Changing only the phase tag moves the checked count ledger to the dark detector. 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 input and first split stay unchanged
The opposite-phase case keeps the same 8 input photons and the same 4 plus 4 first split.
8→4+4
Opposite phase recombines at the dark detector
Only the phase tag changes. The detector ledger becomes 0 bright and 8 dark.