The explicit completion used here

This prescription is part of the model. It is not an assertion that $\tilde{M}$ is globally isometric to $M$, or that different completions give identical answers.

Every bounded spatial detector region needed below can be accommodated: a sufficiently distant portion of a developing angular sector contains a Euclidean ball of any prescribed finite radius. Translate the whole flat detector package into such a ball. Simultaneous translation of tests, modes, and coherent labels is unitary in the declared flat representations and does not change the numerical pairings.