We ask whether it can be certified algorithmically that a self-modifying computational system preserves a safety property at its next step (preservation) and along its whole evolution (persistence). One step of self-modification is a total computable transformation
Φ of program indices, and preservation is the elevated property
ΛΦ(P)={x∈P:Φ(x)∈P}. When
Φ is extensional,
ΛΦ(P) is behavioural and Rice's theorem applies. When
Φ reads the code,
ΛΦ(P) is no longer behavioural, yet under uniform disruption (an inert wrapper encoding
K) the s-m-n reduction that proves Rice's theorem works inside a single behavioural fibre, and
ΛΦ(P) inherits the halting degree: one pullback of Rice, at two scales. One step never exceeds the degree of
P; persistence can be
Π20-complete for
Σ10 properties, even for extensional
Φ. We then isolate the mechanism shared by rewriting, supervision and system comparison: the semantic elevation operator, which wraps a base system and reacts to one finite event anchored to
K, entering or leaving the property. For this class the elevated property is
P∩Sa or
P∖Sa, determined by trigger and polarity alone; it inherits
K or its complement; and the safe region is not recursively enumerable. The Rice-Shapiro theorem restricts the polarity: a finite trigger can only enter a
Σ10 property and only leave a
Π10 one. Four axes (functional, deductive, conformance to a reference, monitoring) are verified instances, and towers of supervisors do not lower the barrier. We exhibit
K-hard intensional operators outside the class and state the open characterisation problem.