Theorem: Fixed-regulator X-B release

For the frozen scalar-only field content, ramp, switch, background space, insertion representatives, and algebra ${\mathfrak{g}}_U$:

1. The bare dressing coefficients are admitted through ${\kappa }^2$, both by the joint cone test and on the scalar-only smooth linear domain.

2. The single normalized map $R_1(V_{1,g};\cdot )$ exists on the frozen off-shell tests, has smooth compact retarded output, is continuous, and has no positive-$\hslash$ correction.

3. The completed coefficient ${\hat{A}}_1$ is invariant in ${\mathfrak{A}}_0$ on ${\mathfrak{g}}_U$. This is the quantum background identity through ${\kappa }^1$, not through ${\kappa }^2$.

4. The second-order bare dressing and retained mixed term are available, with their stated transformations. The classical missing term $\int f{\rho }_{2,g}$ is displayed, but no full second-order quantum dynamical completion or Ward identity is constructed.

No sharp limit, quantum metric response, loop triangle, or two-point response is included.

Proof. Item (i) is Propositions prop:joint and prop:linear. Item (ii) is Theorems thm:retarded and thm:hbar. Item (iii) is Theorem thm:ward. For (iv), use (eq:truncation), (eq:classical-A2), and Proposition prop:mixed-transformation. The pure-gauge diagnostic explicitly exhibits the missing second-order cancellation.