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

Jacobsons entanglementequilibrium 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 volumepreserving variations. In this work, every assumption of this argument beyond established mathematics is isolated as one of seven explicit hypotheses H1 to H7 on a family of quantum network states, and it is proven that, given H1 to H4 and H6, the Einstein equation at a point is equivalent to the equilibrium hypothesis H5. An exactly solvable composite family on compact surfaces is constructed: a product of twoqubit 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 it is shown that: (i) H1 holds 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 holds for the field sector on the Weyl algebra, uniform in the network size in physical units, with explicit velocity; (iii) a Lorentzian-limit theorem holds 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 holds with an explicit Newton constant; (v) 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 an independent mismatch; (vi) static equilibrium corresponds to the Hamiltonian constraint, with a nogo statement for static vacuum geometries; (vii) exact local lattice energy conservation holds; and (viii) exact reflection positivity holds for a kernel ghost-layer ensemble. Numerically, the discrete area law reproduces a continuum nonlocal-perimeter functional to $2\times10^{-2}$ on spheres and hyperbolic planes, while the matter first law is not confirmed in the accessible range: the response ratio against the local conformal kernel is $R_{1}=0.994(11)+0.405(33)/(\omega l/c)$. Hence the model realizes H5 only in a restricted sense (static observers, 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 to H7.