Remark: Faithfulness for the specializations

For ${\mathcal{G}}_{\mathrm{background}}$ the coefficient class is $\mathbb{Q}[\sigma \text{-jet}(y)]\otimes \mathbb{Q}[w^1]\chi (w)$. Diagonal evaluation $F\mapsto F(w,w)$ is injective on this class: at a point $w$ with $\chi (w)\ne 0$, the $\sigma$-jet at $w$ ranges over all values independently of $w^1$, because $\sigma$ is arbitrary. The induction above therefore applies with ${U_N}^{\mathrm{background}}$ computed from the specialized expansion. The same holds for the time-only class, whose remaining symbols are independent, and for the plateau class, which is a sector restriction. For ${\mathcal{G}}_{\mathrm{fixed}}$ the coefficients are fixed functions (the imported ${\sigma }_{\epsilon }$, ${\chi }_{\mathrm{I}}$). In general one only has $\ker {\mathcal{G}}_{\mathrm{formal}}\subseteq \ker {\mathcal{G}}_{\mathrm{fixed}}$. Equality with the formal kernel is not automatic from formal $D\ge 1$ vanishing; it requires a separate injectivity analysis. Stage 2 records sufficient criteria (Proposition prop:fixed-crit) and a region ledger (Proposition prop:fixed-regions); the time-only graph map being injective does not by itself settle the time-only collapse/Helmholtz problem; that problem is settled in Section sec:to-timeonly, the transfer to the concrete XII-C ramp is Proposition prop:to-actual (explicit hypotheses), and regions where the actual ${\chi }_{\mathrm{I}}$ is non-constant remain Unresolved.