Lemma: The ramp is not the declared dynamical clock

With ${\sigma }_{\epsilon }$ held fixed:

1. its variation with respect to $q$ and $T$ is zero;

2. $\mathrm{d}{\sigma }_{\epsilon }=0$ on both plateau regions, so it fails the nondegeneracy criterion there;

3. the same ramp is compatible with different scalar-clock lapses;

4. it is not the harmonic relational scalar or the incidence register update.

Proof. The first claim follows from (eq:ramp), which contains no dynamical field. The second follows from the constant values of $\chi$.

For the third, keep $q=\eta$ and the same ramp, and use ${T^{(a)}}_e$ from Lemma lem:scalar-lapse. The ramp is unchanged while the lapse at the detecting point is $1/(c+ea)$. Lemma lem:factor excludes determination of that lapse by the ramp alone.

For the fourth, changing $T$ changes ${\mathcal{R}}_{\mathrm{H}}$ even at $q=\eta$, whereas it does not change the ramp. The register is a map on a finite code set with a declared update; a fixed smooth function on a time slab is not that map or its update.