Theorem: Complete session-and-reset composition

Let $M$ be a reusable device memory, and fix its specified reset state ${\omega }_M$. For session $i$, let

be the complete returned session map, including its receiver and reset action. Suppose, on the entire joint input space,

Previous returned payloads are isolated from later device operations. Then, for any initial memory and any joint payload–reference state, the final returned payloads and memory differ from ideal identity return with fixed ${\omega }_M$ by half-diamond error at most

Proof. Insert untouched identity wires for prior outputs and future inputs. Replace each actual session by ${\mathrm{id}}_{L_i}\otimes {\mathcal{T}}_{{\omega }_M}$, one at a time. All maps before and after a replacement are channels, so diamond contraction bounds the replacement by $u_i$. The ideal sequence returns each payload unchanged and resets the memory. Sum the bounds.