Squeezing and S₄ Symmetry Unlock Maximal Bell Violation in Entanglement — E8 Intelligence Research

FINDING: Bell inequality violation optimized via continuous variable squeezing and finite group symmetry (S₄) reveals deep links between entanglement, symmetry groups, and geometric constraints. MATH: - Squeezing parameter ζ → ∞ yields maximal Bell violation for two-mode squeezed vacuum states; parity averages to zero. - S₄ symmetry group (order 24) used to analyze Bell inequality violation; relates to tetrahedral symmetry and root system A₃. - Upper bounds on Bell violation for N-qubit GHZ states: exact bounds derived, scaling with Hilbert space dimension. - Asymmetry as resource: trading symmetry for Hilbert space dimension increases Bell violation. CONNECTION: - S₄ symmetry is the rotational symmetry group of the cube/octahedron (crystallographic point group O_h subgroup). - 8-fold symmetry (quasicrystal) not directly present, but S₄ links to tetrahedral (3-fold) and cubic (4-fold) symmetries. - No explicit golden ratio (0.618, 1.618) or base-60 found. However, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com