Theorem: Fixed-regulator X-C release

On the frozen X-B-v3 sheet, with the scalar quantum and $h,k$ classical:

1. The maps

   $$R_1(V_{2,g};F_c)=\hat{\varphi }(-{\mathsf{L}}_{2,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c),\frac{1}{2}{R}_2(V_{1,g},V_{1,g};F_c)=\hat{\varphi }({\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c)$$

   exist on the frozen off-shell linear tests. Their outputs are smooth compact tests supported in $\mathrm{supp}g\cap J^{-}(\mathrm{supp}c)$, continuously dependent on the backgrounds, with no positive-$\hslash$ remainder.

2. Their sum on $F_f$ equals the copied $\int f{\rho }_{2,g}$.

3. The completed ${\hat{A}}_2$ is background-gauge invariant in ${\mathfrak{A}}_0$ on ${\mathfrak{g}}_U$, with the free field held fixed. Together with ${\hat{A}}_1$, it makes the completed observable invariant through ${\kappa }^2$.

The pure-gauge remainder is cancelled with ${\mathcal{O}}^{(2)}$ retained. No sharp limit, quantum metric, graviton tree, triangle, ${\mathsf{T}}_2$, ${\mathsf{T}}_3$, nonlinear-insertion $R_2$, or two-point response is constructed.

Proof. Theorem thm:maps gives existence, support, continuity, and normalization. Theorem thm:yf and Proposition prop:hbar give the restricted linear-field characterization and quantum order. Proposition prop:variation is the proof of record for the variation, and Theorem thm:ward cancels its defects on the frozen algebra. Diagnostic diag:puregauge verifies the missing cancellation by an explicit operator computation.