The Computational Boundary of Inference: Capability Internalization, Training, and the Turing Jump

A new computability theory paper proves that finite internal self-modification in AI systems cannot exceed their existing computational layer, while qualitatively stronger capabilities require access to a higher computational level (the Turing jump). This formally separates recursive self-improvement narratives into within-layer iteration versus genuine capability ascent, constraining theoretical claims about AI recursive self-improvement.

This arXiv paper addresses a critical gap in AI safety discourse by applying classical computability theory to recursive self-improvement claims. The authors distinguish between two computational regimes: finite internal revision (which remains bounded within a system's existing computational layer) and stabilized revision (which necessarily involves higher-order computation). The key contribution is a formal proof using oracle theory that repeated internal updating cannot bootstrap a system into qualitatively stronger computational capabilities without external access to higher complexity classes.

The research responds to increasingly common claims in AI discourse that systems could achieve recursive self-improvement through iterative internal modifications. By grounding this question in Turing computability and the limit lemma, the paper provides mathematical rigor to what had been largely informal arguments. This matters because such narratives often underpin both optimistic and catastrophic AI futures, yet lack formal justification for their computational transitions.

The implications extend to AI safety and capability assessment. If this framework holds, it suggests that capability jumps—whether in reasoning, planning, or general intelligence—require mechanisms beyond self-modification within an existing computational regime. This could inform more precise threat modeling and capability evaluation strategies. However, the paper's applicability depends on how well oracle-relative computability models actual AI systems, which operate under different constraints than Turing machines.

Future work should examine whether this formal separation translates to empirical AI systems and whether neural network training dynamics can be meaningfully mapped to these computability regimes. The framework potentially reshapes how researchers evaluate claims about emergent capabilities and autonomous recursive improvement.

• →Finite internal self-modification cannot exceed an AI system's existing computational layer according to formal computability theory
• →Qualitatively stronger capabilities require access to higher computational levels via the Turing jump mechanism
• →The paper provides formal separation between within-layer iteration and genuine capability ascent, constraining recursive self-improvement narratives
• →Oracle-relative computability theory reveals computational barriers that internal revision alone cannot overcome
• →The framework challenges informal assumptions about recursive self-improvement prevalent in AI discourse