  We consider the binary fragmentation problem in which, at any breakup event,
one of the daughter segments either survives with probability $p$ or disappears
with probability $1\!-\!p$. It describes a stochastic dyadic Cantor set that
evolves in time, and eventually becomes a fractal. We investigate this
phenomenon, through analytical methods and Monte Carlo simulation, for a
generic class of models, where segment breakup points follow a symmetric beta
distribution with shape parameter $\alpha$, which also determines the
fragmentation rate. For a fractal dimension $d_f$, we find that the $d_f$-th
moment $M_{d_f}$ is a conserved quantity, independent of $p$ and $\alpha$. We
use the idea of data collapse -- a consequence of dynamical scaling symmetry --
to demonstrate that the system exhibits self-similarity. In an attempt to
connect the symmetry with the conserved quantity, we reinterpret the
fragmentation equation as the continuity equation of a Euclidean
quantum-mechanical system. Surprisingly, the Noether charge corresponding to
dynamical scaling is trivial, while $M_{d_f}$ relates to a purely mathematical
symmetry: quantum-mechanical phase rotation in Euclidean time.
