Proposition: Fixed tangency witness

There exist real compact-data $\varphi$, compact smooth $\xi$, and nonnegative $f\in {C_c}^{\infty }(M)$, $\int f=1$, such that

For ${\varphi }_{\kappa }=\varphi +\kappa X_{\xi }\varphi$,

Proof. Take $\beta \in {C_c}^{\infty }({\mathbb{R}}^3)$ equal to one near the origin and scalar data

For $a\in {C_c}^{\infty }(\mathbb{R})$ equal to one near zero, set

At the origin,

Thus $P(X_{\xi }\varphi )(0)=1$. It exceeds $1/2$ on a neighborhood. A normalized nonnegative test supported there proves the first claim; $P\varphi =0$ proves the second.