Paper CQ-X: Wormhole Membrane Structure, Dark Matter Entanglement Network, and the Hierarchy Problem in Four-Sector Cosmology

Abstract We extend the Four-Sector Cosmological (FSC) framework (Papers A–F, CQ-I through CQ-IX) in three principal directions. First, we reinterpret Einstein–Rosen (ER) bridges within FSC: wormhole traversability is forbidden for baryonic matter not by rapid collapse but by the fundamental t→iτ transformation required at the Sector-I/II boundary, while Sector-III dark energy naturally provides the exotic matter supporting the throat. Second, we establish that Sector-II dark matter maintains a cosmic quantum entanglement network (QEN) with decoherence time τ_D(II) = 285 t_universe, since electromagnetic, strong, and weak interactions are completely suppressed in Sector-II, leaving only gravitational decoherence with ε_grav ∼ 4×10⁻⁶. From this QEN we derive the observed ratio Ω_DM/Ω_b = 5.47 without parameter tuning. Third, we reformulate the hierarchy problem: the comparison m_H ≪ M_Pl is meaningless because these scales belong to categorically distinct physical entities (Sector boundary vs. Sector interior); the FSC boundary condition m_H²(M_Pl) = 0 provides a natural starting point from which the renormalization group generates m_H ∼ O(100 GeV) without fine-tuning. We classify universal invariants into three layers: (i) all-Sector universals l_P, G, ℏ; (ii) Sector-specific constants c_EM, m_H, α_EM; (iii) inter-Sector relational invariants t→iτ and e^{iπ}=-1. Observationally, FSC achieves χ²/dof = 0.029 against DESI DR1 BAO data. Key predictions include: w_a ≈ +0.003 (decisive test via DESI DR5, ~2026); A_L = 0.9967 (CMB-S4, 6.6σ); gravitational-wave polarization rotation Δψ = π/4 for wormhole-traversing signals (LISA).