Vacuum Entanglement Geometry and Emergent Spacetime: Connections to Entropic Gravity, Tensor Networks, and Causal Sets

We position the Vacuum Time Geometry (VTG) framework—developed in Papers I and II as a nonlinear σ-model on M = Sp(56,R)/U(28) with Fisher–Bures metric—within the broader landscape of emergent gravity programs. We demonstrate that three principal results in the “gravity from thermodynamics” program are limiting cases of the VTG dynamics: Jacobson’s derivation of Einstein’s equations from δQ = TδS on local Rindler horizons (the local Rindler limit), Verlinde’s entropic force F = T∇S (the static adiabatic limit), and Padmanabhan’s holographic equipartition dV/dt = l2P (Nsur − Nbulk) (the FRW-symmetric limit). The hierarchy VTG ⊃ Jacobson ⊃ Verlinde is exact. We then identify the σ-model map λa(x) as a continuum tensor network, establish the structural parallels and precise differences with the “it from qubit” program (MERA, HaPPY code,Ryu–Takayanagi formula, Van Raamsdonk’s entanglement–connectivity argument), and show that the quantum geometric tensor (QGT) is the common mathematical origin of both programs. The discrete tick structure Δtmin = tP/˜η is compared with the causal set Hauptvermutung. No new free parameters beyond those of Papers I and II are introduced.