A Local Resolution of the Problem of Time. IX. Spacetime Constructability

Assuming Temporal and Configurational Relationalism, GR as Geometrodynamic's DeWitt supermetric alongside local Lorentzian Relativity with its universal finite maximum propagation speed arises as one of very few options from Feeding Families through a Dirac-type Algorithm for Consistency. This amounts to Spacetime Construction from prior assumptions about space and dynamics alone. The other alternatives, arising as cofactors' likewise strongly vanishing roots are, firstly, Galileo-Riemann Geometrostatics with Galilean Relativity's infinte propagation speed. Secondly, Strong Gravity with Carrollian Relativity 's zero propagation speed. If none of these vanish, constant mean curvature of the spatial slice is enforced, paralleling previous work on decouping GR's constraints and addressing its initial-value problem. Assuming just Temporal Relationalism, spatial 3-diffeomorphism Configurational Relationalism is enforced as an integrability as one of very few options. The alternatives here are, once again, Galileo-Riemann Geometrostatics and Strong Gravity, for which spatial 3-diffeomorphisms are thereby optional, and local volume preserving diffeomorphisms. Options are few in each case above due to Rigidity kicking in. We furthermore demonstrate that such Rigidities are more generally Lie rather than specific to Dirac-type Algorithms. We do this by deriving the conformal versus projective ambiguity in top-geometry by feeding the general quadratic generator into the Lie Algorithm for Consistency. Both Feeding Families through a Brackets Consistency Algorithm and Rigidity are thus additionally of relevance to the Foundations of Geometry.