Spacetime and Gravity as Emergent Quantum Entanglement — A Proof of Concept: Paper II — Lorentzian Metric, Modular Flow, and the Einstein Equations from Entanglement

This the second paper of a series of three papers under the title Spacetime and Gravity as Emergent Quantum Entanglement — A Proof of Concept by Alvaro Lozano Rodriguez. This paper establishes the Lorentzian and dynamical structure of the holographic model introduced in Paper I, deriving physical time, black hole thermodynamics, and gravitational field equations entirely from the entanglement geometry of the binary tree perfect tensor network. Three results are proved: First, the Euclidean AdS₂ metric of Paper I admits a canonical Wick rotation yielding the Lorentzian metric dsL2=R2 (−dt2+dz2) /z2ds²L = R² (-dt² + dz²) /z² dsL2=R2 (−dt2+dz2) /z2 with constant Ricci curvature K=−2 (ln⁡2) 2/δ2K = -2 (2) ²/² K=−2 (ln2) 2/δ2, fixed entirely by the binary entropy structure and the UV lattice scale with no free parameters. Second, via the thermofield double construction and the thermal time hypothesis of Connes and Rovelli, the modular flow of the boundary state is identified with Lorentzian time translation; the KMS condition establishes the thermodynamic arrow of time; and the Hawking temperature TH= (ln⁡2) / (2πδ) =1/ (2πR) TH = (2) / (2) = 1/ (2 R) TH= (ln2) / (2πδ) =1/ (2πR) follows as a parameter-free prediction. Third, the entanglement first law δS=δ⟨KA⟩ S = KA δS=δ⟨KA⟩, combined with the Casini–Huerta–Myers modular Hamiltonian and the Lashkari–McDermott–Van Raamsdonk theorem, yields the linearised Jackiw–Teitelboim dilaton equation at first order in the metric perturbation, verified exactly for three independent matter distributions. The complete logical chain — quantum network to Euclidean metric to Lorentzian metric to physical time to thermodynamic arrow to Hawking temperature to Einstein equations — is established within a single self-consistent model, with every step either proved or explicitly attributed.