Theorem: Classical auxiliary-coordinate equivalence

For every smooth metric jet $(h,k)$, the constraints (eq:auxiliary-constraints) have exactly one solution. Projection from their solution graph to $(h,k)$ is bijective, and substituting that solution into

gives the original sharp ${\mathcal{O}}_f$.

The same statement holds for the prepared density equations, with their corresponding coefficients and zero data.

Proof. The first equation is a linear Cauchy problem with prescribed zero data. Its unique solution is $w={\mathsf{E}}_{0}Ch$. With that solution inserted, the second is another linear inhomogeneous Cauchy problem, whose unique solution is (eq:coordinate-coefficients). The observable equality is substitution into its formal inverse-map expansion. The prepared equations are handled in the same order.