Lemma: The incidence parameter is not the matter lapse

The declared parameter $s$, the finite reading $j$, the branch-normal speed $L_s$, and the matter-defined relational lapse ${\mathcal{N}}_{\mathrm{E}}$ are not one functional. In particular:

1. a parameter value is not its deformation speed;

2. graph ordering and cone matching alone do not determine a scalar-clock lapse;

3. the reported source parameter $s_c=-1/2$ is not any of the branch speeds $3,5/2,1$;

4. the finite update supplies no missing scalar or rod field.

Proof. The first statement follows from the distinct formulas: $s$ is the independent family argument, whereas $L_s$ depends on ${\partial }_{s}F$ and $\nabla F$. Proposition prop:retiming also shows their different dependence on a parameter change.

For the second, augment the same regular cone patch and graph family by either scalar $T_c=cU$ with $c>0$. Every such scalar is a timelike clock, and its raw lapse is $1/c$. With its corresponding homogeneous reference and Cartesian rods on the regular flat patch, its relational lapse is also $1/c$. The prescribed graph ordering and matching data are unchanged as $c$ varies. They therefore cannot determine this additional matter-defined lapse. This is a counterexample to an inference from geometry alone, not an identification with the cosmological background of Em02-E.

The third assertion is a distinction between the named parameter value and the displayed branch values. For the fourth, $U_{\mathcal{J}}$ is defined on finite codes and has no scalar field or slice-elliptic rod equation in its definition.