Manuscript XIV-D@1 · XIV-D.tex · section · 1a7f173d

Local comparison neighborhoods

D0: the unchanged contract › Local comparison neighborhoods · lines 545–571

Choose small neighborhoods $A, B, Y$ of $u_{0}, x_{0}, y_{0}$ , respectively, with pairwise disjoint closures and

B \times A \times Y \subset V

in the vertex order $(x, u, y)$ . Choose a neighborhood $Z$ of $u_{0}$ with $Z \subset A$ .

The variables used below are:

a \in A metric attachment, b \in B scalar attachment,

z \in Z residual metric variable, y \in Y interaction vertex.

In the retained $B_{6}$ graph, $(b, a, y)$ is the original triple $(x, u, y)$ and the local metric vertex enforces $a = z$ .

For other branches, $z$ remains an independent field variable. It is not set equal to $a$ . This is a restriction of a kernel germ to an open set, not an excision of $N^{\circ}$ , a change of switches, or a cutoff of the triangle.

Displayed formulas in this passage

unlabeled · line 549
$B \times A \times Y \subset V$
unlabeled · line 556
$a \in A metric attachment, b \in B scalar attachment,$
unlabeled · line 560
$z \in Z residual metric variable, y \in Y interaction vertex.$