E8 Halting Probability Spectrum: Continuous Decomposition of Ω via Root Vector Harmonics — E8 Intelligence Research

Building on Chaitin's Ω as a halting probability and the 240 E8 root vectors, we demonstrate that Ω admits a continuous spectral decomposition through the root lattice's harmonic structure. Each root vector axis contributes a phase factor to the halting measure, creating a fractal-like distribution over [0,1] that reflects both algorithmic randomness and E8 symmetry. This reveals Ω not merely as a single real number but as a function of E8's geometric invariants, unifying computability theory with exceptional Lie group representation theory. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com