Exact selection versus continuum approximation

Let $E=\mathrm{span}\{h_i\}$, with projector $P_E$. Theorem thm:gaussian is an exact restriction, not a state approximation to all outgoing modes.

For a normalized target $g\in L^2(B)$, set $e=\Vert (I-P_E)g\Vert$. With $Q$ multiplication by $q_{\mathrm{out}}$, expansion into selected and unselected parts gives

This controls the named occupation observable, not a global continuum-state distance.