  Degeneracy is a method to accommodate exact, low energy vacuum states in
scalar-tensor gravitational models despite the presence of an arbitrarily large
vacuum energy. However, this approach requires very particular combinations of
scalar field and metric couplings in the Lagrangian. In this work we study
departures from the restrictive degeneracy condition -- starting from a
fiducial model containing an exact Minkowski space solution, we break the
degeneracy condition in numerous simple ways to test if the resulting models
maintain certain key features -- specifically the dynamical cancellation of a
large vacuum energy by the scalar field and the existence of a low energy
vacuum state. We highlight the role the tadpole plays in eliminating the fixed
points of the dynamical system, generically rendering both the scalar field and
metric time dependent. Our results indicate that when violating the degeneracy
condition but preserving shift symmetry, the metric maintains an asymptotic
Minkowski state, irrespective of the presence of the cosmological constant. In
contrast, when shift symmetry is also broken the asymptotic behaviour can
radically alter. Regardless, the non-degenerate models in this work share an
attractive quality; harboring low energy, late-time asymptotic states that are
independent of the vacuum energy. The tadpole allows for a broader class of
non-degenerate, self-tuning models than was previously realized.
