The Topological Hooke's Law and Emergent Cosmology in Quantum Graph Emergent Field Theory (QGEFT)

We present a comprehensive computational study on the classical limits of emergent gravity within a background-independent dynamic tensor network. By simulating binary "matter" systems across seven distinct vacuum topologies—including random regular graphs, open 3D Euclidean grids, and periodic 3D tori—we tested the hypothesis that a 3D initial geometry would yield Newtonian gravity ($F \propto 1/r^2$). The Metropolis-Hastings dynamics universally rejected this premise, converging instead to a Topological Hooke's Law where the effective force scales linearly with topological distance ($\alpha \approx +1$). This behavior mirrors QCD quark confinement via flux tubes, suggesting that at the fundamental scale, spacetime topology is strictly hyperbolic and elastic. We further demonstrate that this $F \propto D$ scaling naturally flattens simulated galactic rotation curves, offering a purely topological alternative to Dark Matter. Finally, we establish a stable De Sitter inflation phase through a localized edge-subdivision mechanism, proving that cosmic expansion can emerge as a thermodynamic response to topological stress.