Proposition: Isolated non-tip neighborhoods

The following two kinds of marked incidences admit neighborhoods on which the complete internal family equals the indicated single row:

1. A $47$ graph point with $\beta -A>0$ and $Z\ne 0$: the complete family equals $47$.

2. A $5+$ point with $ϵ>0$ and $\omega \ne \mathbf{n}$: the complete family equals $5+$.

The equalities are equalities of internal distributions restricted to the neighborhood; hence they hold for the regularized vertex on tests supported there.

Proof. At the first graph, with $u=0$,

Both $K_5$ placements require a null residual attachment. The supported rows require $r=0$. A small neighborhood excludes all these closed supports.

At the positive-placement point, $r=-ϵj_{\omega }$, and

This excludes the $47$ and negative-placement seeds, which require $b-z$ to be null. Since $r\ne 0$, it excludes every supported row. Differentiation does not enlarge seed support. Small compact neighborhoods preserve the strict inequalities.