Proposition: Ordering transport of the generated sector

Let $H^{\prime }=H+d$, where $d$ is smooth, symmetric, and compatible with the declared reality convention. Define

Then

For the prepared first and second coordinate coefficients,

where $c_d$ is a scalar and $L_d$ is a compact smooth linear field smearing.

Consequently the generated sector, including ordinary local polynomials and smooth directional descendants, is transported onto the corresponding sector generated by the unchanged prepared formulas.

Proof. The self-contraction generator obeys

Exponentiating the commuting operators proves the intertwining identity and gives inverse ${\alpha }_{-d}$.

A quadratic self-contraction is a scalar. A cubic has only a linear correction. For a projected linear kernel to have a nonzero wavefront covector $p$, the original $Z_3$-bounded kernel would need a tuple $(0,0,p)$. The zero-sum condition forces $p=0$. Thus the correction is smooth; compactness follows from proper projection.

Smooth directional differentiation commutes with the self-contraction operator. Local polynomial ordering corrections are again local polynomials with smooth coefficients. These and compact linear fields are already among the generators. The inverse transport gives the reverse inclusion.