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arXiv · 2603.06012

Diagonalizing Through the $\omega$-Chain: Iterated Self-Certification on Bounded Turing Machines and its Least Fixed Point

Miara Sung
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Abstract

Bounded self-certification in Turing machines fails because self-simulation necessarily incurs a strictly positive temporal overhead. We translate this operational constraint into a domain-theoretic framework, defining an operator that advances a finite halting observation from time bound $i$ to $i+1$. While no bounded machine can achieve a fixed point under this operator, the iterative process forms an ascending $\omega$-chain. The Scott limit of this chain resolves to the least fixed point of the operator, representing an unbounded computation that fully captures the machine's halting behavior. Our construction provides a novel perspective on the halting problem, framing the transition from finite observability to the least fixed point as the continuous deferral of the diagonal.

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Miara Sung. 2026-03-06. Diagonalizing Through the $\omega$-Chain: Iterated Self-Certification on Bounded Turing Machines and its Least Fixed Point. https://arxiv.org/abs/2603.06012

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