Theorem: Scope of the coarse covariance conclusion

The conditional action in Theorem thm:action is an action on supports, labelled acquisition events, laws, and the stated relations. From it alone one cannot infer:

1. a limit action on quantum output instruments or residual states;

2. a limit action by automorphisms of a dynamical observable net;

3. isometries between the microscopic word metrics;

4. closure of a physical device class under unrestricted inter-frame composition;

5. an identification of $c$ with a measured physical constant.

Proof. The defined objects contain none of the extra data in (i)–(iv). For (i), the channels $\rho \mapsto |0⟩⟨0{|}_K\otimes \rho$ and $\rho \mapsto |0⟩⟨0{|}_K\otimes \tau \mathrm{Tr}\rho$ have the same record law but different quantum outputs and complements. Thus their equality after forgetting the output data cannot recover those data.

At each finite scale the word-metric stable body is a finite polytope, not the Euclidean ball; a generic rotation does not preserve it. A Lorentz transformation preserves the declared limiting Minkowski quadratic form, but no map on the microscopic metrics has been constructed.

Finally, construction (eq:family) is possible for every chosen $c>0$ by selecting $\Delta t_n={\epsilon }_n/c$. The formalism does not select a physical time calibration or a unique dimensional speed.