Theorem: Double-plateau scale tensor: top component and full tensor

Assume Gate V. In the double plateau, the cutoff-independent $D=4$ component is variational and Lorentz covariant, but not linearized-gauge invariant. The full scale tensor has cutoff-dependent lower-degree components and need not be variational. In detail:

1. (i)

   Top component. The $D=4$ component $\mathrm{Collapse}(C_{\mathrm{top}})$ of $\mathrm{Collapse}(C_{\mathrm{scale}})$ (the slots with $|\gamma |=N$; independent of $\mu$ and $\varpi$) is nonzero and variational, with

   $${I^{\mathrm{top}}}_{\mathrm{scale}}=\frac{i}{16{\pi }^2}\int (-\frac{71}{120}{h}_{\mu \nu }{\square }^2{h}^{\mu \nu }-\frac{27}{80}h{\square }^{2}h-\frac{11}{10}({\partial }^{\mu }{h}_{\mu \nu })\square ({\partial }_{\rho }{h}^{\rho \nu })+\frac{19}{30}h\square {\partial }^{\mu }{\partial }^{\nu }{h}_{\mu \nu }-\frac{13}{15}({\partial }^{\mu }{\partial }^{\nu }{h}_{\mu \nu }{)}^2)$$

   (mod total derivatives; the prefactor is $i{\pi }^2{c}^2$ with $c=1/(4{\pi }^2)$). It is Lorentz covariant (a combination of Lorentz scalars), but ${\partial }_{\mu }{E}^{\mu \nu }\ne 0$ (not linearized-gauge invariant), and it is not in $\mathrm{span}(E_{R^2},E_{{\mathrm{Ric}}^2})$: in the basis above ${R_{\mathrm{lin}}}^2\mapsto (0,1,0,-2,1)$ and ${\mathrm{Ric}}_{\mathrm{lin}}^2\mapsto (\frac{1}{4},\frac{1}{4},\frac{1}{2},-\frac{1}{2},\frac{1}{2})$, the rank goes from $2$ to $3$ when the scale vector is adjoined, and Proposition prop:wpB-sep gives an exact separating functional. In particular it is not a multiple of $E_{R^2-(188813/88456){\mathrm{Ric}}^2}$, of $E_{\mathrm{EH}}$, or of an ultralocal.

2. (ii)

   Full tensor. The lower-degree components ($D=4-k$, $k=1,\dots ,4$) depend on the cutoff through ${\Lambda }_k$, i.e.\ through ${\nu }_k[\varpi ]$ of (eq:wpB-nu). The full tensor is variational iff ${\Lambda }_1={\Lambda }_3=0$, ${\Lambda }_2$ lies in the $49$-dimensional and ${\Lambda }_4$ in the $26$-dimensional self-adjoint subspaces; so it need not be variational. Lorentz covariance of the full tensor is not established: the lower-order terms depend on the possibly noncovariant $\varpi$, and the Lorentz tests apply to the top order only.

3. (iii)

   Curvature span. For every $\mu$ and every admissible $\varpi$ the full collapsed scale tensor lies outside the span of the M.5 curvature targets: collapse is homogeneous in $D$, and its fixed $D=4$ component already lies outside $\mathrm{span}(E_{R^2},E_{{\mathrm{Ric}}^2})$, while the targets of degree $D=0,2$ lie in other blocks.

The whole tensor is retained as unmatched relative to the curvature ledger.

Proof. (i) Exact double-plateau collapse and adjoint defect of $C_{\mathrm{top}}$ (wpB/higher_grade_scaling.py, Section D; full_scale_collapse_certificates.json, readings.double_plateau.top_D4): adjoint defect $0$, so Lemma lem:helm gives the potential; its coordinates in the five-term Lorentz-scalar basis are the displayed coefficients; the divergence is nonzero exactly; the rank statement is exact over $\mathbb{Q}$. regression_M.py independently recollapses $C_{\mathrm{top}}$ from _wpB_state.json and checks the pure-gauge coefficient of Proposition prop:wpB-sep. (ii) Theorem thm:wpB-B0(2) and the lower-order envelope above; the Lorentz and derivative identities hold for the top order only (paragraph “Exact top-order data”). (iii) Homogeneity in $D$ (Lemma lem:to-comp and the paragraph above the table) and (i).