Proposition

The sector is graded, unital, closed under ${\star }_H$, and $s_0$-stable. Every coefficient admits a compact microcausal lift.

Proof. These claims first hold for finite words by Theorem thm:product. The differential preserves the linear fields, maps ${\mathcal{B}}_f$ to its included image, annihilates that image, and annihilates the scalar-only tree. It is a quantum derivation. Products and this linear differential preserve every formal order, so the claims pass to the stated completion.