas the yeast eats its food, and above a threshold heat kills it at a rate that roughly doubles every few degrees. Once the gas stretches the gluten close to its limit, a smooth tearing term weakens the gluten and lets gas out, and the dough sinks.

A long step is cut into a few pieces by how much gas it makes. A test holds one spot's end and peak within 3 % of very fine steps, for an hour's rise and for three days.

Oven spring is worked out for each spot on its own. The spot warms on a short ramp from room temperature towards the oven's heat, held at boiling. The gas swells by the real ideal-gas ratio of absolute temperatures, through the same closed form as the yeast's gas, so only what the gluten keeps counts. Steam adds a flat extra factor on top.

Lamination. Cold butter laid on dough adds sheets of fat. A fold stacks them; rolling spreads the same count over a wider sheet. Sliding smears them, much faster once the butter is past its melting point, and too many sheets in too little height fuse into one. In the oven the layers lift the dough, and enough layers over a big enough area count as pastry.

The fizz enters through the same rule as the yeast's gas, so the gluten decides what it keeps.

State lives per sprout, not per spot, in a fixed pool with room kept back for hand sowing. Each sprout is brought up to the clock in closed form, a few times a second in play and once per time skip, so a three-day wait costs no more than a moment of play.

• A seed counts damp hours, from the paper's wetness, standing water or watery goop under it. After enough of them it sprouts. Each kind has its own range of hours and heights: cress is quick, grass slower and taller, moss spreads sideways, and chia and flax take longer to become cress.

• Damp sprouts grow and gain health; dry ones lose it slowly, which you see as yellowing and drooping.

• A seed that drank and then stayed dry for a day dies, in a time skip only.

• Heat roasts unsprouted seeds, and fire burns plants down to stubble.

Blades are painted sprites bent by the state; their sway in the wind is drawn only.

Every skip is one job given a temperature, a number of hours and, for the sunny window, how dry the air is. The buttons differ only in those few numbers, and the waits add their own extras on top. Coming back to the app is a wait at room temperature for as long as you were away.

Most of it is closed form. Reactions run on a clock that doubles for every 10 °C, the chemist's familiar rule of thumb; mould is a formula; plants take one update; drying is an exponential. Only the flow of runny things is played out, for a number of passes that grows with the wait up to a cap, with the same fixed amount of work per pass on every device. Dough is proved in steps (below). So at the same sheet size a slow tablet gets the same result, only later: a test runs a two-hour wait at three detail levels from one identical page and checks that the results hash the same.

This is adaptive time-stepping of the simplest kind. The planner starts at half an hour and doubles the step each time the dough barely changed in the last one, while making sure even the longest wait fits within a fixed number of steps. Because the yeast's gas has a closed form, a long step costs no more than a short one and stays accurate.

The dough spreads once a step, so while it rises it spreads as it would with even half-hours. When the wait is over, the message says the dough rose or sank only when the change is big enough to see.

Dough that's the same comes out the same: within a step, identical spots are worked out once and the answer is reused. A kneaded lump is mostly identical spots, so this saves most of the work.

A skip is written as a resumable job that can pause at every safe stopping point and carry on later with nothing lost. In the Web Worker it runs until a budget of about 6 ms is used up, hands back to the message queue, and picks up again. Strokes that arrive meanwhile are held and land after the wait. The curtain stays down for at least a couple of seconds, so even a quick wait reads as time passing. The 6 ms is a budget; we have no measured figure for a whole skip on a device, so we print none.