  We present a new, falsifiable, quantum theory of gravity, which we name
Non-commutative Matter-Gravity. The commutative limit of the theory is
classical general relativity. In the first two papers of this series, we have
introduced the concept of an atom of space-time-matter [STM], which is
described by the spectral action in non-commutative geometry, corresponding to
a classical theory of gravity. We used the Connes time parameter, along with
the spectral action, to incorporate gravity into trace dynamics. We then
derived the spectral equation of motion for the gravity part of the STM atom,
which turns out to be the Dirac equation on a non-commutative space. In the
present work, we propose how to include the matter (fermionic) part and give a
simple action principle for the STM atom. This leads to the equations for a
quantum theory of gravity, and also to an explanation for the origin of
spontaneous localisation from quantum gravity. We use spontaneous localisation
to arrive at the action for classical general relativity [including matter
sources] from the action for STM atoms.
