The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation
Updated 7 h ago · first seen 11 Sept 2026
paper_01M294G599SVRCNXE3JWK7S9ZW
- Published
- 11 Sept 2026
- T1 · 7 h ago
- arXiv
- 2609.11326
- T1 · 7 h ago
- Category
- cs.LO
- T1 · 7 h ago
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 {\Lambda}{\Phi}, which turns the static question "does x satisfy P?" into the dynamic question "is P preserved after x is transformed by {\Phi}?". We prove that when {\Phi} 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 {\Pi}02-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.
Authors 1
Jose Pascual Gumbau Mezquita
Specification
- Official page
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
- Arxiv announce type
- cross
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
- arXiv id
- 2609.11326
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
- Categories
- cs.LO, cs.AI, cs.CL, math.LO
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
- Primary category
- cs.LO
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
- Published
- 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 7 h agohigh
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9
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T19
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7 h ago
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- Authors
- Jose Pascual Gumbau Mezquita
As of
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Claim history
Official pageofficial_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/abs/2609.11326 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Abstractabstract1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 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 {\Lambda}{\Phi}, which turns the static question "does x satisfy P?" into the dynamic question "is P preserved after x is transformed by {\Phi}?". We prove that when {\Phi} 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 {\Pi}02-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. | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Arxiv announce typearxiv_announce_type1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| cross | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
arXiv idarxiv_id1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 2609.11326 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Categoriescategories1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| cs.LO, cs.AI, cs.CL, math.LO | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
PDFpdf_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/pdf/2609.11326 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Primary categoryprimary_category1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| cs.LO | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Publishedpublished_at1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 11 Sept 2026 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
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- New paperPaperThe Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation
New paper: The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation
arxiv
| Source | Document | Type | Tier | Last observed | Snapshots |
|---|---|---|---|---|---|
| arXiv (Atom API + RSS) | rss.arxiv.org/rss/cs.CL | feed | T1· Official | 5 h ago | 1 |
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