Remark

None of (H1)–(H3) is verifiable from the record: no $F_0$ exists (Proposition prop:F0); the record does not say that the permitted family is affine or that its corrections take values in ${\mathcal{C}}_{\mathrm{part}}$ (Proposition prop:classstatus(ii)); and (H3) requires a set equation for $U_{\mathrm{can}}$, which is refused in favor of a role, and the unsupplied lift of ${\Delta }_I$ to $W$. By Lemma lem:recrule(iv), if $\mathsf{Rec}$ is linear over chart functions then $\ker \mathsf{Rec}$ contains, besides XIV-J's relation, the relations $q{\delta }^{(j)}(q){{\partial }^{\beta }}_{r}R+j{\delta }^{(j-1)}(q){{\partial }^{\beta }}_{r}R$ and $r^{2}R(r)$ for every presentation realizing these coefficients; the descent test must then be run on each. No obstruction and no cancellation is asserted.