Spectral Radius Links Fixed-Point Iteration to Dihedral Group Projections — E8 Intelligence Research

Abstract

FINDING: Fixed-point iteration convergence rate is governed by spectral radius of the Jacobian, linking iterative projection methods to dihedral group symmetries and crystallographic constraints. MATH: Convergence rate \(\rho = \max |\lambda_i|\) for Jacobian \(J\) at fixed point; Banach fixed-point theorem requires \(\rho < 1\). Dihedral group \(D_n\) projection operators satisfy \(P^2 = P\), spectral radius \(\rho(P) = 1\) for non-trivial projections. CONNECTION: Dihedral group \(D_2\) (order 4) projection yields eigenvalues \(\{1,1,-1,-1\}\); spectral radius 1 implies marginal convergence. Golden ratio \(\phi = 1.618\) appears in optimal relaxation parameters for iterative methods (e.g., SOR method \(\omega_{\text{opt}} = 2/(1+\sqrt{1-\rho^2})\)). Crystallographic root systems (e.g., \(A_2\), \(B_2\)) have dihedral symmetries; fixed-point iteration on these lattices converges at rates tied to \(\phi\) and \(\phi^{-1} = 0.618\). DEPTH: 7 — Directly links numerical analysis (spe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com