= Solution
A closed spatial <Cauchy hypersurface> is compact and has no boundary. Thus there is no asymptotic surface charge, and the complete gravitational <Hamiltonian> is the integral of the <Hamiltonian constraint> and <momentum constraint> derived above. Both constraints vanish on solutions, giving
$$
\boxed{H_{\rm total}=0.}
$$
This is the <vanishing canonical Hamiltonian of a closed universe>: canonical evolution is generated by constraints and there is no asymptotic time translation whose boundary charge could define an <Arnowitt-Deser-Misner energy>. It does not require local matter energy densities or local curvature to vanish. With matter included, the corresponding total constraints still vanish.
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