The three compounds
Fish waste and decaying food become ammonia (toxic at low ppm). Nitrosomonas bacteria eat ammonia and excrete nitrite (also toxic). Nitrospira eat nitrite and excrete nitrate (tolerable, managed by changes). A "cycled" tank hosts colonies of the second and third — established over 4–6 weeks — converting poison into fertilizer continuously.
Why new tanks kill
A new tank has no bacterial colonies: the first fish's ammonia accumulates with nowhere to go, burning gills and fins. "Cycling with fish" works only with daily testing and changes — fishless cycling (pure ammonia dosing) is kinder and faster. The patience of six weeks is the entry fee; every calculator here assumes it is paid.
Maintaining the colony
The bacteria live on surfaces (filter media, substrate), not in the water — which is why replacing filter cartridges monthly throws the colony away with the cartridge. Rinse media in removed tank water (never tap — chlorine kills the colony), and the filter becomes the tank's permanent chemistry engine.