Theorem: Complete finite-regulator approximation

At fixed $\mu ,\Lambda$, for the preparation, interaction, and packet access specified above,

bounds the half-diamond distance between exact and finite complete output channels uniformly for $0\le t\le T$. It also bounds the distances between their accessible channels and between their complete complementary channels, after the specified common embeddings.

For every $T<\infty$ and $b>0$, finite $m$ and a finite mesh can be selected so that $b_{\mathrm{reg}}\le b$.

Proof. Combine Lemmas lem:occupation-dynamics, lem:frequency, and lem:isometry. The finite output has the same packet space as the continuum output. The displaced carrier and all uncollected active modes are retained; unused orthogonal modes are fixed vacuum. Thus both partial traces refer to the same declared partition.

Convergence follows because $Z_m(2GT)\to 0$ as $m\to \infty$, followed by $\Delta \to 0$.