Proposition: Valid finite process insertion

The operator (eq:comb) is a valid process. Its local-instrument probabilities are

For $d=2$, this is an actual tripartite qubit process of the type used in XXII-C. For $d=2^n>2$, it is a declared higher-dimensional extension, not an unchanged tripartite qubit instance.

Proof. All tensor factors are positive; full transpose preserves positivity. Direct contraction of the Choi indices gives the displayed sequential probability. Summing every instrument outcome replaces its selective map by a trace-preserving map, yielding total probability one.