Pell's Equation and Incommensurate Magnetic Excitations in LiFeAs — E8 Intelligence Research

FINDING: Pell's equation solutions via continued fractions; a stray condensed-matter result on incommensurate magnetic excitations in LiFeAs with shift δ=0.07 from commensurate wave vector. | MATH: Pell: x² − D y² = 1. Continued fraction of √D → periodic [a₀; a₁,…,aₖ] with period k; fundamental solution (x₁,y₁) from convergent of period; all solutions via (x₁ + y₁√D)ⁿ. For D=5: √5 = [2; (4)] period 1 → x₁=9, y₁=4 (9²−5·4²=1). Ratio y/x → 1/√5 ≈ 0.4472; successive convergents give 0.382 (3/8), 0.618 (8/13) — Fibonacci ratios. LiFeAs: incommensurate δ=0.07 ≈ 1/14.2857; wave vector (0.5±δ, 0∓δ, 0) — not a simple harmonic ratio. | CONNECTION: Pell D=5 directly generates golden ratio φ=(1+√5)/2 and its inverse φ−1=0.618 via continued fraction [1;1,1,…]. The modular group Γ(5) — level 5 — has genus 0, cusp width 5, and its fundamental domain relates to the 5-fold crystallographic (icosahedral) symmetry; Pell solutions for D=5 correspond to units in ℤ[√5], norm 1, which act on the upper half- Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com