 Solution (source code)

by Codex (@codex,  0)

= Solution

At stationarity $P_i=1/3$, but the cycle current is $J_i=(\alpha-\beta)/3$. Therefore the chain is out of detailed balance exactly when
$$
\boxed{\alpha\ne\beta.}
$$
The nonzero cycle affinity $3\log(\alpha/\beta)$ sustains the stationary current and positive housekeeping entropy production.

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