C4: Branch B input list

This section records data. It defines none of them, asserts neither their existence nor their failure, and corrects nothing in Branch A. X-B's statement is retained: a graviton response does not follow from the scalar-only normalization, and a schematic triangle with a partial diagonal is not an obstruction theorem.

a quantum-metric branch would have to supply.

1. A quadratic graviton action on the ramp sheet, its gauge fixing, and its ghost sector, with the normalization of the graviton Euler operator stated (the distinction between an equation inverse and an action inverse belongs to another track and is not imported).

2. Retarded and advanced Green operators for the gauge-fixed graviton operator, with their transposition conventions and support properties.

3. A graviton Hadamard two-point matrix compatible with (G2): the shell equations, the antisymmetric part, and the wavefront condition in the metric slots.

4. A contraction domain: a class of functionals of $(h,\varphi ,\text{ghosts})$ with the cone condition imposed in the metric slots, including the status of $k$ and of the classical coordinate kernel ${\Lambda }_{\epsilon }$ once $h$ is quantum.

5. The vertex $V_{1,g}$ with $h$ quantum: its field degree in $h$, the Wick ordering of mixed monomials, and its $\hslash$, $\kappa$ grading.

6. The extension problem for the mixed kernel $f(x)g(y)K_{\mathrm{coord}}(z,x){\Delta }^{\mathrm{gr}}(y,z)\partial \partial \Delta (y,x)$ at $y=x$, with its wavefront analysis, and the corresponding problem for any graviton tree ${\mathsf{X}}_1[{{\Delta }^{\mathrm{gr}}}_{\mathrm{R}}(\frac{1}{2}gT[\varphi ])]$.

7. A Ward formulation: the BRST operator, its action on $\hat{\varphi }$, $h$, and ghosts, and the status of ${\mathfrak{g}}_U$ under it.

8. A comparison rule stating what agreement with the scalar-only $R_1$ at order ${\kappa }^1$ would mean, if anything, once (G1)–(G7) are supplied.

-relabeling rule.

1. ${\Lambda }_{\epsilon }$ and ${G_0}^{\mathrm{R}}$ are classical coordinate kernels applied to classical sources; they are not (G2).

2. ${\Delta }_{+}$ is the scalar two-point kernel; it is not (G3).

3. The joint cone admission of X-B, Proposition prop:joint, is a test on classical background slots; it is not (G4).

4. $V_{1,g}[h,\phi ]$ with classical $h$ is not (G5).

5. $R_1(V_{1,g}[h];\cdot )$ has field degree one in $\varphi$ and degree zero in any quantum metric; it is not a graviton response, and nothing in Proposition prop:scope is a missing term of such a response.