devoted to building viable models and to constraining them from observations (see, for instance, the reviews [@Amendola2010; @Clifton:2011jh; @Kunz:2012aw]). The models are usually characterized by a small number of properties at background and linear-perturbation level (equations of state, speeds of sound, masses, coupling strengths, etc.) which then have some particular effect on the phenomenology (evolution of the scale factor and perturbations of matter and DE itself). In this paper, we employ a different approach, aiming to answer the following questions:

1.  Assuming the ideal case of cosmological observations of the expansion history and linear perturbations in a range of redshifts and scales which are as precise as is required, which physical properties (such as e.g. the Hubble rate $H(z)$, the perturbation normalization $\sigma_{8}$, the perturbation growth rate $f$, etc.) could actually be reconstructed if we were to refrain from any parameterization of dark energy?

2.  Can we use these observable quantities to rule out not just some particular cosmological model but the entire class of viable single scalar-field models?

We are of course not the first to attempt to study dark energy cosmologies in a model-independent way. In Ref. [@Stebbins:2012vw] the author argues for a reconstruction method which does not 