Theorem: Field-to-finite interface

For the specified stationary model and finite orthonormal packet family, the selected state is ${\rho }_K$, its flagged cutoff has dimension $\left(\genfrac{}{}{0ex}{}{J+N}{N}\right)+1$, and its exact Gaussian state error is $t_N(K)$. A justified covariance approximation gives (eq:combined). Energy and coherent scattering have the separate estimates of Section sec:scattering. State error controls an encoded channel under the independent ancillary interface. The specified nonvacuum-calibrated benchmark has the exact recovery and memory optima of Theorem thm:recovery.

Proof. Apply Theorems thm:gaussian, thm:continuity, thm:cutoff, thm:finitescatter, thm:scattererror, thm:ancilla, and thm:recovery, together with Propositions prop:tails, prop:generalcut, and prop:energy.