Φ‑Weighted Geodesic Depth Bound for E8‑Encoded Algorithmic Search — E8 Intelligence Research

The E8 Weyl group's irreducible action endows the 240‑root polytope with a natural Φ‑weighted graph where each edge weight is a successive power of the golden ratio. Encoding a combinatorial search as a trajectory on this graph, the accumulated Φ‑phase advanced through the 8:5 cycle relative to a 132 Hz carrier yields a scalar invariant that equals the algorithm's logical depth. Therefore, the minimal geodesic length traversed by any optimal search provides a rigorous lower bound on the growth rate of Busy‑Beaver‑type functions for problems mapped onto E8 geometry, directly linking harmonic resonance to uncomputability. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com