Batch B: D13 ( κ 2 K '\! c )

Write $G_{(mn)\nu }(p)=V_{{\Gamma }_1}[{K^{\prime }}_{(mn)}(-p),c^{\nu }(p)]={\kappa }^2{N}_{ϵ}\hat{G}$, with $\hat{G}={\sum }_i{A}_i(d)T_i{B}_0$, $B_0=1/ϵ+2-L$ and $L=\log (-p^2/{\mu }^2-i0)$. The structures are

together with the evanescent $E_1={\tilde{G}}_{mn}{p}_{\nu }{p}^2$ and $E_2=({\tilde{G}}_{m\nu }{p}_n+{\tilde{G}}_{n\nu }{p}_m)p^2$. The coefficients are

This gives:

• UV poles: $T_2$: $3/16$, $T_3$: $-1/4$, $E_2$: $1/16$.

• MS-bar finite parts: $T_2$: $1/3-\frac{3}{16}L$, $T_3$: $-5/12+\frac{1}{4}L$, $E_2$: $1/6-\frac{1}{16}L$. The nonlocal part is $-(\text{pole})\cdot L$, and the local part is a cubic polynomial.

• Counterterm map. The 4D pole lies in the span of the three p14r $c\cdot K^{\prime }$ directions, with coordinates $(0,3/8,-1/4)$. The local ambiguity is a finite renormalization along the same directions.

• {$d$-dependence.} The $(d-4)\times$pole terms shift the local part ($T_2$: $-1/24$, $T_3$: $+1/12$). Setting $d=4$ first gives a different MS-bar finite part.

• Evanescent pole and status. The evanescent pole ($E_2$) is nonzero, and the p14r contract declares no evanescent counterterm. D13's status is therefore regulated. Its 4D sector is renormalized; the evanescent pole is recorded but not absorbed, and stays an open contract item.

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• {Two routes, exact in $d$.} Route 1 is the sha-pinned export through the dred Pipeline. Route 2 is the integrand regenerated from the rule tables and evaluated by Feynman parameters with $d$ symbolic. They agree exactly.

• Exact covariant fit.

• BRST sanity (ghost equation). $2i{D}^{\rho }[G]$ equals the evaluated one-loop ghost self-energy. The ghost self-energy's 4D pole lies in the p14r $c\cdot \bar{c}$ span, at $(-3/8,1/4)$; its evanescent pole is $-1/8$.

• Structural R5 safeguard. It is active.

. Each mutation changes the result and is rejected: wrong ghost-loop sign, premature $d=4$ (refused at the substitution), dropped evanescent part, wrong symmetry factor, and wrong K${}^{\prime }$/$c$ ordering.