Spacetime Geometry from Entanglement Networks: Metric Reconstruction, Einstein's Equation from a Single Equilibrium Hypothesis, and a Dynamical Cosmological Constant

I study a pregeometric model in which spacetime geometry, gauge structure and matter are collective properties of the entanglement pattern of a finite-dimensional quantum network. I separate rigorous, conditional and conjectural statements explicitly and make five contributions. (i) I construct a metric on the network from mutual information, prove its metric axioms, and show that, together with a Lieb-Robinson causal order and node counting, the Lorentzian metric is fixed by established reconstruction theorems. (ii) I show that the Ryu-Takayanagi relation holds as a rigorous inequality for every tensor network and as an equality for random tensor networks at large bond dimension, so that "area" is defined combinatorially without reference to AdS. (iii) Following Jacobson's entanglement-equilibrium argument, I reduce the nonlinear Einstein equation to one explicitly stated, numerically testable hypothesis (maximal vacuum entanglement). (iv) In this framework a constant vacuum energy drops out of the null-projected field equation, so the cosmological constant is an integration constant; with a finite Hilbert space and Poisson-type volume fluctuations its natural size is N-1/2~10⁻¹²² in Planck units. This is a consequence of stated assumptions, not a derivation of A, and it yields the falsifiable prediction of a fluctuating, evolving A. (v) I state the conditions the matter sector must satisfy (symmetric chiral gapping, anomaly freedom implying right-handed neutrinos) and a geometric origin of Yukawa hierarchies. I list falsifiable predictions and the numerical programme that would confirm or refute the framework.