Theorem: Local complete-family failure

Suppose $U$ contains a marked positive-$K_5$ incidence

with $0<\alpha <\beta$, $ϵ>0$, and $\omega \ne \mathbf{n}$. Assume $f(b)g(y)\ne 0$ and $M_2(\mathbf{n})\ne 0$ at that incidence.

There is an isolated neighborhood $O⋐U$ and a fixed ${\Phi }_{\ast }\in \mathcal{D}(O;S^{\ast })$ for which

Consequently the complete family has no raw distributional limit on $U$.

A nonempty class of admissible normalized real profiles satisfies $M_2(\mathbf{n})>0$ for every $\mathbf{n}$; it is given in Corollary cor:profile-class.