"It's for emergencies"
The card is insurance. The boiler, the car, the vet. You keep it for the day something breaks, and the balance is just the ordinary use of a thing that's really there for the extraordinary.
Look at the two lines. When the boiler goes, what's available? Twelve hundred and forty pounds, if you haven't bought the coat. A thousand and twenty if you have. The card that's for emergencies has three thousand nine hundred and eighty pounds of non-emergencies on it already, and the room left for the boiler is the room the non-emergencies left. A card at its limit, or near it, isn't cover. It's the absence of cover, with a green number on it.
Abigail Sussman and Adam Alter, in 2012, showed across seven studies that people predict their ordinary spending fairly well and underestimate their exceptional spending, because each exceptional expense gets treated as a one-off that won't recur, when the category recurs every month. An earlier book in this series uses this for saving. It applies here because the boiler is exactly the kind of expense people don't plan for, and the card is where unplanned expenses go, and a card that's already carrying last year's unplanned expenses has no room for this year's. The emergency fund you think you've got is the gap, and the gap is what you spend on coats.
A card at zero would be cover. Five thousand pounds of room, costing nothing until it's used, for the day the boiler goes, and then paid off fast because there's nothing else on it. That's the emergency card the belief describes, and it's the one you don't have. What you have is a card that's mostly full of ordinary things, and when the emergency comes it goes on top, and the balance goes to the limit, and the minimum rises, and now there's no room for the next one.
The method's answer is chapter 21's cushion, a few hundred pounds in an account that isn't the card, built from the fixed amount once the dearest card is clear. That's insurance that doesn't charge twenty per cent, and doesn't fill up with coats, and is there when the boiler goes because nothing else has been put on it.