Proposition: Finite-regulator admission

For every $\tau >0$, the kernel (eq:t-reg) is compact and defines an even, ghost-number-zero quadratic polynomial. Its wavefront set is contained in $Z_2$.

Proof. The numerical coefficient in (eq:Pi) is smooth and compactly supported in the vertex variables. The two delta attachments are well defined: each has its own external variable, so a nonzero covector cancellation between them would require a nonzero covector in an external variable where no other factor contributes.

Their wavefront covectors are balanced between an external variable and its attached vertex. Multiplication by the smooth coefficient and the finite differentiations preserve that bound.

The support is compact because the vertex variables are compact and the delta attachments place each external variable at a vertex. The projection is therefore proper. Its wavefront condition sets all integrated vertex covectors to zero, leaving $p_1+p_2=0$ in the two external slots.

The sum-zero bound excludes both all-causal cones. Finite tensor sums and species symmetrization preserve the bound. The remaining species are even and have ghost number zero Hormander.