= Spin and branch counting in saddle-point optical spectra
A spin factor and disconnected contour branches count different independent contributions. For the transition energy $E_s+\hbar^2(k_x^2-k_y^2)/(2m^*)$, one branch contributes $g_s m^*I/(4\pi^2\hbar^2)$, with $I=\int dk_y/\sqrt{k_y^2+\delta/A}$. 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.
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