Lemma: Elementary non-identifications

In the present model class:

1. even a harmonic scalar profile does not determine the lapse;

2. a finite pointer is not the all-real coordinate $T$;

3. an externally specified density ramp does not, by its specification alone, supply an independent dynamical clock;

4. a slicing-family parameter is not identified with $T$ without an additional calibration.

Proof. For (i), take any two positive constant lapses $N=a$ and $N=b$. The scalar $\xi =T$ is harmonic for both metrics: the coefficients and volume density are independent of $T$, so ${\square }_{g_N}T=0$. The scalar profile is identical, but

depends on the lapse.

For (ii), the pointer has at most $D$ distinct eigenvalues, whereas $T$ takes every real value on $M$.

For (iii), a fixed regulator $r_{\epsilon }(T)=r(T/\epsilon )$ adds a prescribed function of external labels, not a new state variable or evolution equation. It can be used in a calibration if further data are supplied; that does not make an independent dynamical degree of freedom appear.

For (iv), a family of slices may be reindexed by any bijection of its parameter interval without changing the collection of slices. Neither the background metric nor the finite clock instrument selects one such indexing.