The Computational Boundary of Inference: Capability Internalization, Training, and the Turing Jump
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911721563619328 |
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| author | Lu, Chien-Ping |
| author_facet | Lu, Chien-Ping |
| contents | Claims about recursive self-improvement in AI often slide from repeated internal revision to the possibility of qualitatively stronger capability without clearly distinguishing the underlying computational regimes. This paper gives a formal separation result in classical computability theory that blocks that move under a precise modeling assumption. For an oracle $A$, let $\mathcal{C}(A)=\{B : B \leq_T A\}$ be the corresponding computational layer. We prove that finite internal self-modification remains inside $\mathcal{C}(A)$, while stabilized revision is governed instead by the jump $A'$ via the relativized limit lemma. Together with a local closure versus escape theorem, this yields a clean formal separation between within-layer iteration and ascent to a stronger relative level. The point is not that stronger layers never arise, but that they are not explained by finite repetition inside one already settled layer. The resulting separation gives a computability-theoretic limit on a broad class of recursive-improvement narratives in which repeated internal updating is treated as sufficient for qualitative capability ascent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27381 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Computational Boundary of Inference: Capability Internalization, Training, and the Turing Jump Lu, Chien-Ping Computational Complexity Artificial Intelligence Claims about recursive self-improvement in AI often slide from repeated internal revision to the possibility of qualitatively stronger capability without clearly distinguishing the underlying computational regimes. This paper gives a formal separation result in classical computability theory that blocks that move under a precise modeling assumption. For an oracle $A$, let $\mathcal{C}(A)=\{B : B \leq_T A\}$ be the corresponding computational layer. We prove that finite internal self-modification remains inside $\mathcal{C}(A)$, while stabilized revision is governed instead by the jump $A'$ via the relativized limit lemma. Together with a local closure versus escape theorem, this yields a clean formal separation between within-layer iteration and ascent to a stronger relative level. The point is not that stronger layers never arise, but that they are not explained by finite repetition inside one already settled layer. The resulting separation gives a computability-theoretic limit on a broad class of recursive-improvement narratives in which repeated internal updating is treated as sufficient for qualitative capability ascent. |
| title | The Computational Boundary of Inference: Capability Internalization, Training, and the Turing Jump |
| topic | Computational Complexity Artificial Intelligence |
| url | https://arxiv.org/abs/2605.27381 |