E2: decoupling from the ch-projected slot (proof)

In p15 D13 has an evanescent structure $E{2}_{(mn)\nu }=({\tilde{g}}_{m\nu }{p}_n+{\tilde{g}}_{n\nu }{p}_m)p^2$ with $A_{E2}(d)=3/(16(d-1))$ and UV pole $1/16$; p14r declares no evanescent counterterm. D13 enters the ch projection only through the row $(S_0,{{\Gamma }_1}^{[2]})s_{1}h:h^2$:

where $E$ is the operator of the $h^2$ monomials of ${S_{\Psi }}^{(0)}$ (linearized Einstein–Hilbert). Its exact covariant form is fitted from the frozen density and verified on all components; the fit is unique.

• L1. $p_m{E}^{(mn)(rs)}=0$ as a covariant identity. Only $g_4\cdot p=p$ is used; no trace is taken, so the identity holds exactly in $d$.

• L2. $E2$ is longitudinal in $(mn)$.

• L3. Therefore $E2\cdot E=0$ identically. The slot value is $0$ for every external assignment (Q4S, QdS, mixed $\hat{}/\stackrel{~}{}$ components), exactly in $d$ and not merely $O(ϵ)$.

• L4 (control). The non-longitudinal $T1$ and $E1$ give nonzero slot values, so the map is not trivially zero. $A_{T1}=A_{E1}=0$ for D13, so D13's whole ch-slot contribution is $0$.

Mutation controls: the gauge-fixed operator $q^{2}P$ in place of $E$, $E1$ in place of $E2$, and a wrong index slot each make the $E2$ slot nonzero and are rejected. At one loop the ch projection is linear in ${\Gamma }_1$, so no product of evanescent poles arises.

Outcome: DECOUPLED (proof). No evanescent counterterm is added and no contract file changes for E2. The $E2$ pole remains a UV pole of the $ϵ$-scalar component of the $K^{\prime }\cdot c$ Green function (R6: retained and recorded). Its renormalization matters only outside this projection, and it is not claimed here. D13 stays regulated, with a ch-slot contribution of $0$.