Equation ideals are not metric admission

The compact representatives in (eq:limit-ideal) can be used as equation-ideal identities by a successor that declares the same scalar quotient. They do not prove admission after making a metric slot quantum. X-F's separate use of equation-ideal representatives or fixed-width tree constructions has its own domains and field content XF. It does not inherit this sharp scalar limit as a metric-admission theorem.

No inherited characteristic quantum coefficient is admitted here. This qualification does not deny the smooth scalar classes and their convergence that have actually been constructed.