Local conical interpretation

Pull the coefficient fields and their fixed chart data back to the regular laboratory. The background chart is an isometry, and the defining differential equations and compact evaluations are identical in developed coordinates. This establishes the same local identities in that specified presentation.

It does not assert that the chartwise coefficients $w^{\mu }$ are global vector components or that the selected inverse patches consistently across the conical gluing. Such a global statement would require additional construction.