Proposition: Clock basis changes

Changing the clock reading basis changes the pointer observable and generally the clock instrument. It does not change the coordinate function $T$, rods, or metric.

Proof. For a unitary $U$ on $C$, the new projectors are $U{P}_k{U}^{†}$. Unless this is an appropriate relabelling of the old projectors, the branch effects and conditional residuals change. The classical components $M,g,T,\chi$ do not occur as outputs of this operation and remain fixed.