Absorption and drain-free turnover

For initial sector $K$, let ${\pi }_t(n)=\mathrm{tr}({\Pi }_n{\rho }_{C_t})$, $s_t=1-{\pi }_t(0)$. Then

Branch effects are scalar and propagation does not change the core reduction. Distinct core energies have distinct emitted complementary energies, eliminating inter-sector coherences. Thus this recursion is independent of internal gates and loss.