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

Miara Sung

TL;DR

The 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.

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 $ω$-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.

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

TL;DR

The 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.

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 to . While no bounded machine can achieve a fixed point under this operator, the iterative process forms an ascending -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.
Paper Structure (4 sections, 6 theorems, 7 equations)

This paper contains 4 sections, 6 theorems, 7 equations.

Table of Contents

  1. Setup
  2. The Operator
  3. The Fixed Point
  4. Remarks

Key Result

Lemma 1

Let $A$ be an arbitrary Turing machine and let $D$ be any Turing machine (it is possible that $D = A$ by Kleene's recursion theorem) that correctly decides whether $A$ halts within $T$ steps. Then $D$ requires at least $T+1$ steps.

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Proposition 1: Monotonicity and Continuity
  • proof
  • Theorem 1: No bounded fixed point
  • proof
  • Theorem 2: Least fixed point and unboundedness
  • proof
  • Corollary 1: Bounded self-certification vs. Scott limit
  • proof
  • ...and 2 more