Möbius Cross-Ratio Orbits and Modular Group Fixed Points — E8 Intelligence Research

Abstract

FINDING: Cross-ratio under Möbius transformations forms a 6-element orbit tied to the modular group's action on the Riemann sphere, with fixed points at e^{±iπ/3} (primitive 6th roots of unity). | MATH: For four points z1,z2,z3,z4, the cross-ratio λ = (z1−z3)(z2−z4)/((z1−z4)(z2−z3)). Under permutations of the four points, λ takes 6 values: {λ, 1/λ, 1−λ, 1/(1−λ), λ/(λ−1), (λ−1)/λ}. These are the orbit of the anharmonic group (isomorphic to S3). The modular group PSL(2,Z) acts on these; fixed points of the action occur when λ satisfies λ = e^{±iπ/3} = 1/2 ± i√3/2, i.e., λ³ = −1, λ² − λ + 1 = 0. These are the elliptic points of order 3 in the modular group's fundamental domain. | CONNECTION: The 6 values correspond to the 6 vertices of a regular octahedron's dual (or the 6 permutations of 4 points), linking to the tetrahedral symmetry (A4) and the root system A3 (crystallographic). The fixed points e^{±iπ/3} have modulus 1 and arguments ±60°, which in base-60 arithmetic are exact (1/6 of Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com