The Golden Rectangle Model: Strain Eigenvalue Ratios from Icosahedral Geometry

A geometric model derives the strain eigenvalue ratios observed in turbulent flows from three mutually perpendicular golden rectangles — the construction that generates the icosahedron. When one rectangle dominates and a second opposes at strength α=1/12=|ζ(−1)|, the resulting traceless strain tensor predicts three independent observables confirmed by DNS of the Taylor-Green vortex at Re=1600 on a 128³ grid: eigenvalue ratio r=λ₂/λ₁=0.0797 (DNS median: 0.0801, Δ=0.0004), Steinbach third-invariant ratio R_S/(−Q_S)^{3/2}=0.0760 (DNS: 0.0764, Δ=0.0004), and tube-dominant strain topology (DNS: 60% tubes). The Steinbach prediction and the golden rectangle prediction are unified as different instruments measuring the same icosahedral geometry. The value α=1/12 connects the strain eigenvalue ratios to the kissing number k(3)=12, the Riemann zeta function ζ(−1)=−1/12, and the spectral gap 168=12×14=k(3)×(k(3)+2) of the Poincaré homology sphere. The prediction preceded the measurement.