McNaughton (1959), Theorem 4.1: on identical parallel machines, every preemptive schedule can be turned into one without preemption whose total weighted completion time is not larger. The proof is an exchange: in a schedule with $b$ split times, the jobs around the last one are rearranged in one of two ways, whose changes of the cost are $c(X - Y)$ and $-e(X - Y)$ for sums of weights $X$ and $Y$; one of them is not positive, and it leaves $b - 1$ split times.

Without preemption a schedule is also a preemptive one, so $P \mid\mid \sum w_jC_j$ is a special case of $P \mid pmtn \mid \sum w_jC_j$ field by field. The theorem gives the converse, for every bound on the number of machines: the same kernel instance, the same data and parameters in every field except interruption (which changes from preemption to none and carries no data), and the same threshold. The interruption field cannot be kept: a reduction in place in the sense of $Parameters.InPlace$ would keep every invariant field parameter, and whether an instance is read with preemption is one.

Theorem. For every bound on the number of machines, the decision version of $P \mid pmtn \mid \sum w_jC_j$ reduces to that of $P \mid\mid \sum w_jC_j$ with the same kernel instance, the same parameters of every field other than interruption, and the same threshold; in particular the zoo's $P \mid pmtn \mid \sum w_jC_j$ and $P2 \mid pmtn \mid \sum w_jC_j$ reduce to $P \mid\mid \sum w_jC_j$ and $P2 \mid\mid \sum w_jC_j$.