  We study a coarsening process of one-dimensional cell complexes. We show that
if cell boundaries move with velocities proportional to the difference in size
of neighboring cells, then the average cell size grows at a prescribed
exponential rate and the Poisson distribution is precisely invariant for the
distribution of the whole process, rescaled in space by its average growth
rate. We present numerical evidence toward the following universality
conjecture: starting from any finite mean stationary renewal process, the
system when rescaled by $e^{-2t}$ converges to a Poisson point process. For a
limited case, this makes precise what has been observed previously in
experiments and simulations, and lays the foundation for a theory of universal
asymptotic states of dynamical cell complexes.
