Counterterm domain and solve

Route G applies ${\sigma }_0=(S_0,\cdot )$ to the 11 p14r domain densities: h.h#0–4, c.cb#0–1, c.K#0–2, $\varphi .\varphi$. Route C computes the covariant ch images exactly in $d$. Both give rank $P_{ch}{\sigma }_0=3$.

• Over $\mathbb{Q}(i)$ (these counts are the real dimensions under a compatible real structure; wording corrected in p20): rank ${\sigma }_0{|}_D=5$, $\dim D\cap \ker {\sigma }_0=6=\dim D\cap \mathrm{Im}{\sigma }_0$, cohomology 0, and the quotient has dimension 5 (complement h.h#0–2, c.cb#0–1).

• The h.h + c.K ${\Gamma }_1$ data, without the ghost self-energy and in route-G (c,cb) labels, have quotient coordinates pole $(0,0,0,-3/8,1/4)$, coefficient of $L$$(0,0,0,3/8,-1/4)$ and local part $(0,0,0,-2/3,5/12)$. The h.h pole is transverse, i.e.\ ${\sigma }_0$-exact. These coordinates equal $-[\Sigma ]$: $\Sigma$ itself has $(+3/8,-1/4)$, and the data including $\Sigma$ have quotient $0$ [clarified in p20].

• The linear solve $\Delta +P_{ch}{\sigma }_0(x)=0$ gives $x=0$ (outcome ZERO); no residual and no obstruction.

• The MS-bar counterterm has zero ch image. The h.h + c.K part alone has a nonzero full ${\sigma }_0$ image in the c.b sector; adding the c.cb ghost self-energy counterterm in the correct orientation closes it [corrected in p20, see the p20 erratum note].

and their definitions.

• projected_defect_computed: $\Delta$ is assembled from all evaluated inputs, exactly in $d$, in all parts and all seven assignments.

• projected_identity_verified: $\Delta =0$ for the regulated ${\Gamma }_1$.

• projected_restoration_verified [REDEFINED in p20]: full ${\sigma }_0$ closure, in every ghost $+1$ sector (h.c, c.b, c.cbs), of the complete one-loop ${\kappa }^2$ MS-bar counterterm $S_{ct}$ (h.h + c.K + c.cb, ghost self-energy held fixed, derived relative factor $+1$, orientation converted). The p19 definition ($P_{ch}{\sigma }_0(S_{ct})=0$, $x=0$) follows from the identity and locality; it is kept only as a labelled route-consistency control.

Scope of the restoration flag: the one-loop ${\kappa }^2$ counterterm only; not bulk symmetry, not O.1. The release gate re-checks each definition from the certificate content. bulk_symmetry_restored is false regardless.