An Exactly Solvable Pair-Entanglement Network: Exploring Candidate Mechanisms for Entanglement Equilibrium

In a previous paper [1] I reduced the Einstein equation for geometries emerging fromentanglement networks to an entropy statement, but I exhibited no explicit family of statesin which the seven hypotheses H1–H7 hold. Here I construct a family in a restricted sectorand determine precisely how far it goes. The states are products of two-qubit pure statesattached to pairs of points sampled from a compact Riemannian manifold, the entropyof a pair decaying exponentially with the Riemannian distance. I prove: (i) the mutualinformation law of H1 holds with vanishing error, so that the metric-reconstruction lemmasof [1] apply with explicit rates; (ii) the state is the unique ground state of a commutinglocal Hamiltonian that satisfies Lieb–Robinson bounds uniformly, but whose causal coneis degenerate; (iii) in the iterated limit of vanishing vertex spacing followed by vanishingcorrelation length, the entropy of every region with C2 boundary equals its area divided by4Gℏ with G explicit, so that H3 holds with Smat = 0; (iv) consequently the equilibriumhypothesis H5, for static observers, holds at a point if and only if the scalar curvature of theslice equals 2Λc, i.e. the Hamiltonian constraint, and the scalar curvature can be read off fromentropy differences of volume-matched balls. Numerical evaluation of the continuum entropyfunctional indicates that finite-correlation-length corrections to the volume-matched entropydifference are of relative order (ξ/ℓ)2. The model contains no matter, so H4 and H6 holdonly vacuously; its Lorentzian limit H2(b) is not realized; H7 is not addressed. I also correcta statement in [1] on the scaling of Newton’s constant. The result is a consistency proof forthe geometric half of the hypothesis system, not a derivation of the Einstein equation.