Goldstone's theorem states that a system in which a continuous symmetry is spontaneously broken necessarily has gapless excitations. (A hand-waving "proof" of Goldstone's theorem can be given by noting that very large wavelength spatial variations in the order parameter should have very small energy cost).
On the other hand, there appears to be another way to obtain protected gapless excitations, namely by having a (possibly emergent) local gauge invariance where the gauge group is continuous, so that there is a gapless gauge boson. What is the most general statement that one can make in this case, analogous to Goldstone's theorem? Clearly, the gauge charges must remain deconfined, since we know that confined/Higgs phases of gauge theories are gapped. Is the existence of deconfined gauge charges (for a continuous gauge group) a sufficient condition to ensure gaplessness, and if so, how would one prove this?