Theorem: Finite convoy with arbitrary joint input

For session $i$, suppose the complete preparation is independent of all payloads and other session resources, and its decoded channel satisfies

Suppose the actual independent-session output channel is ${⨂}_i{\Lambda }_i$. Then

The bound preserves every joint payload–reference correlation.

If session $i$'s device event has probability at least $1-p_i$, all session certificates hold jointly with probability at least $1-{\sum }_i{p}_i$, when this lower bound is positive. Device-event independence is unnecessary for this assertion.

Proof. Replace the tensor factors by identity one at a time. Each difference has diamond norm at most $2e_i$, since the remaining tensor factors are channels of diamond norm one. The triangle inequality proves (eq:convoy). Its diamond formulation already includes mutually entangled inputs. The probability statement is the union bound.