A phase tag on a dual-rail qubit carries alongside the classical photon-presence ledger without changing it; only a swap moves the bars. 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 untagged rail counts are the reference row
Chapter four's rail cases never carried a phase tag. With no tag, logical zero reads as 1 upper and 0 lower, outputting logical 0.
(Nupper,Nlower)=(1,0)
A plus tag rides along without moving either bar
Adding a plus phase tag to the same untouched, unswapped rail changes nothing in the classical count ledger: the upper and lower bars and the output logical value are identical to the untagged row. The rail's photon-presence counts cannot see the tag at all.
tagnoneplusNupper11Nlower00out00
Only the swap, never the tag, moves the bars
The final row carries a minus tag and is also swapped. The output flips to logical 1 because it is swapped, exactly as chapter four's swap-mirror case predicts -- the minus tag riding along changes nothing about which bar carries the count. A tag is metadata carried alongside the ledger, not a term inside it; a real photonic phase only becomes measurable once two tagged paths actually interfere, as chapter three's bright and dark ports already showed.