  We derive the equations of motion that arise from the one-loop effective
action for the geometry of 3+1 dimensional quantum branes in the IKKT matrix
model. These equations are cast into the form of generalized Einstein
equations, with extra contributions from dilaton and axionic fields, as well as
a novel anharmonicity tensor C_{\mu\nu} capturing the classical Yang-Mills-type
action. The resulting gravity theory approximately reduces to general
relativity in some regime, but differs significantly at cosmic scales, leading
to an asymptotically flat FLWR cosmological evolution governed by the classical
action.
