  To incorporate quantum nonlocality into general relativity, we propose that
the preparation and measurement of a quantum system are simultaneous events. To
make progress in realizing this proposal, we introduce a spacetime geometry
that is endowed with particles which have no distinct points in their
worldlines; we call these particles 'pointons'. This new geometry recently
arose in nonnoetherian algebraic geometry. We show that on such a spacetime,
metrics are degenerate and tangent spaces have variable dimension. This
variability then implies that pointons are spin-$\tfrac 12$ fermions that
satisfy the Born rule, where a projective measurement of spin corresponds to an
actual projection of tangent spaces of different dimensions. Furthermore, the
$4$-velocities of pointons are necessarily replaced by their Hodge duals, and
this transfer from vector to pseudo-tensor introduces a free choice of
orientation that we identify with electric charge. Finally, a simple composite
model of electrons and photons results from the metric degeneracy, and from
this we obtain a new ontological model of photon polarization.
