Varied-action derivation: Route II

Route II gives a variational derivation of the dynamics' covariance and, applied to an auxiliary coordinate action, of the coordinate operator covariance. It does not infer observable invariance from action invariance alone: the fixed Cauchy data, uniqueness, and relational composition remain necessary.

Temporarily let $\xi ,\zeta$ be arbitrary compact interior vector fields, not necessarily plateau-supported. For a scalar density $r$, define

Then

This is the Taylor expansion of density pullback. All currents have compact support in the gauge support.