Input statement: X-G: two separate releases

On the X-E/X-F normalization $P_{\mathrm{gr}}=-\frac{1}{4}\mathcal{P}\square$, ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}=-4\mathcal{P}{E}_0^{\mathrm{R}}$, the free multiplet with independent $b$, gauge fermion ${\Psi }_0=\int \bar{c}(C-b)$, $s\bar{c}=b$, and $b=C[h]/2$ is constructed; off-shell nilpotence uses the independent auxiliary field. Separately, an exact source-preserving variational upgrade of X-F's mixed map $k_{\mathrm{mix}}[w;u]=-{{\Delta }^{\mathrm{gr}}}_{\mathrm{R}}({\sigma }_{\epsilon }{Q}_{w}u)$ is Proved obstructed: a compact pure-gauge ramp background gives the reciprocity pairing

on free metric solutions. Source switching is not density switching. The null-collinear set $\mathcal{N}$ is a failed sufficient product criterion, not nonexistence. The unconditional one-loop background identity is Unresolved XG.