To incorporate quantum nonlocality into general relativity, I will introduce a new spacetime geometry where the points along the worldlines of dust particles are identified without contracting the worldlines to 0-dimensional points. This new geometry recently arose in the study of nonnoetherian coordinate rings in algebraic geometry. I will show that on such a spacetime, metrics are degenerate and tangent spaces have variable dimension. This variability then implies that dust particles are spin-1/2 fermions that satisfy the Born rule, where a projective measurement of spin corresponds to the actual projection of tangent spaces of different dimensions.