Topological Network Effective Field Theory: A q-Deformed Variational Model and Asymptotic Galactic Dynamics

We propose a phenomenologically motivated effective field theory (EFT) incorporating a background 1-form field and an effective network current. A generalized q-deformed capacity parameter, motivated by Tsallis non-extensive statistics, mediates the background interaction. Treating the phase index q as an auxiliary modulus field, the theory admits distinct topological vacuum phases. Through an asymptotic expansion near the macroscopic limit (q ≈ 1), the model suggests a mechanism for asymptotically flat rotation curves with a logarithmic correction. The framework provides a potential phenomenological alternative to dark matter descriptions, subject to the weak-field and spherical symmetry approximations discussed herein.