Bulk retained identity, counterterms, closure

XVI-O's definitions, quoted verbatim:

• ST4a: ${\Gamma }_{1,\mathrm{local}}=\sum z_i{C}_i$ (gh $0$);

• ST4b: $P_{\mathrm{ret}}(S,C)=-P_{\mathrm{ret}}{A}_1$ (gh $1$);

• retained identity: $(S,{\Gamma }_1)={\mathcal{A}}_1$ at one loop and order ${\kappa }^2$ in the retained sector.

Rows. All 23 one- and two-field monomials of ghost number $0$ and $1$ are classified, each with its own justification. Two independent one-loop enumerations agree with XVI-P: $E=2$ gives 9 graphs and $E=1$ gives 3.

Counterterms (ST4a). The MS-bar counterterm, $-$pole, in the 11-direction declared p14r domain over $\mathbb{Q}(i)$:

• $h\cdot h$: $\frac{49i}{480},\frac{41i}{160},-\frac{49i}{240},-\frac{41i}{80},\frac{43i}{120}$;

• $c\cdot \bar{c}$: $-\frac{3}{8},\frac{1}{4}$, in the route-G orientation; these equal the D06 poles up to the orientation sign $-1$;

• $c\cdot K^{\prime }$: $0,-\frac{3}{8},\frac{1}{4}$;

• $\varphi \cdot \varphi$: $0$. P19 carried this coordinate as a placeholder; it is now earned.

The p22 evanescent counterterms ($K^{\prime }c$ E2: $-\frac{1}{16}$; $\bar{c}c$ SE: $\frac{1}{8}$; $hh$ sum) subtract the evanescent poles only. They lie outside the p14r 4D domain as a disclosed contract revision. They are not needed for the identity: the ch defect is zero with $x=0$.

ST4b and closure.

• $c\cdot h$: the defect is zero exactly in $d$ (P19).

• $c\cdot b$: this is the ghost equation, and its $\Sigma$ equals D06.

• $c\cdot {\bar{c}}^{\ast }$: unreachable.

• $c\cdot \varphi$: zero by parity.

Full ${\sigma }_0$ closure of the complete MS-bar counterterm holds in every gh$+1$ sector. The ${\sigma }_0$ rank on $\varphi \cdot \varphi$ is $0$, and the cohomology of the domain is $0$. Outcome: cancellable, with no residual.