Diagnostics with exact scope

-gauge background. At $h=0$, $T_{\mathrm{mix}}=0$, but Theorem thm:jet gives

for every smooth free $r$. This checks the sign of the cancelling term without transforming the free carrier.

free graviton. On $r=0$, the mixed and self classical values vanish. The matter coefficient reduces exactly to X-E-ST's tree, with its $-2$ normalization and full scalar response retained.

scalar. Every displayed observable tree vanishes at $u=0$. This is a statement about these coefficients, not vacuum energy or a graviton state.

-fixed versus invariant shell. A free ${\mathsf{P}}_{\mathrm{gr}}r=0$ field can have $G_{\mathrm{lin}}[r]\ne 0$. The exact F1 identity and background gauge correction are both necessary. No proof step replaces the gauge-fixed shell by the linearized Einstein shell.

tadpole. The fluctuation-independent terms removed from (eq:fluctuation) would induce classical shifts of the fluctuation. Their retarded completion and its background bookkeeping are not part of this release. No second copy of the classical two-jet law is added.