Question and exact scope

XIX-A separated time-related constructions and left the VI–XVI-D coordinate identification undecided XIXA. Its displayed density equation was a comparison prescription. It was not, merely by its label, the registered VI coordinate construction.

The present paper makes one specified comparison and reaches the mismatch alternative. The comparison is:

1. retain XVI-D's response tensor $R_{\lambda }=-{\mathcal{P}}_{+}{V}_{\lambda }$;

2. regard that smooth response as a metric argument of the first harmonic-coordinate map;

3. transport the metric signature explicitly;

4. compare the resulting first harmonic coefficient with XVI-D's $X_{\mathrm{cand}}$, on the same slab and with the same ordinary coordinate-time zero data.

All mathematical definitions used in this comparison are printed below. Neither a separation edge nor an unavailable companion proof is a premise.

decisive distinction. The supplied VI prescription is unswitched:

The supplied XVI-D candidate uses a prepared density forcing:

Equality must be tested after linearization and sign normalization, not inferred from the words “harmonic”, “retarded”, or “Cauchy”.

result is coefficientwise. The comparison parameter below probes the coordinate prescription. It does not regrade the native gravitational coupling order of the response $R_{\lambda }$, which is bilinear in its two tensor inputs. If a different comparison places that response in VI's second metric coefficient, the additional second-order terms must be included. That different comparison is not decided here.

context. Relational-coordinate constructions and perturbative invariant observables are established subjects BFHPR,Frob. Their existence does not identify two particular boundary or preparation prescriptions. Likewise, Cheung's operational characterization of measurement concerns measurement strategies, not the Cauchy forcing studied here CheungEnter.

false identification of the obstruction. VI's compact-switch defect concerns a switched scalar assignment. It is not an obstruction to existence or uniqueness of VI's coordinate jet. Section sec:switch proves the corresponding classical identity separately; no interaction switch $g$ enters the coordinate mismatch theorem.