Intensity scales as amplitude squared, so the transmitted intensity follows I = I0 * cos^2(theta), the familiar Malus result. The squaring is what turns a signed field component into a non-negative power. The exact cos^2 law holds for an ideal polarizer and a single polarization state; measured intensities carry detector noise and the input beam is never perfectly polarized.
The split is checked as a closed sum of field components. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The split is a ledger, not a loose component sketch
The same axis split closes a squared-component ledger from the checked projection source.