  Previous work of the authors establishes a criterion on the fundamental group
of a knot complement that determines when Dehn surgery on the knot will have a
fundamental group that is not left-orderable. We provide a refinement of this
criterion by introducing the notion of a decayed knot; it is shown that Dehn
surgery on decayed knots produces surgery manifolds that have
non-left-orderable fundamental group for all sufficiently positive surgeries.
As an application, we prove that sufficiently positive cables of decayed knots
are always decayed knots. These results mirror properties of L-space surgeries
in the context of Heegaard Floer homology.
