The ultimate extension of Penrose’s Spin Geometry Theorem is given. It is shown how the local geometry of any curved Lorentzian 4-manifold (with \(C^2\) metric) can be derived in the classical limit using only the observables in the algebraic formulation of abstract Poincaré-invariant elementary quantum mechanical systems. In particular, for any point q of the classical spacetime manifold and curvature tensor there, there exists a composite system built from finitely many Poincaré-invariant elementary quantum mechanical systems and a sequence of its states, defining the classical limit, such that, in this limit, the value of the distance observables in these states tends with asymptotically vanishing uncertainty to lengths of spacelike geodesic segments in a convex normal neighbourhood U of q that determine the components of the curvature tensor at q. Since the curvature at q determines the metric on U up to third order corrections, the metric structure of curved \(C^2\) Lorentzian 4-manifolds is recovered from (or, alternatively, can be defined by the observables of) abstract Poincaré-invariant quantum mechanical systems.
