Lemma: Finite adaptation within a fixed skeleton

A finite adaptive instruction tree with fixed finite outcome sets, declared finite interfaces, and causal branch connections can be represented by a finite typed skeleton of complete classically controlled instruments.

Proof. Unroll the finite tree. Reserve fixed output slots and an activity flag for every possible continuation. At a branching vertex, use classical control to place the active quantum factor into its selected continuation slot by a permutation; fill inactive slots with fixed blank states. The quantum factor is not copied. Classical outcome and activity flags may be copied locally and transported along their declared edges.

At a merge, the local flag selects the active input slot by a complete controlled channel. Every inactive or failure branch is retained in the full interface. The tree is finite, so this uses finitely many vertices and factors. All controls are represented locally.