The House Is Never at Temperature: Comfortable Settings Form a Set of Measure Zero, with 118,414 Instrumented Room-Days and a Crossing Timetable

You turn the heating up because the room is cold, and an hour later you are too warm and turn it back down. The setting you are looking for is not missing from your house: it is missing from the dial. A domestic heating system is a switch with a delay, and a switch with a delay cannot hold a temperature — it can only circle one. Two rhythms govern the circling: the boiler’s own minimum-cycle timer and the thermal lag of the building. Wherever their ratio locks, the delivered swing is wider than the band an occupant will sit still in, which we measure from a held button rather than from a questionnaire. Comfort therefore requires an unlocked ratio. Sweeping a 4,096-position dial in an occupied test dwelling, we find that the unlocked settings are nonempty and have measure zero: a set of box dimension $0.87$ (95% CI $0.85$–$0.89$). No dial can select from such a set, and a finer dial does not help — a concealed trial spanning a hundredfold range of setpoint resolution separates its arms by nothing. Households are comfortable for forty-one minutes of an occupied day, and every minute of it is spent on the way between two temperatures the house can hold and nobody will accept. Houses will be sold with a timetable where the thermostat used to be.