Theorem: Tensor-free ancestral factorization

For the specified layers,

There is a unique UCP channel

whose inclusion equals the displayed restriction of $T_{s:T}$. The terminal snapshot therefore depends only on the initial state restriction to $\mathfrak{A}(N_s(Y))$, including joint restrictions with any independent matrix reference.

Proof. Apply Theorem thm:onelayer successively in reverse tick order. The set recursion is exactly (eq:backwardsets). Theorem thm:comparison gives the local channel and snapshot factorization, and Corollary cor:reference gives amplification.