Lemma: Spatial rods are not harmonic Cauchy coordinates

On the cosmological background, original slice-harmonic spatial rods are not interchangeable with spatial harmonic coordinates having fixed zero perturbative Cauchy data.

Proof. The background spatial functions $x^i$ are both slice harmonic and spacetime harmonic. This is a background coincidence. Choose the perturbation $D=-b(t)\Delta \psi$ with $b(t_0)\ne 0$. The original rods have

The perturbative spacetime-harmonic coordinates with fixed zero Cauchy positions have ${W_1}^i(t_0,\mathbf{x})=0$. They therefore differ at the initial surface. Existence of the corresponding hyperbolic coefficient problem follows from the Cauchy theorem for the smooth background BGP; only the fixed initial position is needed for the contradiction.