Theorem: Residual witness

For every $m\ge 0$ and $T>0$, there are compact smooth scalar Cauchy data, a nonnegative $f\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3)$, and a nonzero residual pure gauge $K\xi$, with compact smooth vector-wave Cauchy data, such that

Proof. Choose real $F\in {C_c}^{\infty }({\mathbb{R}}^3)$ with ${\partial }_{1}F\not\equiv 0$, and put $\beta ={\partial }_{1}F$. Let

and let the only nonzero component of the vector wave satisfy

Finite propagation gives spacelike compactness, and $(K\xi {)}_{01}{|}_0=\beta$, so $K\xi \ne 0$.

Set $w={\xi }^1{\partial }_1\varphi$. Its value and first time derivative vanish at zero. Equation (eq:boundarygauge) therefore gives $S_0(K\xi )\varphi =w$. Locally uniformly in space,

It is positive on some open set compactly contained in $0<t<T$. Choose a nonzero nonnegative $f$ supported there. Then $\int fw>0$. Both harmonic prescriptions fix $K\xi$, by (eq:residualsurvives).