  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.
