  We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds
on the spectral degeneracy of quantum systems with higher (Gauge Like)
symmetries with rather generic physical boundary conditions in an arbitrary
number of spatial dimensions. Contrary to applying twists or equivalent
adiabatic operations, we exploit the effects of modified boundary conditions.
When a general choice of boundary geometry is immaterial in approaching the
thermodynamic limit, systems that exhibit non-commuting Gauge Like symmetries,
such as the orbital compass model, must have an exponential (in the size of the
boundary) degeneracy of each of their spectral levels. We briefly discuss why,
in spite of the proven large degeneracy associated with infrared-ultraviolet
mixing, some systems may still exhibit conventional physical behaviors, i.e.,
of those of systems with non-extensive degeneracies, due to entropic "order by
disorder" type effects.
