Entanglement Equilibrium on Quantum Networks: A Hypothesis-Explicit Reduction with Intermediate Results and a Reflection-Positive Poisson–Delaunay Ensemble

I study under which conditions smooth spatial geometry, causal structure and Einstein’sequation can be read o from the entanglement structure of nite-dimensional quantumnetworks. Every assumption going beyond established mathematics is isolated as an explicithypothesis on the family of network states. I prove: (i) the mutual-information shortest-pathdistance is a metric, and the LiebRobinson bound induces a causal partial order with explicitspeed; (ii) the tensor-network min-cut bound, and a quantitative lemma showing that thereconstructed metric converges uniformly to a Riemannian distance; the path-approximationproperty needed there is derived from a range condition rather than assumed; (iii) the xedvolume area variation of geodesic balls, relative to a at or a maximally symmetric referenceball, and the evaluation of the conformal modular-Hamiltonian kernel; (iv) an equivalencetheorem: under hypotheses H1H4 and H6, vacuum entanglement equilibrium (H5) holdsat a point if and only if Gµν + Λpgµν = 8πGTµν holds there, and Λp is a global constant;(v) exact re ection positivity for a PoissonDelaunay ensemble with a stochastic ghost layeron the re ection plane, for complex-valued observables. I state precisely which results aretheorems and which are hypotheses, and I state openly that the equivalence in (iv) makesthe result a reduction of the Einstein equation to an entropy statement, not a derivation ofit.