A shot of espresso is a race against time. Water pulls flavour from the grounds — first the bright acids, then the sweet sugars, and if you go too long, the bitter tannins. There is a window, and it is only seconds wide. Slide the shot time (or tap) and find it.
Espresso extraction as a rate curve. Soluble flavour dissolves fast at first and slows as it runs out, so the extraction yield (mass dissolved Ć· dry dose) rises and plateaus: Y(t) = YāĀ·(1 ā e^(āt/Ļ)). The barista's target window is 18ā22% yield ā below it the shot is under-extracted (sour, thin), above it over-extracted (bitter, harsh). The curve and the current yield are computed live; the ideal band is marked. A fail-loud self-check throws unless yield increases monotonically with time and passes through the 18ā22% window.
A single-exponential model of a complex, multi-compound extraction (real shots depend on grind, pressure, temperature, dose ratio); the 18ā22% window is the specialty-coffee convention. The monotone rise and the band are computed exactly.