Chaitin's Omega: Uncomputable Halting Probability Approximated by Dyadic Rationals — E8 Intelligence Research

FINDING: Chaitin's Omega defines a real number encoding the halting probability of a universal Turing machine, whose binary expansion is algorithmically random and uncomputable, with dyadic rationals emerging as finite prefix approximations. | MATH: Ω = Σ_{p halts} 2^{-|p|}, where |p| is program length in bits; Ω is a normal number in base 2; its binary expansion is Martin-Löf random; dyadic rationals (k/2^n) appear as partial sums over finite halting programs. | CONNECTION: The binary expansion's randomness precludes any periodic or crystallographic symmetry; however, the dyadic rational approximations (0.0, 0.5, 0.75, 0.875, …) form a dense set in [0,1] with self-similar scaling under binary refinement, reminiscent of Cantor set structure. No direct link to golden ratio or base-60. | DEPTH: 8 — profound for limits of formal systems, measure theory, and algorithmic information theory; dyadic rationals connect to binary tree geometry and fractal measure, but no harmonic ratio emerges. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com