Oriented connector crossing count ($dN/dS=n|\mathbf{R}\cdot \mathbf{n}|\psi d^{3}R$)

For a locally uniform distribution of labelled connectors, a connector crosses a small surface when its midpoint lies in a slab of thickness $|\mathbf{R}\cdot \mathbf{n}|$. This gives the displayed unsigned count per unit area. Omitting the absolute value gives a signed count, or requires a restriction to one orientation. The signed spring force transmitted from the positive side to the negative side is $H\mathbf{R}\mathrm{sgn}(\mathbf{R}\cdot \mathbf{n})$, so multiplication by the unsigned count yields traction $nH⟨\mathbf{R}{\mathbf{R}}^T⟩\mathbf{n}$. Each labelled dumbbell is counted once.