An ideal half-silvered mirror splits the photon-count ledger evenly between two paths, so the output counts add back to the input. The half split is exact only for the stated ensemble and ideal splitter; real optics add loss, imperfect 50/50 ratios, detector inefficiency, dark counts, alignment error, and finite-shot uncertainty.
An ideal half-silvered mirror is modeled as a checked fifty-fifty ensemble splitter. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
A balanced splitter needs an even ensemble
The checked input count is 8 photons. The ideal balanced splitter can therefore claim an exact half split.
Nin=8
The two path counts add back to the input
The output ledger is 4 photons in the upper path and 4 photons in the lower path.