Emergent Spacetime and Gravity from Timeless Renormalization Consistency: A Quantum Gravity Framework

We develop a quantum gravity framework in which spacetime geometry and gravitational dynamics emerge from a quantum system governed by three fundamental principles: (i) a global Hamiltonian constraint enforcing timelessness, (ii) a variational principle based on stationary action, and (iii) resolution independence, implemented as consistency under renormalization (coarse-graining). Adopting an algebraic formulation of quantum theory, where subsystems are defined as subalgebras of observables and states as linear functionals, we show that entanglement structure provides a natural basis for the emergence of geometry. In a regime characterized by approximate renormalization locality and smooth correlation structure, a metric tensor arises from variations of mutual information between subsystems, while curvature is identified with the nontrivial response of correlations under local deformations. We further demonstrate that imposing simultaneous consistency with the variational principle and renormalization invariance leads to a stability condition on the entanglement structure. In the continuum and near-equilibrium limit, this condition yields Einstein's field equations. Beyond linear order, fluctuations of entanglement naturally generate higher-curvature corrections, suggesting a quadratic gravity structure as a candidate ultraviolet completion. This approach provides a unified perspective in which spacetime, gravity, and their quantum corrections arise as consequences of consistency across scales and stationary dynamics, rather than as independent fundamental inputs.