  Dynamical cancellation frameworks present a potential means of mitigating the
effect of a large vacuum energy, that would otherwise ruin the late-time, low
energy dynamics of the Universe. Certain models in the literature, such as the
Fab Four and Well Tempering, realize this idea by introducing some degeneracy
in the dynamical equations. In this paper, we introduce a third potential route
to self-tuning, and infer the existence of a new, exact Milne solution in the
simplest tadpole plus cubic-Galileon scalar-tensor theory. We study the
dynamics of the scalar field and metric in the vicinity of the Milne coordinate
singularity, and find that the vacuum solution belongs to a more general family
of Milne-like metrics. By numerically evolving the field equations for a range
of initial conditions, we show that the Milne solution is not an attractor, and
varying the initial scalar field data can lead to completely different
asymptotic states; exponential growth of the scale factor, a static
non-spatially flat metric or a severe finite-time instability in the scalar
field and metric. We generalise the Milne solution to a class of FLRW
spacetimes, finding that the tadpole-cubic Galileon model admits
perfect-fluid-like solutions in the presence of matter. Finally, we present a
second Horndeski model which also admits an exact Milne solution, hinting at
the existence of a larger undiscovered model space containing
vacuum-energy-screened solutions.
