Corollary: Dense-gate variant

The characterization remains valid when exact CNOT is available and finite available gate circuits are dense, up to phase, in the unitary group on every finite register, with all other assumptions retained.

Proof. Remote invariance first holds for available circuits. At fixed dimension, approximate an arbitrary remote unitary and use $(\mathrm{K}{)}^{\star }$ to extend the invariance to its limit. This supplies the transformations required for locality and uniform correlations. Witness rays are admissible inputs regardless of exact preparability.

Record terms use exact CNOT. In the update proof, take available $V_j$ converging to a unitary mapping ${\phi }_{\kappa }$ to $|0⟩$. The equal squared-amplitude identities pass to the limit. No continuity of the unknown update map is needed.