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

• URL: http://arxiv.org/abs/2609.11326v1
• Date: Thu, 10 Sep 2026 09:56:17 GMT
• Title: The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation
• Abstract summary: Self-modifying systems require analysing not whether a property holds now, but whether it is preserved when the system rewrites itself.<n>We formalise this transition through a semantic elevation operator, which turns the static question "does x satisfy P?" into the dynamic question "is P preserved after x is transformed by?"<n>We prove that when is intensional, the elevated property remains undecidable even though it breaks the extensionality that Rice's theorem requires.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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 the extensionality that Rice's theorem requires; the proof rests on Kleene's recursion theorem, not on Rice. Consequently the class U of non-verifiable properties is closed under the elevation operator. Unbounded iteration of the operator climbs the arithmetical hierarchy -to Π02-completeness- consolidating non-verifiability as a structural fact. We further show that the supervisory regress does not terminate: no fnite tower of increasingly capable verifiers yields an unconditional certificate. A categorical reading of these results in the efective topos, in which elevation appears as an instance of Lawvere's fxed-point theorem, is left as a direction for future work.

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