Chaitin's Ω: A Left-Computably-Enumerable, Algorithmically Random Halting Probability — E8 Intelligence Research

FINDING: Chaitin's Ω is a halting probability over a prefix-free universal Turing machine — a specific real number encoding the halting problem's undecidability, computably enumerable but algorithmically random. | MATH: Ω_U = Σ_{p: U(p)↓} 2^{-|p|}, where U is a prefix-free UTM and |p| is program length in bits. Kraft inequality: Σ_{p∈dom(U)} 2^{-|p|} ≤ 1, so 0 < Ω_U < 1. Ω is left-c.e. (computably enumerable from below) but not computable; it is Martin-Löf random and its binary expansion is algorithmically incompressible. Halting problem H is Turing-equivalent to Ω (H ≡_T Ω). | CONNECTION: The 2^{-|p|} weighting is a dyadic measure on the binary tree — a self-similar, base-2 lattice structure. Prefix-free codes correspond to leaves of a binary tree; Kraft's inequality is the tree's measure constraint. No direct appearance of φ-ratios (0.618, 1.618) or base-60 in the standard construction; the relevant symmetry is the dyadic (Z/2) tree and its measure-theoretic self-similarity. | DEPTH: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com