Proposition: Signs, multiplicities, and degrees

At positive cutoff, (eq:g1223)–(eq:g1323), with the correspondingly regulated kernels and residual fields, are the three two-edge placements. Each has sign $+1$ from the two insertion signs and assignment multiplicity $2$. Their residual field degree is two. The fully contracted placement has coefficient

Proof. The allowed nonzero edges are the sharp–middle metric edge $12$, the scalar edge $13$, and the middle–sharp metric edge $23$. Each choice of two leaves exactly two field slots.

For $12+23$, both middle slots are used, leaving the two scalar slots. There are two assignments of the middle slots. For $12+13$, either middle slot can be used on edge $12$; the remaining middle slot and third-vertex metric slot survive. The case $13+23$ is the exchanged case. Both sharp insertions contribute one minus sign, so all three coefficients have positive sign.

The inherited tensor cancellation selects the zero vector component on every sharp–middle metric edge. This gives $S_{0\nu }$ and $S_{\mu 0}$ in the latter two placements. Full contraction uses both metric edges and the scalar edge, giving the inherited triangle and multiplicity IXB (prop:unique, eq:T).