Proposition: Detection at a fixed scalar leg

For the real witness selected in Theorem thm:mismatch, there are a homogeneous real free Klein–Gordon scalar $\varphi$ and a real compact test $f$ for which replacing $w$ by $X_{\mathrm{cand}}$, while keeping the scalar first coefficient fixed, changes the first-order smearing.

Proof. Write $d^0=\mathrm{Re}(\zeta {d_{\lambda }}^0)$, and choose a detecting time $t_c$ at which $d^0(t_c)>0$. Take

Then $P\varphi =0$ and $\dot{\varphi }(t_c)=m>0$. For any common smooth first scalar coefficient ${\Phi }_1$, the difference of the two displayed expressions is

Choose a nonnegative nonzero compact test where both factors are positive. This integral is strictly negative.