The next necessary construction

For the proposed quantum response, the next step is not to search for a separator among guessed columns. It is to choose and justify a switched gauge-compatible realization of the observable.

A possible completion must specify one of the following, or another equally explicit alternative:

1. a restricted gauge-support class with a proved intertwining identity;

2. an enlarged background space in which the switch transforms, together with the resulting Ward identity;

3. a switch-dependent relational prescription whose complete coefficient action is proved;

4. a local-coupling construction with a precisely controlled switching-region breaking term and a demonstrated way of extracting the desired physical response.

These possibilities are not asserted to be equivalent.

After that choice, the required two-vertex and composite products must be constructed, including their coincident-point and boundary extensions. Only then can a complete finite-renormalization response space and its matrix be evaluated.

The adiabatic limit, a localized probe, higher radiative orders, and dense-TT representability remain separate successor problems.