Flat galactic rotation curves and the Baryonic Tully-Fisher relation have resisted explanation for five decades. Dark matter and modified gravity each resolve the problem by importing something external — either undetected matter or a modified law. A third route exists.A preceding empirical programme showed that the condition that total kinetic energy is conserved along galactic streamlines — the constant-Lagrangian (CL) condition — fits 175 observed rotation curves using only two physically meaningful parameters, the bulge radius and enclosed mass, without dark matter. That condition was a postulate. This paper derives it.Starting from a single first-order field equation for the gravitational vacuum flow, a channel decomposition automatically produces Newtonian gravity, the FRW expansion scalar, and the disk's rotation curve as three independent projections. The CL condition emerges as an exact internal output of the norm channel. Flat rotation curves, the Tully-Fisher scaling, and the cosmological structure of galactic disks follow simultaneously, with no free parameters.