An exact classical sector description

Let ${\lambda }_j=2j-1$, so $Q{P}_j={\lambda }_j{P}_j$. The branches satisfy

Thus, on states diagonal in the source basis, the full evolution is exactly the classical Markov chain on $(g,j)$ with fixed $j$ and transition probabilities $p_a(j)$.

For an arbitrary initial qubit state at one label, its geometric history law is the mixture of those two classical sector laws with weights ${\rho }_{00},{\rho }_{11}$. Off-diagonal blocks have zero trace at every history and do not contribute. Outcomes are independent conditional on $j$, not generally independent after mixing the sectors.