Proposition: The contact template is not the frozen parametrization

Equations (eq:Dtemplate)–(eq:contacttemplate) define a parameterized local contact realization. They do not identify the actual frozen $D_{\Gamma }$, $w_{\Gamma }$, or $S_{\Gamma /I}$. They prove neither exhaustiveness of the frozen class nor an identity between that class and a presentation-level correction space $\mathcal{A}$.

Proof. The source roles specify that there is a graph operator, an admissible external weight, and an inherited spectator. They do not supply the parameters in (eq:Dtemplate), (eq:wtemplate), and (eq:Stemplate). A family indexed by unspecified parameters does not identify one prescribed member.

Exhaustiveness would require a theorem that every permitted frozen change is represented by the displayed parameter family, together with a statement of its admissible parameters. Neither is supplied. Finally, the values in (eq:contacttemplate) are distributions. A map in $\mathcal{A}$ would require a value for each presentation, and possibly additional linearity, continuity, covariance, and tensor restrictions. No such assignment follows from a class of values.