Entanglement Equilibrium on a Solvable Gaussian Network: The Contact Term Identified, Geometry Recovered from Entanglement Data, Causal Structure Bounded, and Equilibrium Explained

Jacobson's entanglement-equilibrium argument derives the Einstein equation from (i) an area law with finite Newton constant, (ii) a first law for the matter modular Hamiltonian, and (iii) stationarity of the vacuum entropy of small geodesic balls at fixed volume. We test these ingredients on a solvable family of network states: a pair sector whose entanglement encodes a Riemannian metric, combined with the Gaussian ground state of a scalar field on the same weighted graph, for which entropies and modular Hamiltonians of arbitrary regions are computed exactly. Every assumption is isolated as a hypothesis and classified as proved, derived, computed, or imposed by construction. (1) Derivation (conditional). The wave-packet quadratic form of the exact massless modular Hamiltonian of a ball, computed by Longo and Morsella, is identical to the bulk form K_bulk = 2π ∫ κ T_00^(min) + (2π ξ_c n / ℓ) ∫ φ^2 exhibited by the lattice data; the contact-term coefficient χ = 2π ξ_c and the response R = 1 + ξ_c n(n+2)/(ωℓ/c)^2 are identified, not fitted. The step from the continuum identity to the lattice is a modular-convergence statement (Conjecture C1) which we do not prove: a proved proposition on the non-analyticity of the entropy function at the band edge, together with a heuristic accumulation argument, shows why the spectral convergence of the matter sector does not imply it. The lattice data support C1: q_∞ = 1.02 ± 0.02 in n = 2 (up to 1.09 in n = 3 at the accessible radii ℓ ≤ 8). (2) Geometry (recovered, not emergent). The metric is imposed on the pair sector by construction; what is tested is the inverse map. An identifiability lemma shows the entropic distance matrix determines the manifold up to isometry; on S^2 the blind pipeline recovers the dimension (c_2 = 3.13 ≈ π), detects curvature through the negative tail of the MDS spectrum, and measures the curvature radius to 4%. (3) Causal structure (bounded, then computed). For the oscillator sector we prove a field-commutator cone bound with explicit constants, valid on arbitrary point sets, with cone velocity v_φ = 3.05 = 3.1 c_lat; the exact field commutator shows the physical front propagates at c_lat, with Lorentz violation controlled by β = γ(6,m)/16γ(4,m) = 0.384 as (kξ)^2. (4) Equilibrium. The equilibrium postulate is proved equivalent to the reconstructed metric having constant scalar curvature (Einstein only in n = 2); passivity supplies an exact constrained second-variation identity: at fixed geometry and fixed modular energy the vacuum is a local maximum of the generalised entropy along every constrained matter direction. The entropic flow whose fixed points are the equilibrium geometries is defined, not derived; in n = 2 it coincides with the normalised Ricci flow, whose convergence is Hamilton's theorem. (5) Spectral convergence (leading order, black box). For deterministic quasi-uniform sequences on curved manifolds the form-consistency step is proved explicitly, and eigenvalue convergence is obtained only conditionally by citing the graph discretisation of the Laplace–Beltrami operator as a black box. The curvature renormalisation of the speed of sound, c_eff^2 = c^2(1 - R_s γ(n+4,m)ξ^2 / 6n γ(n+2,m)), is verified on S^2 to 1%. For non-uniform random point sets the raw operator converges to a drift operator, and a blind density correction computed from entropic ball counts repairs it, recovering the manifold spectrum to within 0.2–2.3% plus a counting bias of order 1 / N̄_(r_b). All numbers are generated by the programs distributed with this work.