Theorem: Fixed-width symbol release

For the conventions and support class below:

1. The bilinear insertion ${\mathcal{B}}_f=-\int f{\mathsf{X}}_1[\gamma ]\cdot \partial \varphi$ has a compact mixed kernel satisfying the polynomial microcausal exclusion.

2. The one-graviton retarded contraction is a well-defined continuous cubic classical scalar polynomial with a compact microcausal kernel.

3. Its intermediate metric uses precisely ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. For free scalar input and a switch above the ramp, its coordinate response is the double retarded integral displayed in Theorem thm:normalization.

4. The classical restricted second-order candidate defined in Section sec:boundary has an explicitly nonzero background defect for suitable free inputs.

These statements do not establish an admission map into ${\mathcal{A}}_0$, a quantum second-order Ward identity, or an interacting differential.