Emergent Spacetime Geometry from Entanglement Networks: Min-Cut Bounds, the Entanglement Equilibrium Hypothesis, and Local Euclidean Invariance

I present a pregeometric framework in which spatial geometry, causal structure, andgravitational field equations emerge from the entanglement patterns and statistical geometryof a finite-dimensional quantum network. The theoretical formulation is strictly segregatedinto proved theorems, conditional results, and explicit hypotheses. First, I construct ametric space from mutual information and demonstrate that spatial distances satisfy thestandard metric axioms, while Lorentzian causal structures follow from Lieb-Robinson causalbounds and Poisson sprinkling reconstructions. Second, I establish that the Ryu-Takayanagiarea-entropy relation arises as a rigorous, model-independent upper bound for arbitrarytensor network states with fixed bond dimension χ, saturating for random tensor networks.Third, I decompose the local entanglement equilibrium argument for Einstein’s field equationsinto four isolated components: I prove the exact fixed-volume geodesic ball area variationδA|V = −KG00 for time-symmetric spatial slices in arbitrary dimensions (verified numericallyon n-spheres), demonstrate the validity of the lattice entanglement first law in 1+1 dimensions,and prove that random tensor networks maximize all region entropies within their class.The derivation of non-linear general relativity is thus shown to be strictly conditional ona single unproved hypothesis: the stationarity of ball entropy under microscopic networkdynamics. Fourth, to resolve the conflict between lattice discreteness and continuous spacetimesymmetries, I introduce Euclidean Poisson-Delaunay networks (Route C). I prove that theensemble of these networks is exactly E(d)-invariant and local with finite average vertexdegree, and demonstrate numerically that the ensemble-averaged scalar kinetic operatorachieves isotropy at the sub-percent level in d = 2,3,4. The Lorentzian continuation of thisroute is formulated precisely in terms of an explicit reflection positivity hypothesis.