  We analyze the semiclassical polymer dynamics of the inhomogeneous Mixmaster
model by choosing the cubed scale factor as the discretized configurational
variable, while the anisotropies remain pure Einsteinian variables. Such a
modified theory of gravity should be regarded as the appropriate framework to
describe the behavior of quantum mean values, both in polymer quantum mechanics
and in Loop Quantum Cosmology. We first construct the generalized Kasner
solution, including a massless scalar field and a cosmological constant in the
dynamics. They account for a quasi-isotropization, inflationary-like mechanism.
The resulting scenario links a singularity-free Kasner-like regime with a
homogeneous and isotropic de Sitter phase. Subsequently, we investigate the
role of the three-dimensional scalar curvature, demonstrating that a bounce of
the point-universe against the potential walls can always occur within the
polymer framework, also in the presence of a scalar field. However, the absence
of the singularity implies that the curvature is bounded. Therefore, the
point-universe undergoes an oscillatory regime until it oversteps the potential
walls (if the Big Bounce is not reached before). After that, a final stable
Kasner-like regime will last all the way toward the Big Bounce. Thus, the
present study demonstrates that, as soon as a discretization of the volume of
the universe is performed, the generic cosmological solution is non-chaotic and
free from singularities. It is likely that the same result can be achieved also
in the loop quantum cosmology approach.
