Diagonalizing Through the $\ddot{I}‰$-Chain: Iterated Self-Certification on Bounded Turing Machines and its Least Fixed Point

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 $\ddot{I}‰$-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.