Proposition: Records do not uniquely determine the geometric dictionary

Finite record laws and complete matter channels in this construction do not determine the rod frame, the lapse, or the global deficit.

Proof. First, apply a spatial Euclidean isometry to the chosen developing chart and all prescribed carrier locations. The abstract name permutations and channels are unchanged, while their numerical rod coordinates change.

Second, retain the same event labels, sites, and scheduled channels, but compare the constant lapses $N=1$ and $N=2$. Every edge satisfying $\Vert \Delta x\Vert \le c\Delta T$ is causal for both. The same matrices produce exactly the same complete channels and record probabilities. Nevertheless, a static interval of coordinate duration $\Delta T$ has proper duration $\Delta T$ in the first calibration and $2\Delta T$ in the second.

Third, any two deficits in $(0,2\pi )$ admit regular developing patches containing isometric sufficiently small Euclidean balls. Install the same finite experiment in such balls. The complete channels coincide, although the global circumference-to-radius ratios and conical gluing angles differ.