Proposition: Same local channels, different conical completions

The finite experiment does not determine the global conical deficit.

Proof. Take two deficits ${\mu }_1\ne {\mu }_2$. Both regular products contain sufficiently small developing patches isometric to the same flat laboratory. Install the same supported classical data, matrices, dictionary, and causal routes in those patches. All local equations, evaluation tables, channels, and record laws agree.

The completed backgrounds nevertheless have different circumference-to-radius ratios $2\pi -{\mu }_1$ and $2\pi -{\mu }_2$. Hence these operational data do not determine the deficit.