Guard stability is a separate obligation

An exact clock guard $j=0$ is stable under an exact finite representation, but not under arbitrary perturbation of a real parameter. Similarly, if a length is $c|s|$, a floor report at resolution $h$ is zero whenever $0<c|s|<h$. It cannot serve as an exact event detector.

Theorem thm:stability applies to already specified channels on the same finite dictionaries. It does not convert a discontinuous geometric guard into a uniformly continuous map of real geometric data. Such a claim requires a separate margin or event-resolution theorem, as emphasized in VII-E and VII-F CheungVIIE, CheungVIIF.