We identify the gravitational rapidity potential chi-g as the primary scalar field of a Lorentz-rotor formulation of gravity. Unlike conventional approaches, which take either the Newtonian potential or the spacetime metric as fundamental, the present framework constructs the local Lorentz rotor directly from chi-g. The gravitational four-velocity, the Newtonian potential, the Painlev'e--Gullstrand (PG) coframe, and the corresponding spacetime metric are recovered as successive projections of this underlying Lorentz geometry. The resulting projection hierarchy preserves the exact hyperbolic structure of local Lorentz boosts until the final metric representation. The Schwarzschild and cosmological horizons appear as the two asymptotic limits of a single rapidity field, while the physical velocity remains bounded by construction through v=c tanh(chi-g). Applying the previously derived constant-Lagrangian (CL) condition D/Dt(Gamma)=0 to the gravitational four-velocity yields a geometric interpretation of stationary galactic flows. The conserved quantity Gamma=cosh(chi-r)cosh(chi-phi) defines an isorapidity circle on the Poincar'e disk, from which the flat galactic rotation curve, the universal sigma-channel residual, and a weak galactic geodetic precession emerge as complementary projections of the same hyperbolic geometry. The same Lorentz construction extends continuously into the strong-field regime of Schwarzschild accretion disks. Exact composition of the radial vacuum inflow and azimuthal orbital rapidity yields a fully relativistic expression for geodetic precession, recovering the weak-field Gravity Probe~B limit while predicting a universal ISCO frequency ratio of 0.374, together with a set of observational tests that distinguish this Lorentz-rotor formulation of geodetic precession from conventional spin-dependent frame-dragging models.