Entanglement as Topology: Hopf Linking as the Geometric Origin of Quantum Correlation

Abstract

Title: Entanglement as Topology: Hopf Linking of Soliton Preimage Curves Author: Alexander Novickis (alex.novickis@gmail.com) We propose that quantum entanglement has a geometric origin: the topological linking of soliton field configurations within a shared Hopf fiber bundle. In the Hopf soliton framework of Papers I–III, particles are topological solitons governed by the Hopf fibration S¹ → S³ → S², and each soliton defines a family of preimage curves in S³. When two solitons' preimage curves are linked, the resulting topological inseparability manifests as quantum entanglement. The central insight is that there is no "spooky action at a distance": along the shared Hopf fiber, the distance between entangled particles is exactly zero, and the apparent spatial separation is a projection artifact from S³ to ℝ³. This identification reproduces all essential properties of entanglement — nonlocality, quantization, monogamy, no-signaling, and the impossibility of local creation — as consequences of topology. Key original results include: The Born rule P = cos²(θ/2) derived as a geometric identity of the Hopf map combined with U(1) gauge ergodicity — structurally parallel to Gleason's theorem but operating in the fiber bundle rather than the Hilbert space (§10.8) A no-cloning theorem from topological conservation of linking — cloning would require creating linking with distant solitons via a local operation, which is topologically impossible (§6.7) The spin-statistics connection from Hopf charge parity: exchange phase (−1)ᴴ = (−1)²ˢ from the linking acquired during the exchange braid (§3.5) Entanglement entropy bound S ≤ log(|n|+1) from Seifert surface intersection counting, with the vacuum channel derived from π₁(S²) = 0 (§5.4–5.5) Exponential Hilbert space from Milnor μ̄-invariants: 2ⁿ − n − 1 independent linking invariants for n solitons (§12.4) Stepwise entanglement sudden death from integer linking as a falsifiable prediction (§9.6) Quantum teleportation as topological reconnection — four Bell outcomes correspond to four reconnection topologies (§3.4.6) Continuous-variable entanglement from linking sector superposition — the two-mode squeezed vacuum as Σ cₙ(r)|Lk=n⟩ (§5.8) A five-step path integral proof deriving the singlet correlation E(â,b̂) = −cosθ from the CP¹ path integral through Chern-Simons theory, with SU(2)₁ CS emergence as one well-characterized conjecture (§13) Surgery calculus resolution: U(1)₁ Chern-Simons fails (1D Hilbert space), while SU(2)₁ succeeds (maximally entangled state) — the entanglement is carried by the parafermion sector (§13.8.11) Unification of entanglement with quark confinement (Paper III) as two instances of the same topological inseparability A geometric realization of the Maldacena–Susskind ER=EPR conjecture, identifying the microscopic wormhole with the linked Hopf fiber The paper systematically compares the framework with Copenhagen, many-worlds, Bohmian, and QBist interpretations, and shows it satisfies the PBR theorem as a ψ-ontic nonlocal realistic theory. Twenty original contributions are enumerated with honest assessment of limitations. Keywords: quantum entanglement, Hopf fibration, topological linking, Bell inequality, Born rule, nonlocality, soliton, Chern-Simons theory, monogamy, EPR paradox, CP1 path integral, fiber bundle

The authors' abstract, as published at the source. Open MIND, 2026 · DOI ↗