Dynamic Vacuum Theory (DVT v26) – Mathematical Appendix: First-Principles Field-Theoretic Derivation of the Tanh Order Parameter from Ginzburg-Landau Vacuum Kinetics

A central question in phenomenological modifications of galactic dynamics concerns the mathematical origin of the transition function connecting the Newtonian regime to the asymptotic flat-velocity regime. In this foundational Appendix to the Dynamic Vacuum Theory (DVT v26) series, we demonstrate that the transition function Q(u) = tanh(u) is not an ad-hoc or empirical fit. By formulating the physical vacuum as a self-interacting scalar Bose-Einstein Condensate (BEC) governed by a classical Ginzburg-Landau order-parameter equation d^2Q/du^2 = Q^3 - Q, we prove analytically via spatial quadrature integration that the spatial order parameter corresponds strictly to the topological Kink Soliton solution Q(u) = tanh(u/sqrt(2)). Coupling this topological soliton to Euler-Madelung hydrodynamics and a polytropic equation of state (P = g/2 * rho_v^2) demonstrates that gravitational acceleration is an emergent vacuum pressure gradient a_DVT = -(1/rho_v) * grad(P), which directly generates the exact non-linear galactic acceleration law a_DVT = a_N / tanh(sqrt(a_N / a_0)) with zero free parameters. In the deep-acceleration limit, the spatial coordinate cancels identically, rigorously deriving the Baryonic Tully-Fisher Relation from first principles.