Hyperbolic Tiling and the Spectral Constant φ² in Fuchsian Groups — E8 Intelligence Research

FINDING: Poincaré's Polygon Theorem and Fuchsian fundamental domains provide the constructive framework for hyperbolic tiling, where the modular group's fixed points and eigenvalue spectra encode φ² (2.618) as a natural spectral constant. | MATH: Poincaré's Polygon Theorem: A convex hyperbolic polygon with side-pairing isometries generates a Fuchsian group iff the angle sum at identified vertices is \(2π/k\) (k∈ℤ⁺). Fundamental domain: \(D ⊂ H^2\) with \(γ ∈ Γ γ(D) = H^2\), \(γ(D) ∩ D = ∅\) for \(γ ≠ e\). Modular group \(Γ = PSL(2,Z)\): fixed points of elliptic elements of order 2 (eigenvalue \(i\)) and order 3 (eigenvalue \(eiπ/3\)) lie at \(z = i\) and \(z = eiπ/3\) in the standard fundamental domain \(\{|z| ≥ 1, |Re(z)| ≤ 1/2\}\). The hyperbolic distance between these fixed points: \(d(i, eiπ/3) = ⁻¹(3/2) ≈ 0.9624\). The eigenvalue of the Laplacian on Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com