BT‑PEC: Berry-Taro Private Entanglement Cosmology — A Tripartite Coupling Emergent Quantum Gravity Theory

Abstract

This work proposes BT‑PEC (Berry-Taro Private Entanglement Cosmology), a tripartite coupling emergent quantum gravity theory to reconcile general relativity and quantum mechanics and resolve the black hole information paradox. Unlike conventional quantum gravity frameworks built on a dual matter-spacetime system, BT‑PEC promotes the observer subsystem to a fundamental dynamical degree of freedom, standing on equal footing with matter and spacetime. The global universe evolves unitarily. The total Hilbert space decomposes into the matter spin subsystem and an independent quantum observer subsystem. Spacetime geometry, connectivity and metric emerge self-bootstrapped from entanglement strength between lattice sites, rather than being a fixed background structure. Three core operators govern the dynamics: the matter operator B, the spacetime operator T, and the PEC observer operator, which controls observer-matter entanglement exchange, decoherence and quantum-classical crossover. Quantum negativity, instead of entanglement entropy, serves as the primary quantity generating spacetime geometry, filtering out contributions from classical correlations. Under Markovian weak-coupling approximation, partial trace over the observer subsystem yields a Lindblad master equation for matter, preserving global unitarity while only the reduced subsystem evolves non-unitarily. A continuous quantum-classical phase transition arises at a critical entanglement threshold. We introduce the single-anchor dual-picture paradigm: one spatial anchor admits both quantum microscopic and classical geometric descriptions, with observables obtained by weighted interpolation to eliminate discontinuity across the quantum-classical boundary. Taking the continuum limit of discrete spin networks recovers 4-dimensional Riemannian spacetime. Variation of the effective action gives modified Einstein field equations corrected by observer and entanglement stress-energy tensors, which reduces to classical general relativity at low energy and far-field regions. For black hole evaporation, the horizon becomes a smooth transition zone of decaying cross-horizon entanglement without firewall. The whole evaporation process is divided into three stages: information locking, critical transition, and geometric decoupling with information release. The Page entropy rises to its peak at Page time and finally falls back near zero, safeguarding global unitarity and recovering information. Numerical finite-size scaling simulations on 1D spin-chain toy models support the critical threshold and tanh-shaped decay of cross-horizon coupling. This theory provides a self-consistent chain from microscopic entanglement dynamics to emergent spacetime, macroscopic gravity and black hole evolution, offering a numerically testable candidate quantum gravity proposal.