Proposition: The finite incidence data do not determine a matter lapse

The parameter, finite code, normal deformation component, and matter-defined relational lapse are not one functional. In particular, retaining the same carrier, graph family, matching data, register, and update does not determine $\mathcal{N}$.

Proof. Keep all incidence data fixed and add, on a regular flat patch, the scalar clocks

Equation (eq:lapse) gives $N[q_{\mu },T_c]=1/c$. With the matching homogeneous reference clock and identity spatial rods on that patch, the relational map is the identity, so $\mathcal{N}=1/c$. The finite incidence data and $L_s$ are unchanged as $c$ varies. Lemma lem:fibers excludes determination of this additional matter lapse by those data.

The parameter is an argument of $F$; the code is an element of a finite set; $L_s$ is a derivative-dependent field; $\mathcal{N}$ depends on a metric, clock gradient, calibration, and rods. Their declared domains and dependencies are distinct.