Manuscript XIV-C@1 · XIV-C.tex · section · a34d13df

Cycles and collision loci

J1: copied branches, line orders, and collision inventory › Cycles and collision loci · lines 794–847

Branch	Selected locations	Cycle vertices
$B_{1}$	$(u, v)$ or $(v, v)$	${y, u, v}$ or ${y, v}$
$B_{2}$	$(u, u)$	${y, u}$
$B_{3}$	$(u, v)$ or $(v, v)$	${y, u, x, v}$ or ${y, v}$
$B_{4}$	$(u, x)$ or $(v, x)$	${y, u, x}$ or ${y, v, x}$
$B_{5}$	$(u, x)$ or $(v, x)$	${y, u, x}$ or ${y, v, u, x}$
$B_{6}$	$(u, x)$	${y, u, x}$
$B_{7}$	$(u, x)$ or $(v, x)$	${y, u, x}$ or ${y, v, x}$
$B_{8}$	$(v, x)$	${y, v, u, x}$
$B_{9}$	$(v, u)$	${y, v, u}$
$B_{10}$	$(v, v)$ or $(v, x)$	${y, v}$ or ${y, v, u, x}$
$B_{11}$	$(v, v)$ or $(v, u)$	${y, v}$ or ${y, v, u}$

For a collapsed connected subset $I$ , its normal codimension is $4 (∣ I ∣ - 1)$ . The inherited superficial count is $2 E_{I} + D_{I}$ , with numerical internal-edge count $E_{I}$ and derivative count $D_{I}$ .

Type	Edges	Codimension	Superficial count	Count minus codimension
Parallel pair	$2$	$4$	$4 + D_{I}$	$D_{I} \leq 6$
Three-cycle	$3$	$8$	$6 + D_{I}$	$D_{I} - 2 \leq 4$
Four-cycle	$4$	$12$	$8 + D_{I}$	$D_{I} - 4 \leq 2$
Forest, $n$ vertices	$n - 1$	$4 (n - 1)$	$2 (n - 1) + D_{I}$	$D_{I} - 2 (n - 1)$

These are copied power counts, not established scaling degrees of a presently undefined joint distribution.