= Planar exactness of the Zeldovich approximation
For $\mathbf r=a[\mathbf q+b(t)(p(q_1),0,0)]$, put $J=1+bp'$. Before <shell crossing>, mass conservation gives $\rho=\bar\rho/J$. The dust background equation $3\ddot a/a=-4\pi G\bar\rho$ and the growing-mode equation $\ddot b+2H\dot b=4\pi G\bar\rho b$ imply $\nabla_r\cdot\ddot{\mathbf r}=-4\pi G\bar\rho+4\pi G\bar\rho bp'/J=-4\pi G\rho$. Thus planar <Zeldovich approximation> trajectories satisfy the full nonlinear dust force-divergence equation until the map loses invertibility. All three physical coordinates are included in the divergence; the two unperturbed directions still contribute background acceleration.
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