Proposition: E0 fails

The record does not define ${\mathcal{P}}_{\mathrm{int}}$, $\mathsf{Rec}$ on ${\mathcal{P}}_{\mathrm{int}}$, $O$, $m^{\circ }$, ${\mathcal{C}}_{\mathrm{cross}}$, the vertex data, the candidate class, or the correction class. Hence no preliminary lift $F_0:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ with $r_O{F}_0=m^{\circ }\mathsf{Rec}$ can be stated, and no correction space $\mathcal{A}$ against which a null defect could be tested is available.

Proof. A linear map requires a domain, a codomain, and a value on each element of the domain. The domain ${\mathcal{P}}_{\mathrm{int}}$ (D0.1) and codomain (D0.5) are not defined. The value on any presentation of the six entries is the composition of the internal kernel with ${\mathcal{C}}_{\mathrm{cross}}$, the vertex Hessian, and weights (D0.10–D0.13), none of which is defined; the candidates of sec:candidates concern only the internal kernels. The correction class needs D0.16–D0.17. All are in status R, U, or C.