Theorem: Nonzero coefficient on a free scalar

For every nonzero compact switch $g$ with a plateau, there are a smooth free scalar and a compact odd ghost test on the completion for which the coefficient of $\kappa$ in (eq:breaking) is nonzero.

Proof. Take

Then $P_m{\varphi }_m=0$, ${T_0}^{00}[{\varphi }_m]=1/2$, and ${T_0}^{0i}[{\varphi }_m]=0$.

A vertical line through a plateau point has $g=0$ at sufficiently early times and $g=1$ at that point. The fundamental theorem of calculus gives an open region with ${\partial }_{t}g>0$. Choose a nonnegative nonzero $\beta \in {C_c}^{\infty }$ there and a nonzero odd Grassmann generator $\theta$. Put $c=\theta \beta {\partial }_t$.

At the leading metric $q=\eta$, the density difference in (eq:B) is zero, and the free scalar equation term vanishes. Therefore

The even coefficient of $\theta$ is strictly positive.