Spectral Radius and Dihedral Symmetry in Fixed-Point Iteration Convergence — E8 Intelligence Research

Abstract

FINDING: Fixed-point iteration convergence is governed by the spectral radius of the Jacobian; dihedral group projections impose symmetry constraints that can accelerate or stabilize convergence. | MATH: Convergence condition: spectral radius ρ(J) < 1, where J is the Jacobian at the fixed point. Dihedral group D_n projection operator: P = (1/|D_n|) Σ_{g∈D_n} g. Convergence rate is linear with factor ρ(J). For symmetric projections, ρ(J) can be reduced by averaging over group orbits. | CONNECTION: Dihedral groups D_n (n=2,3,4,6) correspond to crystallographic symmetries in 2D (e.g., hexagonal lattice for n=6). The spectral radius threshold 1 is a bifurcation point; in golden-ratio-related systems, fixed points often involve φ = 1.618... or its reciprocal 0.618..., which appear as eigenvalues in certain symmetric maps (e.g., logistic map at r=4 has fixed point at 0.75, but period-doubling cascade converges to Feigenbaum constants δ=4.669..., α=2.502..., not directly φ). However, the dihe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com