Theorem: Open dense failure

There are compact smooth scalar and vector Cauchy data and a nonzero residual gauge $K\xi$ for which the three functionals $B_f,B_R,B_0$ agree and are positive for some nonnegative compact $f$. For those fixed data, nonzero evaluation is open and dense in a specified fixed-support Fréchet space of real smearings.

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

with other vector components zero. Then $(K\xi {)}_{01}{|}_0=\beta$, and

The scalar $w$ has zero initial data, so Proposition prop:subtraction gives $S_0(K\xi )\varphi =w$. It is positive in a small open future region. Choose a nonnegative nonzero $f$ supported there. Both prescriptions fix this residual gauge.

Fix a compact $Q⋐(0,b)\times {\mathbb{R}}^3$ whose interior meets that positive region, and let

The functional $f\mapsto \int fw$ is continuous and nonzero. Its kernel is a proper closed hyperplane, with empty interior. Its complement is therefore open and dense. All Cauchy data used above are literally compact.