= Solution
At $\epsilon=0$ the metric is the constant <Minkowski metric>, whose Christoffel symbols and curvature vanish, so
$$
R^{(0)}_{\mu\nu\rho\sigma}=0.
$$
Insert $R=\epsilon R^{(1)}+O(\epsilon^2)$ into the <Penrose wave equation>. Every curvature-square term is $O(\epsilon^2)$, while the covariant wave operator reduces at first order to the flat <d'Alembert operator>. Thus
$$
\boxed{\partial^\alpha\partial_\alpha
R^{(1)}_{\mu\nu\rho\sigma}=0}.
$$
The <linearized Riemann curvature operator> is unchanged by $h_{\mu\nu}\mapsto h_{\mu\nu}+\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$, because the resulting third derivatives cancel pairwise. The equation therefore requires no gauge choice for $h_{\mu\nu}$.
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