Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

Branch–vertex inventory

Density and vertex audits › Branch–vertex inventory · lines 2336–2371

For every internal row $K_{b}$ , the three terms are

Type	Crossed operator	Coefficient and output
Kinetic	$\partial_{2, μ} H_{I}^{μν}$	$\partial_{y^{ν}} (i g K_{b}^{I, J} \cdot)$ .
Trace kinetic	$\partial_{2}^{ν} H_{I}^{μ λ}$	$\partial_{y^{ν}} (- i g η_{μ λ} K_{b}^{I, J} \cdot /2)$ .
Mass	$H_{I}^{μν}$	Multiplication by $- i m^{2} g η_{μν} K_{b}^{I, J} /2$ .

Their product with the seven rows is the complete finite inventory.

Output derivatives act after multiplication and therefore retain derivatives of $g$ , the internal distribution, and both relative crossed dependences. By contrast, $\partial_{2}$ acts only on the crossed scalar slot before pullback.

The transposed signs are: kinetic $- i$ becomes $+ i$ ; trace-kinetic $+ i /2$ becomes $- i /2$ ; mass remains $- i m^{2} /2$ . Symmetrization introduces no extra physical polarization factor.