= Solution
With vanishing <extrinsic curvature>, the <momentum constraint> is identically satisfied and the vacuum <Hamiltonian constraint> reduces to $\mathcal R=0$. To evaluate that <Ricci scalar>, write $\psi_i=\partial_i\psi$, raise these intermediate indices with $\delta^{ij}$, and use $\gamma^{ij}=\psi^{-4}\delta^{ij}$. The <Levi-Civita connection> is
$$
\Gamma^k{}_{ij}=\frac2\psi\left(\delta^k{}_i\psi_j+\delta^k{}_j\psi_i-\delta_{ij}\psi^k\right).
$$
Substitution into the stated curvature convention gives
$$
\mathcal R_{ij}
=-\frac2\psi\partial_i\partial_j\psi
-\frac2\psi\delta_{ij}\Delta\psi
+\frac6{\psi^2}\psi_i\psi_j
-\frac2{\psi^2}\delta_{ij}|\nabla\psi|^2.
$$
Tracing with $\gamma^{ij}$ cancels the gradient-square terms:
$$
\boxed{\mathcal R=-8\psi^{-5}\Delta\psi.}
$$
Here the <Laplacian> and norm on the right are those of the flat Euclidean metric. Thus the <time-symmetric conformally flat vacuum initial data> constraints become
$$
\boxed{\Delta\psi=0.}
$$
The equivalence uses $\psi\ne0$. This identity is also the three-dimensional specialization of <scalar curvature under conformal rescaling>; locally the conformal exponent is $2\log|\psi|$, so either fixed nonzero sign of $\psi$ gives the same metric.
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