 Spin and branch counting in saddle-point optical spectra

A spin factor and disconnected contour branches count different independent contributions. For the transition energy $E_s+{\hslash }^2(k_x^2-k_y^2)/(2m^{\ast })$, one branch contributes $g_s{m}^{\ast }I/(4{\pi }^2{\hslash }^2)$, with $I=\int d{k}_y/\sqrt{k_y^2+\delta /A}$. The full two-branch saddle doubles this. A formula for one branch with spin degeneracy two can therefore equal a full-contour formula with spin degeneracy one, but those are distinct counting conventions. An extra valley or inequivalent saddle requires its own multiplicity.