  We generalize our previous results (Theorem 1 and Corollary 2 in
arXiv:1412.4114) and Theorem 1 in arXiv:1502.00668) on the existence of an
$L^2$-energy gap for Yang-Mills connections over closed four-dimensional
manifolds and energies near the ground state (occupied by flat, anti-self-dual,
or self-dual connections) to the case of Yang-Mills connections with arbitrary
energies. We prove that for any principal bundle with compact Lie structure
group over a closed, four-dimensional, Riemannian manifold, the $L^2$ energies
of Yang-Mills connections on a principal bundle form a discrete sequence
without accumulation points. Our proof employs a version of our
{\L}ojasiewicz-Simon gradient inequality for the Yang-Mills $L^2$-energy
functional from our monograph arXiv:1409.1525 and extensions of our previous
results on the bubble-tree compactification for the moduli space of
anti-self-dual connections arXiv:1504.05741 to the moduli space of Yang-Mills
connections with a uniform $L^2$ bound on their energies.
