  Positivity bounds provide conditions that a consistent UV-completion exists
for a quantum field theory. We examine their application to Horndeski gravity
models reconstructed from the effective field theory (EFT) of dark energy. This
enables us to assess whether particular phenomenological parameterizations of
the EFT functions reconstruct theories that respect or violate the positivity
bounds. We find that commonly adopted EFT parametrizations, cast in terms of
the dark energy density or power laws of the scale factor, only satisfy the
positivity bounds in non-trivial regions of the parameter space. We then
examine parameterizations of the inherently stable EFT basis, constructed to
avoid gradient and ghost instabilities by default. In stark contrast, in this
basis the positivity bounds either only provide constraints in a-priori
unrealistic regions of the parameter space or do not provide any constraints on
parameter values at all. The application of positivity bounds to common
parametrizations of the standard EFT functions can therefore lead to artificial
conclusions that the region of viable Horndeski modifications of gravity is
highly constrained. Our results provide a strong motivation, in addition to the
default avoidance of theoretical instabilities, for instead adopting
parametrizations of the inherently stable EFT basis when testing dark energy
and modified gravity models with forthcoming cosmological survey data.
