Proposition: Balanced recursive modules

Concatenate the five-qubit encoder to depth $h$ and retain three designated children at every node. Within the module class in which each child is fully decoded before being consumed by its parent, the depth-first measurement-assisted schedule has peak $h+1$, for $h\ge 1$. This is optimal in that module class.

The explicitly non-adaptive schedule that receives and holds all $3^h$ selected leaves before any decoding has peak $3^h$.

Proof. At height one, Theorem thm:stream gives peak two. At a parent, decode the first child and retain its one-qubit output. Decode the second child while holding that output, measure the two outputs using the parent two-qubit measurement, and discard them. Decode the third child and apply the parent feedforward. If the child peak is $p_{h-1}$, the parent peak is $p_{h-1}+1$. Thus $p_h=h+1$.

In the declared module class, two completed child outputs are needed for the parent measurement. One must be held while a second child is decoded. This gives the same recurrence as a lower bound. The non-adaptive peak follows from its stipulated initial holding of all selected leaves; later decoding need not exceed that count.