Rice's Theorem under Self-Modification: Elevation Operators and a Normal Form

RESUMO

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 $\Phi$ of program indices, and preservation is the elevated property $\Lambda_\Phi(P)=\{x\in P:\Phi(x)\in P\}$. When $\Phi$ is extensional, $\Lambda_\Phi(P)$ is behavioural and Rice's theorem applies. When $\Phi$ reads the code, $\Lambda_\Phi(P)$ 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 $\Lambda_\Phi(P)$ inherits the halting degree: one pullback of Rice, at two scales. One step never exceeds the degree of $P$; persistence can be $\Pi^0_2$-complete for $\Sigma^0_1$ properties, even for extensional $\Phi$. 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 $P\cap S_a$ or $P\setminus S_a$, 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 $\Sigma^0_1$ property and only leave a $\Pi^0_1$ 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.