  The aim of this paper is to answer the following two questions: (1) Given
cosmological observations of the expansion history and linear perturbations in
a range of redshifts and scales as precise as is required, which of the
properties of dark energy could actually be reconstructed without imposing any
parameterization? (2) Are these observables sufficient to rule out not just a
particular dark energy model, but the entire general class of viable models
comprising a single scalar field?
  This paper bears both good and bad news. On one hand, we find that the goal
of reconstructing dark energy models is fundamentally limited by the
unobservability of the present values of the matter density Omega_m0, the
perturbation normalization sigma_8 as well as the present matter power
spectrum. On the other, we find that, under certain conditions, cosmological
observations can nonetheless rule out the entire class of the most general
single scalar-field models, i.e. those based on the Horndeski Lagrangian.
