Proposition: Scope of the constructed responses

Every output in Definition def:catalogue:

1. belongs to the homogeneous degree-one sector, with zero allowed;

2. has a smooth compact representative test with no $\hslash$ in it;

3. depends continuously on the classical backgrounds at fixed $\epsilon$;

4. is polynomial in those backgrounds, of degree at most two in $h$ and one in $k$;

5. contains no contraction between two background slots, no quantum metric line, and no gravitational action.

Proof. For the bare coefficients this is Proposition prop:linear. For the response it is Theorems thm:retarded and thm:hbar. The map is linear in $h$ for independent $c$; substituting $c_1[h]$ gives quadratic $M[h]$. The completed coefficient and its differences are sums of the same expressions.

Every quantum output is of the form $\mathbf{\varphi }(d)$. Background slots are evaluated on $h,k$, not contracted through a background two-point kernel. The response uses a scalar Green kernel against a smooth test.