Definition: Admissible inventory, graph support, and ambient span

Let ${\Lambda }_{\mathrm{adm}}$ be the set of $255$ admissible unordered numerical labels $\{a+p,b+q\}$ obtained by enumerating all source-slot budgets of Theorem thm:budget against all vertex placements with $|p|+|q|\le 2$ (Lemma lem:L3comp). Let ${\mathcal{V}}_{\mathrm{adm}}:=\mathrm{span}\{{q^{\circ }}_{\lambda }:\lambda \in {\Lambda }_{\mathrm{adm}}\}$ be the fixed ambient numerical span. Let ${\Lambda }_{\mathrm{graph}}:=\{\lambda \in {\Lambda }_{\mathrm{adm}}:a_{\lambda }\not\equiv 0\}$ be the cancellation-aware support of the assembled coefficients in (eq:QL2) (background-dependent when computed), and let ${\mathcal{V}}_{\mathrm{graph}}:=\mathrm{span}\{{q^{\circ }}_{\lambda }:\lambda \in {\Lambda }_{\mathrm{graph}}\}$. Let ${\mathcal{V}}_{\mathrm{K}}:=\mathrm{span}\{{q^{\circ }}_{\mathsf{k}}:\mathsf{k}\in \mathcal{N}\}$ be the XVI-K span ($25$ generators before relations).

{{0.90} Ambient extension domain. The coherent extension of L.4–L.7 is the linear map

Grading, moments, defects, coherent changes, and the scale family are constructed on this fixed ambient space. Zero coefficients in the ambient sum contribute nothing; the graph uses the appropriate coefficients $a_{\lambda }$ with $\lambda \in {\Lambda }_{\mathrm{adm}}$. }

Notation: the earlier draft symbol ${\mathcal{V}}_{\mathrm{L}}$ (formerly the graph span) is retired; throughout L.4–L.7 the domain of ${\mathfrak{R}}_{\mathrm{L}}$, ${\mathfrak{R}}_{\mathrm{L},\mu }$, $A_{\mu }$, coherent $\Delta {\mathfrak{R}}_{\mathrm{L}}$, and attachment sums is ${\mathcal{V}}_{\mathrm{adm}}$. The symbols ${\Lambda }_{\mathrm{graph}}$ and ${\mathcal{V}}_{\mathrm{graph}}$ are retained only for reporting cancellation-aware support on a stated background.