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

2026-09-25T00:52:20.829Z

• title: The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation → Rice's Theorem under Self-Modification: Elevation Operators and a Normal Form
• summary: The undecidability of a program's static semantic properties is governed by Rice's theorem. Self-modifying systems, however, require analysing not whether a property holds now, but whether it is preserved when the system rewrites itself. We formalise this transition through a semantic elevation operator ΛΦ, which turns the static question "does x satisfy P?" into the dynamic question "is P preserved after x is transformed by Φ?". We prove that when Φ is intensional (depending on the source code, not only on the computed function), the elevated property remains undecidable even though it breaks → 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 $Λ_Φ(P)=\{x\in P:Φ(x)\in P\}$. When $Φ$ is extensional, $Λ_Φ(P)$ is behavioural and Rice's theorem applies. When $Φ$ reads the code, $Λ_Φ(P)$ is no longer behavioural, yet under uniform disruption (an inert wrapper encoding $K$) the s-m-n reduction that proves Rice's