Remark: What the budget does and does not yet give

Combined with the vertex bound $|p|+|q|\le 2$ of XVI-K, Proposition prop:hess, Theorem thm:budget suggests numerical derivative counts at most $2$, $3$, $4$ in the $C_2$, $M_2$, $J_3$ sectors, hence declared orders at most $2$, $3$, $4$. This is proved in L.2–L.3: graph decomposition and placement of source-slot derivatives on the edge kernels (Proposition prop:QL), and product/scaling statements (Proposition prop:Lsd, Theorem thm:L3).

The source-slot derivatives are derivatives in the source variable $y$. By XVI-K, eq:sign, ${{\partial }^a}_y[S(y-w)]=({\partial }^{a}S)(y-w)$ carries no sign, while vertex derivatives carry $(-1{)}^{|p|}$; this is the L.2 sign rule, re-derived for the complete source in Lemma lem:Lsign.

The source is a pointwise polynomial in jets. By (S1) it is the Euler derivative of the printed ${\sigma }_{\epsilon }{\ell }_3$. No replacement of the printed density by an integration-by-parts-equivalent representative is performed. XII-C records that replacing ${\ell }_3$ by ${\ell }_3+{\partial }_{\mu }{B}^{\mu }$ changes the weighted source by $-{\mathcal{E}}_{\gamma }(({\partial }_{\mu }{\sigma }_{\epsilon })B^{\mu })$; no such replacement is made.

Nothing in L.1 concerns the vertex, the edge kernels, the extension, counterterms, scale, or gravitational symmetry.