  M5-branes on an ADE singularity are described by certain six-dimensional
"conformal matter" superconformal field theories. Their Higgs moduli spaces
contain information about various dynamical processes for the M5s; however,
they are not directly accessible due to the lack of a Lagrangian formulation.
Using anomaly matching, we compute their dimensions. The result implies that M5
fractions can recombine in several different ways, where the M5s are leaving
behind frozen versions of the singularity. The anomaly polynomial gives hints
about the nature of the freezing. We also check the Higgs dimension formula by
comparing it with various existing conjectures for the CFTs one obtains by
torus compactifications down to four and three dimensions. Aided by our
results, we also extend those conjectures to compactifications of theories not
previously considered. These involve class S theories with twisted punctures in
four dimensions, and affine-Dynkin-shaped quivers in three dimensions.
