Proposition: Signs and multiplicities

The three two-edge placements have insertion sign $+1$, assignment multiplicity $2$, and residual degree two. Their fully contracted completion has the parent coefficient ${T_{\epsilon }}^{\rho }$.

Proof. The only available nonzero edges are metric edges $12,23$ and scalar edge $13$. Each two-edge choice removes four of the six slots. Both outer insertion signs occur, giving $+1$. For $12+23$, the two middle slots have two assignments. For $12+13$, either middle slot is used, again giving two; the third case is its reverse. The tensor identity fixes index $0$ on each mixed metric edge. Full contraction adds the remaining edge and gives the supplied triangle.