Coordinate prescription and observable

For the formal metric $g_{\kappa }=\eta +\kappa h$,

Thus $x^{\mu }+\kappa w[h{]}^{\mu }$ is the harmonic coordinate map through first order, and its inverse has coefficient $-w[h]$. Equation (eq:observable) is precisely the corresponding first scalar coefficient after composition with that inverse map.

The theorem concerns this coefficient itself. It does not infer a controlled full finite-$\kappa$ observable from a formal first-order expansion.