Manuscript XIII-C@1 · XIII-C.tex · section · e6556893

Mixed channels and larger collisions

C3: scaling degrees and their exact scope › Mixed channels and larger collisions · lines 939–984

XIII-C@1#eq:superficial

The inventory gives an exact derivative-counting rule. For a collision subset $I$ , let $E_{I}$ be its number of internal numerical edges and $D_{I}$ the sum of derivative orders on those edges after the specified placement. Its superficial homogeneous exponent is

ω_{I}^{sup} = 2 E_{I} + D_{I} .

The complete graph has at most six field/line derivatives after including the interaction. Derivatives left on unused fields do not increase the numerical exponent.

Collapsed subgraph	$E_{I}$	Codimension	Superficial exponent	Exponent minus codimension
Parallel two-vertex pair	$2$	$4$	$4 + D_{I}$	$D_{I} \leq 6$
Three-vertex cycle	$3$	$8$	$6 + D_{I}$	$D_{I} - 2 \leq 4$
Four-vertex cycle	$4$	$12$	$8 + D_{I}$	$D_{I} - 4 \leq 2$
Forest with $n$ vertices	$n - 1$	$4 (n - 1)$	$2 (n - 1) + D_{I}$	$D_{I} - 2 (n - 1)$

Equation (eq:superficial) is not advertised as a proved scaling-degree bound for an undefined joint distribution. It is a finite graphwise power count. If a graph product and its uniform scaled limits are established, it supplies the expected degree bound. At the characteristic cancellation of Lemma lem:pinch, the prerequisite product theorem has not been established.

Consequently, the table does not complete C3 for the mixed triangles, four-vertex cycles, or their intersections. Importing the degree of a different two-propagator graph would not fill that gap.

Displayed formulas in this passage

XIII-C@1#eq:superficial · line 945
$ω_{I}^{sup} = 2 E_{I} + D_{I} .$