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

TL;DR

This paper introduces a domain-theoretic framework for understanding bounded self-certification in Turing machines, using an operator that models the transition from finite observations to the least fixed point representing unbounded halting behavior.

Contribution

It develops a novel domain-theoretic approach to model the transition from finite to unbounded computation in Turing machines via an iterative operator and its least fixed point.

Findings

The operator advances finite halting observations by one step.

No bounded machine can reach a fixed point under this operator.

The Scott limit of the chain captures the machine's full halting behavior.

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.