Proposition: Exact separating functional for the displayed scale functional (audit 3)

In the ordered basis $(L_1,\dots ,L_5)=(h_{\mu \nu }{\square }^2{h}^{\mu \nu },h{\square }^{2}h,({\partial }^{\mu }{h}_{\mu \nu })\square ({\partial }_{\rho }{h}^{\rho \nu }),h\square {\partial }^{\mu }{\partial }^{\nu }{h}_{\mu \nu },({\partial }^{\mu }{\partial }^{\nu }{h}_{\mu \nu }{)}^2)$ of Theorem thm:wpB-dp (the order of COV4 in higher_grade_scaling.py and of conventions.json), let

Then $\ell (v_{R^2})=0+1-0-2+1=0$, $\ell (v_{{\mathrm{Ric}}^2})=\frac{1}{4}+\frac{1}{4}-\frac{1}{2}-\frac{1}{2}+\frac{1}{2}=0$, and

so $v_{\mathrm{scale}}\notin \mathrm{span}(v_{R^2},v_{{\mathrm{Ric}}^2})$ exactly. Interpretation. On the pure-gauge configuration $h_{00}=f(t)$, all other components zero ($h_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$ with ${\xi }_0=\frac{1}{2}\int f$), one has $h=h^{00}=f$, ${\partial }^{\mu }{h}_{\mu 0}=f^{\prime }$, ${\partial }^{\mu }{\partial }^{\nu }{h}_{\mu \nu }=f^{\prime \prime }$, $\square ={{\partial }_t}^2$, so modulo total derivatives

($L_3=f^{\prime }{f}^{\prime \prime \prime }\equiv -f{f}^{\prime \prime \prime \prime }$, $L_5=(f^{\prime \prime }{)}^2\equiv f{f}^{\prime \prime \prime \prime }$). Hence $\int {\sum }_j{a}_j{L}_j\equiv \ell (a)\int f{f}^{\prime \prime \prime \prime }$: the curvature squares vanish on pure gauge, as they must, while the scale functional leaves $-\frac{1}{16}f{f}^{\prime \prime \prime \prime }$ times the unit $i/(16{\pi }^2)$; its Euler–Lagrange expression in $f$ is $2\ell (a)f^{\prime \prime \prime \prime }$. This is verified symbolically with sympy in regression_M.py (audit-3 section). Scope. The proposition certifies the displayed functional only; the identification of the graph's collapsed top scale tensor with $E[{I^{\mathrm{top}}}_{\mathrm{scale}}]$ (the graph$\to$functional obligation) is carried by the collapse certificate of Theorem thm:wpB-dp(i); regression_M.py re-verifies it exactly (the recollapse of the archived $C_{\mathrm{top}}$ equals the Euler tensor of the displayed ${I^{\mathrm{top}}}_{\mathrm{scale}}$ entrywise, $490$ entries, and its $f^{\prime \prime \prime \prime }$ coefficient in $E^{00}[f{e}_{00}]$ is $-\frac{1}{8}=2\ell (v_{\mathrm{scale}})$).