Coupling Constant Scaling from Fano Node Superposition: Energy-Dependent Delo- calization of Octonionic Eigenmodes, Running Couplings as Fano Overlap Amplitudes, and Gauge Unication as Complete Node Delocalization at the Planck Scale

We propose and develop the central mechanism behind all coupling constant scaling in the brane-bulk octonionic framework: at energy E, a focal zone cannot be localised to a single Fano lattice node but occupies a quantum superposition of neighbouring nodes, with overlap probability Pij (E) ∝ (E/mPl) ∆ij where ∆ij is the topological distance between nodes i and j on the Fano graph. At the Planck scale, all seven nodes are in complete superposition no focal zone can be resolved as belonging to a single node and all coupling constants converge to a common G2-symmetric value: gauge unication is the complete delocalization of the Fano eigenmodes. As energy decreases from mPl, the eigenmodes localise back onto their characteristic nodes at a rate governed by the Fano Laplacian eigenvalue λi ∈0, 4, 7: higher-λ modes localise faster, giving steeper running. The three gauge coupling constants correspond to the three Fano eigenmode families: αs ↔λ = 7 (fastest localisation, steepest running, asymptotic freedom) ; αW ↔λ = 4 (intermediate) ; αem ↔ λ = 0 (softest mode, slowest running). Asymptotic freedom in QCD emerges because at high energy the H+/H−sector distinction blurs through node superposition, reducing the cross-sector bridge formation amplitude of Paper LVIII and hence αs. The one-loop beta function coecients are identied with the Fano Laplacian eigenvalues weighted by the Fano mode census; the exact coecients require the census computation deferred from Paper LIII, but the ratios of running rates are xed by λ = 7: 4: 0 from the lattice geometry alone. Five new predictions follow (Predictions 110114). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10. 5281/zenodo. 19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M. D. , University of Minnesota. ORCID: 0000-0002-1143-941X.