Keep the same eight-photon input and the same 4 plus 4 arm split, then change
only the phase tag. In this pinned toy model the output moves to 0 bright and 8
dark counts, showing phase as the switch; real dark-port checks are limited by
phase stability, leakage, and counting noise.
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.