Entanglement Equilibrium on an Exactly Solvable Pair–Gaussian Network: Rigorous Reductions, Conditional Results and a Quantitative Test of the Matter First Law

Jacobson’s entanglement-equilibrium argument obtains the Einstein equation from three inputs on the entropy of small geodesic balls: an area law with a finite Newton constant, a first law for the matter modular Hamiltonian, and stationarity of the vacuum under volume-preserving variations. I isolate every assumption of this argument beyond established mathematics as one of seven explicit hypotheses (H1-H7) on a family of quantum network states, and prove that, given H1-H4 and H6, the Einstein equation at a point is equivalent to the equilibrium hypothesis H5. I then construct an exactly solvable composite family on compact surfaces: a product of two-qubit pure states whose entanglement encodes a Riemannian metric, tensored with the exact Gaussian ground state of a conformally coupled scalar field on the same weighted graph, calibrated to unit propagation speed. On this family I prove: (i) H1 with an explicit analytic bound on the contamination of the reconstructed metric by matter, valid under a hierarchy condition involving the regulator mass; (ii) a Lieb-Robinson bound for the field sector on the Weyl algebra, uniform in network size in physical units, with explicit velocity; (iii) a Lorentzian-limit theorem for finite families of smeared probes, including a quantitative bound excluding superluminal chains of probe relations, conditional on spectral convergence of the graph Laplacian; (iv) the geometric part of H3 with an explicit Newton constant; (v) that the improvement term of the conformal stress tensor does not drop out of the first-law integral, so first-law tests with the minimal lattice energy density carry a scale-independent mismatch; (vi) static equilibrium as the Hamiltonian constraint, together with a no-go theorem for static vacuum geometries; (vii) exact local lattice energy conservation; and (viii) exact reflection positivity of a kernel ghost-layer ensemble. Numerically, the discrete area law reproduces a continuum nonlocal-perimeter functional to about 2 x 10^-2 on spheres and hyperbolic planes, while the matter first law is not confirmed in the accessible regime. The measured response ratio against the local conformal kernel is R1 = 0.994(11) + 0.405(33)/(omega l / c). Hence the model realizes H5 only in a restricted sense (static observers and linear order in G) and by construction; it does not derive the Einstein equation, and its matter sector currently obstructs a joint realization of H1-H7.