Persistence of generic non-degeneracy under parameter extension

Consider a family of periodic operators whose parameters include the periodic potential, and possibly additional parameters such as a metric. Generic non-degeneracy means that degenerate critical points do not occur generically in the relevant parameter space. Persistence conjecture. Generic non-degeneracy should survive when the set of parameters is enlarged; in particular, varying the metric in addition to the potential should not make the situation worse. The paper presents this as an expected principle beyond the random-test corollary, but does not establish it in general.