Detailed balance says each transition and its reverse carry equal probability current, p_i * k_ij = p_j * k_ji, so the opposing flows cancel and the distribution stops changing. The arithmetic is exact only for the stated probabilities and transition rates in the ideal ensemble model. Real systems add finite samples, preparation bias, correlations, non-equilibrium effects, rate-estimation error, and measurement uncertainty.
An equilibrium flow claim is a pair of checked directed ledgers. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Detailed balance compares opposing flows
Both directed flows are six in the accepted case.
ΦA→B=6,ΦB→A=6
Equal flows accept the balance claim
The accepted bit is one only because the flows match.