  Extending our previous analysis, we study the interior of a Schwarzschild
black hole derived from a partial gauge fixing of the full Loop Quantum Gravity
Hilbert space, this time including the inverse volume and coherent state
subleading corrections. Our derived effective Hamiltonian differs crucially
from the ones introduced in the minisuperspace models. This distinction is
reflected in the class of homogeneous bouncing geometries that replace the
classical singularity and are labeled by a set of quantum parameters associated
with the structure of coherent states used to derive the effective Hamiltonian.
By fixing these quantum parameters through geometrical considerations, the
post-bounce interior geometry reveals a high sensitivity to the value of the
Barbero-Immirzi parameter $\gamma$. Surprisingly, we find that $\gamma\approx
0.274$ results in an asymptotically de Sitter geometry in the interior region,
where now a cosmological constant is generated purely from quantum
gravitational effects. The striking fact is the exact coincidence of this value
for $\gamma$ with the one derived from the SU(2) black hole entropy
calculations in Loop Quantum Gravity. The emergence of this value in two
entirely unrelated theoretical frameworks and computational setups is strongly
suggestive of deep ties between the area gap in Loop Quantum Gravity, black
hole physics, and the observable Universe. In connection to the latter, we
point out an intriguing relation between the measured value of the cosmological
constant and the observed mass in the Universe from a proposal for a spin
quantum number renormalization effect associated to the microscopic dynamics.
