Proposition: Exact support isolation

Status: Proved for displayed mathematical data.

1. At

   $$b-a=\beta \ell ,r=-ϵ(1,\mathbf{m}),ϵ>0,\mathbf{m}\ne \mathbf{n},$$

   a sufficiently small neighborhood excludes every row except $5a$.

2. If $b-a$ and $b-z$ are future null and nonparallel, a sufficiently small neighborhood excludes every row except $47$.

The complete contraction on tests in either neighborhood equals the surviving row contraction for every smooth crossed array. No nonzero singular coefficient follows from isolation alone.

Proof. In (i),

This excludes the $47$ and $5z$ seed supports; $r\ne 0$ excludes the response rows.

In (ii), write $s=b-a$, $t=b-z$. Then $r=s-t$ and $r^2=-2s\cdot t<0$. Both $5$ supports require $r^2=0$, and the response rows require $r=0$. Differentiation and smooth multiplication preserve these local support exclusions.