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The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation

arXiv · AI, language, vision and robotics · article · Sep 10, 2026 · UTC

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

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