A detailed-balance-flow cross-scan varies the transition rates across rows, showing how the forward and reverse probability currents p_i * k_ij and p_j * k_ji track each other row by row. The arithmetic is exact only for the stated probabilities, rates, and rows in the ideal ensemble model. Real systems add preparation bias, finite samples, degeneracy/model choices, thermalization limits, detector/counting calibration, finite precision, and measurement uncertainty.
Opposing flow rows show that detailed balance accepts only equality. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Reverse-flow row 1
The forward flow stays 6. Reverse rate 1 gives reverse flow 3 and accepted bit 0.
ΦB→A=3⋅1=3,accepted=0
Reverse-flow row 2
The forward flow stays 6. Reverse rate 2 gives reverse flow 6 and accepted bit 1.
ΦB→A=3⋅2=6,accepted=1
Reverse-flow row 3
The forward flow stays 6. Reverse rate 3 gives reverse flow 9 and accepted bit 0.
ΦB→A=3⋅3=9,accepted=0
Only equal opposing flows accept detailed balance
Reverse flow three is too small, six matches, and nine is too large. The accepted bit is one only at the exact match.