Proposition: Corrections on the seeds

For the quadratic and cubic seeds,

Here $c_d(f)$ is a scalar and ${\ell }_d(f,g)\in {C_c}^{\infty }(M)$.

Proof. The quadratic formula is the terminating exponential. The image $s_0{\mathcal{B}}_f$ has species $(c,\varphi )$; its only possible self-contraction is forbidden by ghost number.

For the scalar cubic, the second derivative contributes the factor $3\cdot 2$, canceled in part by the $\frac{1}{2}$ in (eq:alpha). No fourth derivative remains. For the projected kernel ${\ell }_d$, a nonzero output wavefront point would come from $(0,0,p)\in \mathrm{WF}(t_{f,g})$. The sum-zero bound (eq:sumzero) forces $p=0$. Thus ${\ell }_d$ is smooth, and proper projection makes it compact.