  A geometric formalism is developed which allows to describe the non-linear
regime of higher-spin gravity emerging on a cosmological quantum space-time in
the IKKT matrix model. The vacuum solutions are Ricci-flat up to an effective
vacuum energy-momentum tensor quadratic in the torsion, which arises from a
Weitzenb\"ock-type higher spin connection. Torsion is expected to be
significant only at cosmic scales and around very massive objects, and could
behave like dark matter. A non-linear equation for the torsion tensor is found,
which encodes the Yang-Mills equations of the matrix model. The metric and
torsion transform covariantly under a higher-spin generalization of
volume-preserving diffeomorphisms, which arises from the gauge invariance of
the matrix model.
