If the intent of a split is merely enough bees to survive a frame of brood and a frame of honey will most often survive, but they will struggle to reach "critical mass". Once they finally do, then they will start building more rapidly. If the intent of a split is for it to build quickly and be ready to overwinter, then I would not have less than ten deep frames of bees, brood and honey (probably about five of brood and five of honey and pollen and all covered in bees) or 15 or so medium frames of bees etc. This size has a shot at being AT critical mass when they start and so they don't have to struggle to get to that point.
There is a certain amount of overhead just heating, cooling, feeding, patroling etc. that doesnt' change much. It's only after that point is reached that they can prosper.
Think of it as if you're dirt poor struggling hand to mouth. You can't do a good job of anything and you have nothing to spare. Once you get your bills paid, your groceries bought and have some excess income, you can start to prosper. Bee economy has some similarities.