Topological Solitons with Spontaneous Symmetry Breaking: From Derrick's Theorem to Flat Rotation Curves

We investigate the dynamical origin of galactic acceleration scales and flat rotation curveswithin the framework of q-deformed Effective Field Theory (q-EFT). We show that standard3D scalar solitons collapse due to Derrick’s Theorem. To stabilize the topological field, weintroduce Spontaneous Symmetry Breaking (SSB) via a Mexican-hat potential, anchoringthe vacuum expectation value at v. Starting from an axisymmetric soliton ansatz γϕ=A(r) sinnθeiWϕ, this topological anchor enforces a boundary condition A(r→∞) = v.Crucially, wedemonstratethatthisnon-zerovacuumexpectationvalueyieldsarotationalenergy integral that diverges linearly with radius (Erot ∝ R). This geometric propertydictates a linear mass accumulation M(R) ∝R in the galactic halo, naturally deriving thephenomenological flat rotation curves (Vrot = const) from first principles without postulatingparticulate dark matter. Furthermore, the topological core structure prevents the centraldensity cusps typical of ΛCDM. By imposing a quantum symmetry conjecture (n= W), webridge microscopic vacuum topology with macroscopic galactic scaling, recovering MOND-like phenomenology as an asymptotic limit of topological charge saturation.