Abstract

We decide three calculations separately, retaining the graviton inverse ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. First, the fixed-width insertion and classical graviton tree are admitted to a polynomial on-shell Hadamard algebra extending the finite-generator causal algebra of XI-B. The full auxiliary and ghost blocks are derived, their graded antisymmetric parts are checked, and smooth changes of product data give algebra isomorphisms. No state or expectation value is selected.

Second, the complete coordinate word has a first-order coefficient identity for compact ghosts above the ramp. Its ghost-slot defect is localized in the ramp. The scalar-response frame adds switching terms; their advertised support bound requires an observation plateau. The affine ghost witness leaves precisely $\epsilon /2$ after the complete-word cancellation, whereas the bare insertion retains its larger response.

Third, the held-fixed second-order defect cancels against the order-$\kappa$ co-transformation of the metric in the first coefficient. The positive witness pairing verifies the sign. The omitted cubic Einstein response is then named and its classical variation is computed, including a nonzero evaluation.

These are admission and coefficient theorems, not one interacting BRST-closure theorem. Raw causal-Weyl multiplication has a separate obstruction; the Hadamard extension preserves the causal commutation relations, not that undefined contraction prescription. The mixed loop, quantum master equation, sharp-width limit, and empty Einstein jet remain outside the release.