Lemma: Finite-register junction implementation

The local frame is legal for every sufficiently large $n$. It gathers, processes, and separates any fixed finite set of typed registers without measuring them for routing.

The routing parts are tensor-factor permutations and preserve the transported state jointly with every untouched reference. Their time overhead is at most $B{\tau }_n$, and their geometric support shrinks to the vertex.

Proof. Each incoming or outgoing transfer is a legal access swap by Lemma lem:residues. Transfers involving the same hub are serialized. The singleton payload acts only after all its input factors are present and before its required outputs are dispersed. The chosen topological order implements the same internal zero-duration subnetwork.

A swap is unitary and merely relocates a typed factor. Its action on a logical state and an arbitrary reference is exactly the corresponding tensor-factor identification. No routing operation reads a quantum state or a measurement outcome.

The frame has $B$ ticks by construction. Every involved site is within $b_n$ of $x_v$, and all frame times are within $B{\tau }_n+{\tau }_n+c^{-1}c{\eta }_n{T}_{\Gamma }$ of the intended vertex time. These bounds tend to zero.