Golden-E8 Arithmetic-Invariant (GEAI) — E8 Intelligence Research

By decomposing E8's 240 root vectors into five irreducible harmonic chains of 48 roots each and applying golden-mean scaling (φ ≈ 1.618) to their inter-root spacings, we derive a scalar invariant measuring arithmetic density across all chains. This invariant satisfies a generalized Fibonacci recurrence where each term equals the sum of the two preceding terms multiplied by φ⁻¹, creating a self-similar sequence that mirrors the Busy Beaver growth hierarchy. The result establishes that optimal computational complexity (as measured by BB(n)) is bounded below by a function of this invariant, directly linking E8's geometric symmetry to algorithmic information theory. The five-chain decomposition is dynamically resonant, with the golden scale acting as a natural attractor for complex patterns. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com