Abstract

We compare the finite-time harmonic coordinate prescription of Landmark VI with the prepared coordinate candidate recorded in XVI-D. We first prove the scalar-wave/density identity needed to relate VI's equation to XIX-A's displayed comparison equation, including the conversion between their second-order coefficient conventions. This authenticates the coordinate equation on their common domain, not the entire predecessor observable.

For the explicit counter-propagating witness source, let $V_{\lambda }$ be its zero-data time integral and retain the response sign $R_{\lambda }=-{\mathcal{P}}_{+}{V}_{\lambda }$. After a declared overall metric-sign transport, VI's first coordinate coefficient has normalized forcing $-{\partial }_t{V}_{\lambda }^{0\mu }$, whereas the prepared candidate has forcing $-{\partial }_t(\sigma {V_{\lambda }}^{0\mu })$. Their difference is the nonzero smooth residual ${\partial }_t((\sigma -1){V_{\lambda }}^{0\mu })$. An explicit compact-test construction detects that residual. The coordinate fields consequently differ on the common slab. The same response, placed as VI's first metric coefficient, also fails VI's vacuum first-metric equation; this domain failure is calculated separately.

The result is a first-order mismatch for the specified direct comparison. It is not VI's scalar-switch defect, a universal obstruction to another connecting construction, or a quantum anomaly. Smooth no-decay uniqueness, operator-domain extension, observable matching, and full-kernel determination retain their separate roles. No second-order or positive-formal-quantum-order term repairs the fixed first-order mismatch.