A weighted test completion, not a global Fock density

For positive-frequency tests, use the norm

The compact positive-frequency covariance extends by continuity to its completion. The resulting Gaussian characteristic functional remains positive: each finite matrix and each Weyl positivity condition is a limit of the corresponding compact-test condition.

For any finite-dimensional canonical subspace with finite occupation matrix $A⪰0$, the restricted state is normal. Indeed, diagonalize $A$ by a passive finite-mode unitary. The characteristic function becomes a product of one-mode characteristic functions; Theorem thm:marginal constructs their normal product density. Transforming it back constructs the normal finite-mode restriction.

This completion provides the radiation-mode calculations used below. It does not assert a new characteristic restriction theorem for local bulk fields.