Purpose, background, and scope

Relational coordinates can remove a simultaneous coordinate change of fields only when the coordinate prescription itself transforms correctly. Initial data, switching, and regularization are therefore part of the statement, rather than ancillary notation. The field-dependent-coordinate literature provides useful context, but does not prove covariance of the particular switched and regulated problem considered here BFR,FrobLima.

The starting classical prescription is Cheung's X-A: a compact switch multiplies the complete curved-minus-flat scalar density before variation, whereas an independent smooth time ramp regulates the coordinate density XA. We retain this distinction. In particular, neither substituting $gh$ into a metric nor convolving metric arguments defines the prescription below.

The operational background is the Enter calculus, in which destructive readout consumes the measured quantum names and returns a classical record with any unmeasured residual CheungEnter,I. We use ordinary finite Born instruments and prove the channel identities needed in this manuscript. The operational characterization of the Born strategy is not a premise of the classical proof.

The philosophy of Enter emphasizes distinctions between readout and recovery, between an input–output channel and a thermodynamic implementation, and between an analogy and an identified physical model EnterPhilosophy. Here those distinctions prevent three unwarranted conclusions:

1. a finite clock encoding does not define the continuum time coordinate used by the Cauchy problem;

2. a finite record of a classical evaluation does not constitute a measurement of that functional on a field;

3. a prescribed spacetime supporting a protocol is not spacetime derived from the protocol.