Proposition: Schedule compatibility obstruction

Prescribed unconditional one-particle weights $a_j\ge 0$ can be realized by the transfer attempts exactly when

Before exhaustion, a realizing schedule is

After exhaustion all later $a_j$ must vanish, and later ${\lambda }_j$ are immaterial on the actual state.

Proof. The model emits at most one transfer excitation, and ${\sum }_j{w}_j=p_N\le 1$, proving necessity. For sufficiency, use (eq:calibratedschedule) and induction: the pending weight before attempt $j$ is $1-{\sum }_{k<j}{a}_k$, so $w_j=a_j$.