Proposition: Admission, polynomial scope, and normalization

Every $e_j$ is smooth and compact, with

Their background dependence is continuous at fixed $\epsilon$. The retained outputs have field degree one, with zero allowed; $e_1$ is linear in $h$, and $e_2$ is quadratic in $h$ plus linear in $k$.

For every smooth free solution $u$,

The last line is the inherited full classical second relational coefficient, counted once.

Proof. The bare tests are supported in $K$. The support and continuity of every response term follow from Theorem thm:maps and the inherited first-map theorem. Their finite sums have the stated support and topology.

The coordinate coefficients satisfy $w[h]$ linear in $h$ and $v[h,k]$ quadratic in $h$ plus linear in $k$. The same degree statement follows for $c_1,c_2$, and then for (eq:completed). Every output is $\mathbf{\varphi }$ of a smooth compact test, so scalar-only admission needs no further singular-kernel theorem.

Transpose the response terms using (eq:individual-normalizations). For the mixed term,

The bare terms give $O^{(j)}$, and the new sum gives ${\rho }_{2,g}$.