Theorem: A nonzero leading coefficient

Every nonzero compact scalar switch with a plateau admits a free scalar and a compact odd ghost test for which the coefficient of $\kappa$ in (eq:sS) is nonzero.

Proof. Use the spatially homogeneous free solution

It has ${T_0}^{00}=1/2$ and ${T_0}^{0i}=0$. At the leading metric $q=\eta$, the density difference is zero.

Along a vertical line through a plateau point, $g$ rises from zero to one. Therefore ${\partial }_{t}g>0$ on some open interior region. Choose a nonnegative nonzero $\zeta \in {C_c}^{\infty }$ there and an odd Grassmann generator $\vartheta$. Set $c=\vartheta \zeta {\partial }_t$. Then