  Large-N volume independence in circle-compactified QCD with N_f \geq 1
adjoint Weyl fermions implies the absence of any phase transitions as the
radius is dialed to arbitrarily small values. This class of theories are
believed to possess a Hagedorn density of hadronic states. It turns out that
these properties are in apparent tension with each other, because a Hagedorn
density of states typically implies a phase transition at some finite radius.
This tension is resolved if there are degeneracies between the spectra of
bosonic and fermionic states, as happens in the N_f=1 supersymmetric case.
Resolution of the tension for N_f>1 then suggests the emergence of a fermionic
symmetry at large N, where there is no supersymmetry. We can escape the
Coleman-Mandula theorem since the N=\infty theory is free, with a trivial
S-matrix. We show an example of such a spectral degeneracy in a
non-supersymmetric toy example which has a Hagedorn spectrum.
