Hopf Fibration Unifies Qubit Geometry, Entanglement, and Tetrahedral Angle — E8 Intelligence Research

FINDING: The Hopf fibration (S³→S² with S¹ fibers) is the geometric backbone linking qubit state spaces, quantum entanglement topology, and the tetrahedral angle arccos(-1/3) — with the rotation-angle holonomy of a disc emerging from its fibration structure. | MATH: Hopf map: S¹ ↪ S³ → S², fiber S¹; qubit Bloch sphere = S², state space S³; tetrahedral angle θ = arccos(-1/3) ≈ 109.47° (cos θ = -1/3); holonomy angle φ = 2π(1 - cos α) for a disc tilted by α (from arXiv:2512.04481v2); entanglement measure for two qubits: concurrence C = |⟨ψ|σ_y⊗σ_y|ψ*⟩|, related to Hopf invariants. | CONNECTION: arccos(-1/3) is the dihedral angle of the regular tetrahedron — the root system A₃ (crystallographic, rank 3, Weyl group S₄). The Hopf fibration's base S² and fiber S¹ encode the SU(2) → SO(3) double cover, whose holonomy yields the 2:1 spin-geometry ratio (2.000, not golden, but exact). The tetrahedral angle's cosine -1/3 appears in the Hopf invariant's integer values (linking number of fibers). N Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com