Theorem: Uniform output moment domination

For every source density, including every reference extension, its two-pulse reduced field output belongs to ${\mathfrak{S}}_{C_G}$, where

The compact-source output two-point functions retain the Hadamard singularity of the baseline.

Proof. Let

It is a self-adjoint unitary. Direct conjugation gives

On any initial purified source-reference vector times $\Omega$, the norm of this expression is at most

Squaring proves the moment bound without assuming a Gaussian output.

For compact source tests, the exact branch formulas are finite sums of Weyl sandwiches of the initial Hadamard state. Differentiating their characteristic functions twice shows that the change of the two-point function consists of products of smooth causal solutions and contractions of the original two-point function against compact smooth source tests. Such one-variable smearings of a Hadamard bisolution are smooth. The two-point change is therefore smooth.