Remark

Proposition prop:K5chart is the first mutual-consistency check between D0.1–D0.2 and the six entries: the internal expressions of the two $K_5$ placements and of $K_6$ are, on the respective charts, elements of the span described by (eq:Recrule), with the placement $\pm$ selecting the chart—consistent with XIV-J's clause (b) “retain both $K_5$ placements.” The coefficients depend on $q_{\pm }$ (through ${d_{\pm }}^0$ and $f(b)$) and on $r_{\pm }$ (through ${\sigma }_{\epsilon }(z)$); the $q$-dependence can be removed by Lemma lem:taylor at the cost of lower-order terms, the $r$-dependence cannot in general. Whether XIV-I's coefficient class admits this dependence is part of the unreproduced definition; by (F1) it is the natural reading. By the same mechanism $K_9$ and $K_8$ are sums of terms ${\partial }^{\gamma }R(d_{+})$ or ${\partial }^{\gamma }\mathcal{B}(d_{+})$ with $|\gamma |\ge 1$ times ${\partial }^{{\gamma }^{\prime }}\delta (r_{+})={\partial }^{{\gamma }^{\prime }}{\square }_{r}R(r_{+})$ and smooth coefficients, hence also of the form (eq:Recrule) on $V_{+}$ once (eq:Q) is expanded; this is not carried out, since the $q_{\alpha \beta }$ are not reproduced. $K_{47}$ contains the factors $R(d_{+})$ and $R(d_{+}+r_{+})$ and is not of this form in either chart.