Diagnostics

Let ${\xi }^{\mu }=x^{\mu }$, so $K\xi =2\eta$, and ${\mathcal{L}}_{\xi }u=(x\cdot \partial +2)u$ on covariant two-tensors, ${\mathcal{L}}_{\xi }T=(x\cdot \partial +2)T$ on contravariant densities of weight one. Then $[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]u=[x\cdot \partial ,P_{\mathrm{gr}}]u=-2P_{\mathrm{gr}}u$, since $\square$ is homogeneous of degree $-2$ and $\mathcal{P}$ is constant; so (eq:key) predicts ${\mathcal{Q}}^{(1)}[\eta ]=-P_{\mathrm{gr}}$. Directly: $\bar{q}=(1+\kappa )\eta$, and for a constant $c>0$, $\sqrt{-c{q}^{\prime }}R[c{q}^{\prime }]=c\sqrt{-q^{\prime }}R[q^{\prime }]$, so the part of $S_{\mathrm{gr}}[c\eta +\kappa u]$ quadratic in $u$ is $c^{-1}$ times the flat one; $C_{c\eta }[u]=c^{-2}C[u]$ and $\sqrt{-c\eta }c{\eta }_{\mu \nu }=c^3{\eta }_{\mu \nu }$, so $S_{\mathrm{gf}}[u;c\eta ]=c^{-1}{S^{(0)}}_{\mathrm{gf}}[u]$. Hence $\mathcal{Q}[c\eta ]=c^{-1}{P}_{\mathrm{gr}}$ and the $\kappa$-coefficient at $c=1+\kappa$ is $-P_{\mathrm{gr}}$. Likewise (eq:cov2) predicts ${\mathcal{G}}_2(\eta ,u)=-\frac{1}{2}{G}_{\mathrm{lin}}[u]$, and $\mathcal{E}[c{q}^{\prime }]=\mathcal{E}[q^{\prime }]$ gives the same. Both agree.

Let $\xi \in {\mathfrak{g}}_U$, $\zeta =0$, $h_{\mathrm{cl}}=K\xi$, $k_{\mathrm{cl}}=\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$. By (eq:jetlaw) at $h_{\mathrm{cl}}=0$, ${\mathsf{T}}_{\mathrm{mix}}={\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]-{\hat{A}}_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}]\equiv {\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]$ modulo $\mathfrak{J}$, and on X-E's witness pair $h_0=x^1{\delta }^2\otimes {\delta }^2$, ${\varphi }_0=t{x}^1$, $\xi =\chi {\partial }_1$, its classical value is $2\int f{\partial }_1{\partial }_t({{\mathsf{E}}_0}^{\mathrm{R}}{)}^2\chi \ne 0$ for suitable $f,g,\delta$ xe:ward. The mixed tree is nonzero modulo $\mathfrak{J}$ on a pure-gauge background, and it is exactly the term whose absence X-E's Diagnostic Pure gauge with the graviton held fixed recorded as the order boundary. With it, ${\hat{\Phi }}^{\mathrm{F}}(f)[K\xi ,\frac{1}{2}{\mathcal{L}}_{\xi }K\xi ]\equiv {\hat{\Phi }}^{\mathrm{F}}(f)[0,0]$ modulo $\mathfrak{J}$ and ${\kappa }^3$, by Theorem thm:main.

At $h_{\mathrm{cl}}=0$, ${\mathsf{T}}_{\mathrm{mix}}=0$ identically. On $\hat{\varphi }\to t$, $\hat{h}\to 0$, $m=0$, with $g=g_0(t)$ near the dependence region of $K_f$, ${\mathsf{T}}_{\mathrm{self}}$ and ${\mathsf{T}}_{\mathrm{mix}}$ vanish and ${\mathsf{T}}_{\mathrm{mat}}$ gives X-E's number $\int f{{\int }_0}^t(1-g_0(s))G_2(s)\mathrm{d}s{\mathrm{d}}^{4}x$. X-E's Diagnostic Sourced tree number is therefore the complete classical ${\kappa }^2{\hslash }^0$ tree value at trivial background and vanishing graviton.

For a Poincaré generator $\xi$ (not in ${\mathfrak{g}}_U$), ${\mathcal{Q}}^{(1)}[K\xi ]=0$ and $[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]=0$; (eq:key) is $0=0$. For $\xi \in {\mathfrak{g}}_U$, $K\xi \ne 0$ unless $\xi =0$, and ${\mathcal{Q}}^{(1)}[K\xi ]\hat{h}\ne 0$ in general; Diagnostic diag:puregauge exhibits a case where its retarded response is not in $\mathfrak{J}$.