  Vanadium dioxide is one of the most studied strongly correlated materials.
Nonetheless, the intertwining between electronic correlation and lattice
effects has precluded a comprehensive description of the rutile metal to
monoclinic insulator transition, in turn triggering a longstanding "the chicken
or the egg" debate about which comes first, the Mott localisation or the
Peierls distortion. Here, we suggest that this problem is in fact ill-posed:
the electronic correlations and the lattice vibrations conspire to stabilise
the monoclinic insulator, and so they must be both considered not to miss
relevant pieces of the VO$_2$ physics. Specifically, we design a minimal model
for VO$_2$ that includes all the important physical ingredients: the electronic
correlations, the multi-orbital character, and the two components
antiferrodistortive mode that condenses in the monoclinic insulator. We solve
this model by dynamical mean-field theory within the adiabatic Born-Oppenheimer
approximation. Consistently with the first-order character of the
metal-insulator transition, the Born-Oppenheimer potential has a rich
landscape, with minima corresponding to the undistorted phase and to the four
equivalent distorted ones, and which translates into an equally rich
thermodynamics that we uncover by the Monte Carlo method. Remarkably, we find
that a distorted metal phase intrudes between the low-temperature distorted
insulator and high-temperature undistorted metal, which sheds new light on the
debated experimental evidence of a monoclinic metallic phase.
