  Recent measurements in single-walled carbon nanotubes show that, on
resonance, all nanotubes display the same peak optical conductivity of
approximately 8 $e^2/h$, independent of radius or chirality [Joh \emph{et al.},
\emph{Nature Nanotechnology} \textbf{6}, 51 (2011)]. We show that this uniform
peak conductivity is a consequence of the relativistic band structure and
strength of the Coulomb interaction in carbon nanotubes. We further construct a
minimalist model of exciton dynamics that describes the general phenomenology
and provides an accurate prediction of the numerical value of the peak optical
conductivity. The work illustrates the need for careful treatment of relaxation
mechanisms in modeling the optoelectronic properties of carbon nanotubes.
