A cascaded splitter acts only on the selected arm that enters it, so path-local counts must be updated without re-splitting the untouched arm. The arithmetic is exact only for the declared selected path and splitter ratios; real cascades include loss, phase drift, alignment error, imperfect splitting, and detector uncertainty.
A second splitter acts only on the arm that actually enters it. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Only one selected arm enters the next splitter
The first splitter sends 8 photons to 4 upper and 4 lower.
8→4+4
The selected arm splits again
Sending the selected upper count 4 through a second balanced splitter gives 2 and 2, while the other path still carries 4.