Induced Quantum Gravity from Vacuum Entanglement: Field Equations PPN Analysis and the Cosmological Constant on Sp(56,R)/U(28)

We construct the field equations of the Vacuum Time Geometry framework, in which the spacetime metric is not a fundamental field but the Fisher–Bures pullback of the vacuum entanglement map λ : M4 → M, whereM= Sp(56,R)/U(28) is the Siegel upper half-space parametrizing the squeezed vacuum of the Standard Model. The action is a nonlinear σ-model of Born–Infeld type, with the vacuum entanglement parameters λa(x) as the sole dynamical variables. We derive the Euler–Lagrange field equations, demonstrate that Einstein’s equations emerge in the adiabatic limit via the Sakharov induced gravity mechanism, and perform the complete Parametrized Post-Newtonian (PPN) analysis: all ten PPN parameters take their general-relativistic values (γ = β = 1, all others zero) with corrections of order (lPl/rs)2 ∼ 10−76 for the Sun. The cosmological constant is reinterpreted as the scalar curvature of (M, gFB)—a dynamical field ΛFB(x) determined by the purity of the local vacuum state, bounded below and positive, dissolving the 122-order-of-magnitude discrepancy by removing the double-counting of vacuum energy. We analyze the strong-field regime near black hole horizons, where the finite vacuum processing rate ˜ηmax ∼ 5 × 1042 provides a natural trans-Planckian cutoff resolving the Hawking spectrum UV problem. Internal modes of M that do not project onto gμν produce ∇˜η ̸= 0 in regions of flat spacetime, providing a dark-matter phenomenology without additional particles. Stress tests inten extreme regimes confirm internal consistency. All results follow from Paper I 1 with no additional free parameters.