Proposition: Finite composition and frozen incidence

For Definition def:controller:

1. every finite legal command word of length at most $K$ has a unique controller output;

2. during every incidence command, the packet and its deficit remain fixed;

3. if every admitted command preserves complete matching acceptance, every legal prefix preserves acceptance;

4. a supplied token replacement does not acquire a token.

Proof. Compose the declared functions $u_{\ell }$. This proves (i). The incidence rule gives (ii). Induction on word length proves (iii). The replacement function takes a token as an argument; no state-acquisition map occurs in its definition, proving (iv).