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.

ΦBA=31=3,accepted=0\Phi_{B\to A}=3\cdot 1=3,\quad accepted=0
Detailed-balance row 1The fixed forward flow is compared with each reverse flow.forward=6, reverse=3, accepted=0

Reverse-flow row 2

The forward flow stays 6. Reverse rate 2 gives reverse flow 6 and accepted bit 1.

ΦBA=32=6,accepted=1\Phi_{B\to A}=3\cdot 2=6,\quad accepted=1
Detailed-balance row 2The fixed forward flow is compared with each reverse flow.forward=6, reverse=6, accepted=1

Reverse-flow row 3

The forward flow stays 6. Reverse rate 3 gives reverse flow 9 and accepted bit 0.

ΦBA=33=9,accepted=0\Phi_{B\to A}=3\cdot 3=9,\quad accepted=0
Detailed-balance row 3The fixed forward flow is compared with each reverse flow.forward=6, reverse=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.

ΦABkBAΦBAa613062616390\begin{array}{c|c|c|c}\Phi_{A\to B}&k_{B\to A}&\Phi_{B\to A}&a\\6&1&3&0\\6&2&6&1\\6&3&9&0\\\end{array}
Detailed-balance flow cross-scanThe equality row is displayed as the accepted boundary.only 6 matched against 6 gives accepted bit 1