Live fraction compares usable live time with total clock time, so detector availability becomes an explicit timing ledger. The ratio is exact only for the stated live and clock times; real detectors add pulse pileup, recovery tails, threshold drift, clock jitter, calibration uncertainty, and counting noise.
Live fraction is a budget ratio, not a free parameter. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.