Proposition: Compression preserves the proved joint limit

For a fixed finite isometry $V_X$, the regulated cubic matrices converge to (eq:compression) jointly in operator norm. Using a common $B$ strictly larger than the limiting matrix norms, their complete instruments converge in diamond norm.

Proof. Each matrix entry is bounded by Theorem thm:cubicform with two fixed domain vectors. There are finitely many entries, so the convergence is uniform and implies operator-norm convergence.

For all sufficiently refined cutoffs, the effects stay in a common compact interval inside $(0,1)$. Their positive square roots therefore converge in norm by continuous functional calculus. For contractions $M,N$,

Summing the finitely many recorded Kraus branches proves complete instrument convergence.