D5: the copied witness on a parameter commutator

We now specialize the detecting family, after the class and its representation have been frozen.

Choose the admissible XII-F data with a temporal bump that is a square. More explicitly, let

and choose a compact spatial cutoff $\zeta \in {C_c}^{\infty }({\mathbb{R}}^3;[0,1])$ equal to one on the dependence region used by XII-F. Take its temporal bump and spatial realization to be

These are among XII-F's permitted choices.

Choose $W$ to contain the compact parameter support and choose $g$ to satisfy (eq:UCP), as in XIII-A's embedding of the detecting tuple into fixed restriction data. The switch, observation, region, and cutoffs are fixed for the entire test. No parameter subclass is removed afterward.

Define two allowed parameters

where $t_{\ast }$ is any fixed real number. Both lie in ${\mathfrak{g}}_W$.