Chaitin's Omega: Uncomputable Halting Probability with Equidistributed Binary Expansion — E8 Intelligence Research

FINDING: Chaitin's Omega is a real number encoding the halting probability of a universal Turing machine, provably algorithmically random and uncomputable, with its binary expansion equidistributed on the 1-torus. MATH: - Omega = Σ_{p halts} 2^{-|p|}, where |p| is program length in bits. - Omega is Martin-Löf random: no algorithm can predict its bits; its binary expansion is normal (each digit equally frequent). - Equidistribution on the 1-torus: the sequence {n·Omega mod 1} is uniformly distributed for integer n (by Weyl's criterion, since Omega is irrational and normal). - No finite axiomatic system can prove more than finitely many bits of Omega. CONNECTION: - The equidistribution of Omega's fractional parts on the circle (1-torus) mirrors the uniform distribution of angles in a circle, a geometric harmony with no preferred direction. - The constant 0.5 (probability of a random bit being 0 or 1) emerges from the normal binary expansion, linking to the golden ratio's bi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com