M.5 Counterterm-functionals ledger

All statements in this section use Gate V; Input (S-bv) enters through the presentation of M.2 and is certified (Theorem thm:wpA-Sbv). Four ledgers are maintained separately:

1. literal local representatives;

2. equality modulo total derivatives;

3. EOM reduction;

4. field-redefinition equivalence.

EOM and permitted field redefinitions are not defined in Target I. Consequently ledgers (3) and (4) are left empty/Unresolved; work proceeds in (1)–(2). (Defining EOM/redefs is a Target II / M.8 obligation.)

family and Lorentz scalars. The $26$-dimensional ultralocal SA family ($N=4$, $\gamma =()$, all in ${\mathcal{C}}_{\mathrm{reg}}$) consists of Gate V-symmetric multiplication operators $E^{\mu \nu }={\chi }_{\mathrm{I}}{\sigma }_{\epsilon }\sum {K^{\mu \nu }}_{ab}{h}_{ab}$, with matched functionals $I=\frac{1}{2}\int {\chi }_{\mathrm{I}}{\sigma }_{\epsilon }⟨h,Kh⟩$. The SA basis realises $26$ distinct nonzero $K$-patterns (counterterm_functionals.py). Inside $\mathrm{span}\{E_{P_1},E_{P_2}\}\cap \mathrm{im}\mathrm{Collapse}$ there is a single Lorentz-scalar line $P_1-\frac{89}{140}{P}_2$, with preimage $C[N=4,\{33,33\},()]=3/70$.

{$R^2$/${\mathrm{Ric}}^2$ preimages and K-block changes.} In the double plateau, $E_{R2}$ and $E_{RC2}$ are collapse images in ${\mathcal{C}}_{\mathrm{coh}}$ ($286$ nonzero coefficients each). Their supports use $N=2$ slots, hence change the ${\mathcal{V}}_{\mathrm{K}}$ block and are not regression-preserving. In ${\mathcal{C}}_{\mathrm{reg}}$ only the combination $R2-\frac{188813}{88456}RC2$ is an image. At quadratic order $\int ({\mathrm{Riem}}^2-4{\mathrm{Ric}}^2+R^2{)}_{\mathrm{lin}}$ is a total derivative, so $\int {\mathrm{Riem}}_{\mathrm{lin}}^2$ is not a separate probe (ledger (2)).

third-order directions. The eight double-plateau variational directions at $D=3$ (Remark rem:thirdorder) are recorded in ledger (1) as Gate V variational collision tensors. They are not gravitational scalar counterterms.

completion gate.