For stated probabilities only the correct reverse rate closes p_i * k_ij = p_j * k_ji; a wrong rate leaves the opposing flows unmatched, so detailed balance is broken. The verdict is exact only for the stated probabilities and rates in the ideal ensemble model. Real systems add finite samples, preparation bias, correlations, non-equilibrium effects, rate-estimation error, and measurement uncertainty.
A rejected balance claim can still be an exact checked ledger. 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 wrong rate breaks the flow match
The two directed flows are six and three.
ΦA→B=6,ΦB→A=3
The rejected bit is zero
The balance claim fails, but the arithmetic remains exact.