  Following from a question of Wheeler, why does the Hamiltonian constraint
${\cal H}$ of GR have the particular form it does? A first answer, by Hojman,
Kucha\v{r} and Teitelboim, is that using embeddability into spacetime as a
principle gives the form of ${\cal H}$. The present paper culminates a second
Machian answer - initially by Barbour, Foster and \'{o} Murchadha - in which
space but not spacetime are assumed. Thus this answer is additionally a
classical-level resolution of the spacetime reconstruction problem. In this
approach, mere consistency imposed by the Dirac procedure whittles down a
general ansatz to one of four alternatives: Lorentzian, Galilean, or Carrollian
relativity, or constant mean curvature slicing. These arise together as the
different ways to kill off a 4-factor obstruction term. It is novel for such an
alternative to arise from principles of dynamics considerations (in contrast
with the historical form of the dichotomy between universal local Galilean or
Lorentzian relativity). It is furthermore intriguing that it gives constant
mean curvature slicing - familiar from York's work on the initial value problem
-- as a further option on a similar footing. That is related to a number of
recent alternative theories/formulations of GR known collectively as `shape
dynamics'. The original work did not treat this with Poisson brackets and a
proper systematic Dirac-type analysis; we rectify this in this paper. It is
also the first demonstration of how this approach solves the classical
spacetime reconstruction problem via `hypersurface tensor dual nationality' and
what can be interpreted as embedding equations arising.
