Theorem: Ordering equivalence and descent

The map ${\alpha }_d$ preserves ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]]$ and ${\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$, and

It fixes constants and linear fields, is the identity modulo $\hslash$, has inverse ${\alpha }_{-d}$, and satisfies ${\alpha }_{d_1}{\alpha }_{d_2}={\alpha }_{d_1+d_2}$. It induces ${\bar{\alpha }}_d:{\mathcal{E}}_H\to {\mathcal{E}}_{H^{\prime }}$ with

Under the causal presentation, the regular descended map is the identity of ${\mathcal{A}}_0(M)[[\kappa ]]$.

Proof. A smooth self-contraction is multiplication followed by a proper projection. Output wavefront tuples lift with zero entries in the contracted slots, so microcausality is preserved. The self-contraction operators commute and lower field degree by two, proving the inverse and cocycle identities.

Let $D_d=(\hslash /2)⟨d,{\delta }^2⟩$. Graded symmetry of $d$ gives the Leibniz identity

Exponentiating commuting operators proves (eq:intertwining), the smooth-change ordering equivalence FR.

For $F\in {\mathcal{N}}_{\mathrm{sol}}$ and $\Phi \in {\mathcal{C}}_{\mathrm{sol}}$, $F^{(2k)}(\Phi )$ annihilates solution tuples. The kernel made from paired copies of $d$ is a smooth multisolution in all $2k$ slots. Corollary cor:fullseparation therefore makes its pairing with $F^{(2k)}(\Phi )$ zero. Thus ${\alpha }_d{\mathcal{N}}_{\mathrm{sol}}\subset {\mathcal{N}}_{\mathrm{sol}}$; the inverse proves equality. Descent follows. Constants and linear fields have too few slots to self-contract, and every nonidentity term contains $\hslash$. The regular assertion follows because the descended homomorphisms agree on their topological generators.