Assumption: Complete origin-germ certificate

For the fixed complete translation map $\Phi \mapsto {\Omega }_{\Phi }$, assume:

1. A finite set $\mathcal{E}\subset (-\infty ,0]$ of distinct real degrees and integers $J_{\lambda }\ge 0$ is specified.

2. For each $\lambda$, linear maps

   $$H_{\lambda ,j}:\mathcal{D}(U;V_{\mathrm{out}}^{\ast })⟶{\mathcal{D}}^{\prime }({\mathbb{R}}^8),0\le j\le J_{\lambda },$$

   give chains satisfying (eq:chain). For every compact joint support $L$ and compact relative support $B$, there are finite orders $m,n$ and a constant such that

   $$|H_{\lambda ,j}[\Phi ](\psi )|\le C_{L,B}\Vert \Phi {\Vert }_m\Vert \psi {\Vert }_n,\mathrm{supp}\Phi \subset L,\mathrm{supp}\psi \subset B.$$

3. For every compact $L⋐U$, a radius $r_L>0$ and linear continuous-function representatives $W_{\Phi }$ on $B(0,r_L)$ are specified such that

   $${\Omega }_{\Phi }=\sum _{\lambda \in \mathcal{E}}{H}_{\lambda ,J_{\lambda }}[\Phi ]+W_{\Phi }\text{on }B(0,r_L),\mathrm{supp}\Phi \subset L.$$

4. For some $a_L\in (0,1]$, finite order $m_L$, and constant $C_L$,

   $$\begin{array}{rl}|W_{\Phi }(0)|& \le C_L\Vert \Phi {\Vert }_{m_L},\\ |W_{\Phi }(h)-W_{\Phi }(0)|& \le C_L|h{|}^{a_L}\Vert \Phi {\Vert }_{m_L},|h|<r_L.\end{array}$$

The certificate concerns the entire ${\Omega }_{\Phi }$, including all sectors and all joint tests. Its existence is not inferred from the cone rank, the source rule, or moderate growth.