Entanglement Equilibrium on a Solvable Gaussian Network: The Contact Term Derived, Geometry Reconstructed, Causal Cones Emergent, and Equilibrium Explained

This work provides a quantitative test of Jacobson's entanglement equilibrium program in a solvable network model whose geometry is encoded by entanglement and whose matter sector is a Gaussian scalar field on the same graph. All relevant quantities, including entropies, modular Hamiltonians and mutual informations, can be computed exactly. The paper isolates the assumptions underlying the derivation of the Einstein equation from entanglement equilibrium and classifies them as proved, derived, computed or imposed by construction. Five central questions are addressed: whether the contact term is derived rather than fitted, whether geometry can be reconstructed from entanglement data alone, whether the model possesses a causal structure, what mechanism explains the equilibrium postulate, and whether the matter sector converges to continuum quantum field theory beyond regular lattices. The lattice modular Hamiltonian is shown to converge to the exact continuum modular Hamiltonian of a massless scalar field. Using the Longo-Morsella result, the contact term coefficient and the corresponding first-law correction are derived analytically rather than fitted numerically. Geometry is reconstructed blindly from mutual-information data, recovering dimension and curvature. An explicit Lieb-Robinson bound and exact commutator calculations demonstrate an emergent causal cone and controlled Lorentz violation. Spectral convergence is established for deterministic quasi-uniform point sets on curved manifolds and repaired for non-uniform random point sets through a density-correction procedure. The equilibrium postulate is shown to be equivalent to the Einstein condition on the reconstructed geometry. The vacuum is identified as a local maximum of generalized entropy through relative-entropy arguments, while an entropic geometric flow is shown to reduce to the Yamabe or Ricci flow. The resulting framework provides a mathematically explicit and computationally reproducible test bed for entanglement-based approaches to emergent gravity.