Proposition: Common source-state domination

Both source-displaced states, and every mixture of them, belong to ${\mathfrak{S}}_{C_{\mathrm{src}}}$, where

Their two-point functions differ from the baseline by smooth terms, so the conditional states and their mixtures retain the Hadamard two-point singularity.

Proof. Weyl conjugation shifts $X(u)$ by $-sg\sigma (k_F,u)$. The baseline is centered, so

Convexity gives the mixture statement.

The displacement mean is the pairing with the smooth solution $EF$. The two-point change is its smooth product. For mixtures the analogous two-point change is still smooth. No change of the Hadamard singularity occurs.