There needn't be a layer such that all layers below are completely white and all layers above are completely blue. For example, consider a cube. It has four la

There needn't be a layer such that all layers below are completely white and all layers above are completely blue. For example, consider a cube. It has four layers. We could say the bottom-most layer is the one with (0, 0, 0). Above that, there's a layer which contains (0, 0, 1), (0, 1, 0), and (1, 0, 0) [the entries with a single 1]. Above that, there's a layer which contains (1, 1, 0), (1, 0, 1), and (0, 1, 1) [the entries with two 1s]. Above that, there's a layer which contains (1, 1, 1). We could make (0, 0, 0), (0, 0, 1), (0, 1, 0) and (0, 1, 1) white, while making (1, 0, 0), (1, 1, 0), (1, 0, 1), and (1, 1, 1) blue. Note that all the blue values will have a 1 in the first position, while all the white values will have a 0 in the first position, ensuring monotonicity. But both the "single 1" and the "double 1" layers have a mix of blue and white within them. So there's no single transition layer, as you put it.