Conventions

All vector spaces are over $\mathbb{C}$; the real versions are identical. $\mathrm{Hom}$ denotes algebraic linear maps unless continuity is explicitly required. Manifolds are smooth, Hausdorff, and second countable. Distributions on manifolds act on compactly supported smooth densities; in Euclidean charts we use the fixed coordinate density and write tests as functions.

Bundle-valued assertions are interpreted in local trivializations of a finite-rank smooth bundle and proved componentwise. Multi-index inequalities are componentwise, and

For $K⋐X\subset {\mathbb{R}}^d$ and $M\in \mathbb{N}$, write

The notation $K⋐X$ means that $K$ is compact and contained in the open set $X$.