  We prove that, if an isospectral torus contains a discrete Schr\"odinger
operator with nonconstant potential, the shift dynamics on that torus cannot be
minimal. Consequently, we specify a generic sense in which finite unions of
nondegenerate closed intervals having capacity one are not the spectrum of any
reflectionless discrete Schr\"odinger operator. We also show that the only
reflectionless discrete Schr\"odinger operators having zero, one, or two
spectral gaps are periodic.
