Lemma: Lemma A: equation identity on a matched jet

On $M_T$, use the same metric jet, the same initial surface, and the same ordinary coordinate-time data.

Then the density-coordinate equation of XIX-A and the scalar-wave coordinate equation of VI are equivalent coefficientwise through order two. For the conventions used in XIX-A,

The equivalence applies on the shared domain, including XIX-A's spatially compact metric subclass. It does not identify the complete scalar observables, enlarge XIX-A's original domain, or turn its label ${\mathsf{C}}_{\mathrm{VI}}$ into an authentication of every VI construction.

Proof. The zeroth coefficient of ${\rho }_q$ is $1$, so ${\rho }_q$ has an invertible formal power series modulo ${\kappa }^3$. Multiplication by it preserves vanishing modulo ${\kappa }^3$. Equation (eq:density-identity) therefore proves equivalence of the two coordinate equations.

For completeness, in the convention $q={\eta }_{-}+\kappa h+{\kappa }^{2}j$, put

Matrix inversion and determinant expansion give

All mixed indices in (eq:a2) use ${\eta }_{-}$. The density equation for $X=x+\kappa w+{\kappa }^{2}v$ has coefficients

The initial coefficient data are unchanged.

XIX-A wrote the second metric and coordinate coefficients with a factor $1/2$. Substitution of $k=2j$ and $W_2=2v$ therefore yields the displayed conversion. The argument uses equations and data only; it does not compare scalar dynamics or observable domains.