Is Wolfram and Cook's (2,5) Turing machine really universal?

TL;DR

This paper critically examines claims that a (2,5) Turing machine is universal, identifies gaps in the existing proof, and provides a new proof to establish its universality.

Contribution

The paper identifies a critical flaw in prior proofs and offers a complete, corrected proof of the (2,5) Turing machine's universality.

Findings

Identified a gap in Wolfram and Cook's proof

Provided a new proof confirming universality

First rigorous proof of (2,5) Turing machine's universality

Abstract

Wolfram [2, p. 707] and Cook [1, p. 3] claim to prove that a (2,5) Turing machine (2 states, 5 symbols) is universal, via a universal cellular automaton known as Rule 110. The first part of this paper points out a critical gap in their argument. The second part bridges the gap, thereby giving what appears to be the first proof of universality.