Manuscript Em01-B@3 · Em01-B-v3.tex · section · 69189992

Internal placement inventory

Complete branch–vertex inventory › Internal placement inventory · lines 2054–2111

Split $K_{9} = K_{9, der} + K_{9, mass}$ using its two displayed lines. This produces seven bookkeeping rows, not seven new physical branches.

Internal row	Placement or seed	Internal derivative bound	Crossed/output operations
$K_{47}$	$R (b - a) R (b - z)$ ; exact coefficient (eq:K47)	$4$	All three vertex types above.
$K_{5, +}$	$R (b - a) R (a - z)$ ; coefficient (eq:K5plus)	$4$	All three vertex types above.
$K_{5, -}$	$R (b - z) R (z - a)$ ; coefficient (eq:K5minus)	$4$	All three vertex types above.
$K_{6}$	Local $a = z$ , outer $q_{R} = 0$ ; coefficient (eq:K6)	$2$	All three vertex types above.
$K_{8}$	Actual source incidence, $B = R * R$ , and coordinate divergence; (eq:K8)	$3$	All three vertex types above.
$K_{9, der}$	Source response $T$ , derivative scalar root; first line of (eq:K9)	$3$	All three vertex types above.
$K_{9, mass}$	Same $T$ , mass scalar root; second line of (eq:K9)	$2$	All three vertex types above, including the additional crossed-vertex mass type.

Thus another reader obtains the complete finite contraction by taking the product of the seven internal bookkeeping rows with the three explicitly indexed vertex types. There is no unspecified “complete expansion.”